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ity). A singularity z0 of f is a removable singularity if f is bounded near z0. Deﬁnition (Pole). A singularity z0 is a pole of order k of f if \|f (z)\| → ∞ as z → z0 and one can write f (z) = with g : U → C, g(z0) = 0. g(z) (z − z0)k Deﬁnition (Isolated essential singularity). An isolated singularity is an isolated essential singularity if it is neither removable nor a pole. It is easy to give examples of removable singularities and poles. So let’s look at some essential singularities. Example. z → e1/z has an isolated essential singularity at z = 0. Note that if B(z0, ε) \ {z0} → C has a pole of order h at z0, then f naturally deﬁnes a map ˆf : B(z0; ε) → CP1 = C ∪ {∞}, the Riemann sphere, by f (z) = ∞ z = z0 z = z0 f (z). This is then a “continuous” function. So a singularity is just a point that gets mapped to the point ∞. As was emphasized in IA Groups, the point at inﬁnity is not a special point in the Riemann sphere. Similarly, poles are also not really singularities from the viewpoint of the Riemann sphere. It’s just that we are looking at it in a wrong way. Indeed, if we change coordinates on the Riemann sphere so that we label each point w ∈ CP1 by w = 1 w instead, then f just maps z0 to 0 under the new coordinate system. In particular, at the point z0, we ﬁnd that f is holomorphic and has an innocent zero of order k. Since poles are not bad, we might as well allow them. Deﬁnition (Meromorphic function). If U is a domain and S ⊆ U is a ﬁnite or discrete set, a function f : U \ S → C which is holomorphic and has (at worst) poles on S is said to be meromorphic on U. The requirement that S is discrete is so that each pole
in S is actually an isolated singularity. Example. A rational function P (z) Q(z), where P, Q are polynomials, is holomorphic on C \ {z : Q(z) = 0}, and meromorphic on C. More is true — it is in fact holomorphic as a function CP1 → CP1. 39 2 Contour integration IB Complex Analysis These ideas are developed more in depth in the IID Riemann Surfaces course. As an aside, if we want to get an interesting holomorphic function with domain CP1, its image must contain the point ∞, or else its image will be a compact subset of C (since CP1 is compact), thus bounded, and therefore constant by Liouville’s theorem. At this point, we really should give essential singularities their fair share of attention. Not only are they bad. They are bad spectacularly. Theorem (Casorati-Weierstrass theorem). Let U be a domain, z0 ∈ U, and suppose f : U \ {z0} → C has an essential singularity at z0. Then for all w ∈ C, there is a sequence zn → z0 such that f (zn) → w. In other words, on any punctured neighbourhood B(z0; ε) \ {z0}, the image of f is dense in C. This is not actually too hard to proof. Proof. See example sheet 2. If you think that was bad, actually essential singularities are worse than that. The theorem only tells us the image is dense, but not that we will hit every point. It is in fact not true that every point will get hit. For example e 1 z can never be zero. However, this is the worst we can get Theorem (Picard’s theorem). If f has an isolated essential singularity at z0, then there is some b ∈ C such that on each punctured neighbourhood B(z0; ε) \ {z0}, the image of f contains C \ {b}. The proof is beyond this course. 2.7 Laurent series If f is holomorphic at z0, then we have a local power series expansion f (z) = ∞ n=0 cn(z − z0)n near z0. If f is singular at z0 (and the singularity is not removable), then there is no hope we can
get a Taylor series, since the existence of a Taylor series would imply f is holomorphic at z = z0. However, it turns out we can get a series expansion if we allow ourselves to have negative powers of z. Theorem (Laurent series). Let 0 ≤ r < R < ∞, and let A = {z ∈ C : r < \|z − a\| < R} denote an annulus on C. Suppose f : A → C is holomorphic. Then f has a (unique) convergent series expansion f (z) = ∞ n=−∞ cn(z − a)n, 40 2 Contour integration IB Complex Analysis where cn = 1 2πi ∂B(a,ρ) f (z) (z − a)n+1 dz for r < ρ < R. Moreover, the series converges uniformly on compact subsets of the annulus. The Laurent series provides another way of classifying singularities. In the case where r = 0, we just have f (z) = ∞ −∞ cn(z − a)n on B(a, R) \ {a}, then we have the following possible scenarios: (i) cn = 0 for all n < 0. Then f is bounded near a, and hence this is a removable singularity. (ii) Only ﬁnitely many negative coeﬃcients are non-zero, i.e. there is a k ≥ 1 such that cn = 0 for all n < −k and c−k = 0. Then f has a pole of order k at a. (iii) There are inﬁnitely many non-zero negative coeﬃcients. Then we have an isolated essential singularity. So our classiﬁcation of singularities ﬁt nicely with the Laurent series expansion. We can interpret the Laurent series as follows — we can write f (z) = fin(z) + fout(z), where fin consists of the terms with positive power and fout consists of those with negative power. Then fin is the part that is holomorphic on the disk \|z − a\| < R, while fout(z) is the part that is holomorphic on \|z − a\| > r. These two combine to give an expression holomorphic on r < \|z − a\| < R. This is just
a nice way of thinking about it, and we will not use this anywhere. So we will not give a detailed proof of this interpretation. Proof. The proof looks very much like the blend of the two proofs we’ve given for the Cauchy integral formula. In one of them, we took a power series expansion of the integrand, and in the second, we changed our contour by cutting it up. This is like a mix of the two. Let w ∈ A. We let r < ρ < \|w − a\| < ρ < R. 41 2 Contour integration IB Complex Analysis ρ ρ a w ˜γ ˜˜γ We let ˜γ be the contour containing w, and ˜˜γ be the other contour. Now we apply the Cauchy integral formula to say f (w) = 1 2πi ˜γ f (z) z − w dz 0 = 1 2πi ˜˜γ f (z) z − w dz. and So we get f (w) = 1 2πi ∂B(a,ρ) f (z) z − w dz − 1 2πi ∂B(a,ρ) f (z) z − w dz. As in the ﬁrst proof of the Cauchy integral formula, we make the following expansions: for the ﬁrst integral, we have w − a < z − a. So −a z−a = ∞ n=0 (w − a)n (z − a)n+1, which is uniformly convergent on z ∈ ∂B(a, ρ). For the second integral, we have w − a > z − a. So −−a w−a = ∞ m=1 (z − a)m−1 (w − a)m, which is uniformly convergent for z ∈ ∂B(a, ρ). By uniform convergence, we can swap summation and integration. So we get f (w) = ∞ n=0 1 2πi + ∞ m=1 1 2πi ∂B(a,ρ) f (z) (z − a)n+1 dz (w − a)n ∂B(a,ρ) f (z) (z − a)−m+1 dz (w − a)
−m. 42 2 Contour integration IB Complex Analysis Now we substitute n = −m in the second sum, and get f (w) = ∞ n=−∞ ˜cn(w − a)n, for the integrals ˜cn. However, some of the coeﬃcients are integrals around the ρ circle, while the others are around the ρ circle. This is not a problem. For any r < ρ < R, these circles are convex deformations of \|z − a\| = ρ inside the annulus A. So ∂B(a,ρ) f (z) (z − a)n+1 dz is independent of ρ as long as ρ ∈ (r, R). So we get the result stated. Deﬁnition (Principal part). If f : B(a, r) \ {a} → C is holomorphic and if f has Laurent series ∞ f (z) = cn(z − a)n, then the principal part of f at a is n=−∞ fprincipal = −1 n=−∞ cn(z − a)n. So f − fprincipal is holomorphic near a, and fprincipal carries the information of what kind of singularity f has at a. When we talked about Taylor series, if f : B(a, r) → C is holomorphic n=0 cn(z − a)n, then we had two possible ways of with Taylor series f (z) = ∞ expressing the coeﬃcients of cn. We had cn = 1 2πi ∂B(a,ρ) f (z) (z − a)n+1 dz = f (n)(a) n!. In particular, the second expansion makes it obvious the Taylor series is uniquely determined by f. For the Laurent series, we cannot expect to have a simple expression of the coeﬃcients in terms of the derivatives of the function, for the very reason that f is not even deﬁned, let alone diﬀerentiable, at a. So is the Laurent series unique? Lemma. Let f : A → C be holomorphic, A = {r < \|z − a\| < R}, with f (z) = ∞ n=−∞
cn(z − a)n Then the coeﬃcients cn are uniquely determined by f. Proof. Suppose also that f (z) = ∞ n=−∞ bn(z − a)n. 43 2 Contour integration IB Complex Analysis Using our formula for ck, we know 2πick = ∂B(a,ρ) ∂B(a,ρ) bn = = f (z) (z − a)k+1 dz bn(z − a)n−k−1 dz n (z − a)n−k−1 dz ∂B(a,ρ) n = 2πibk. So ck = bk. While we do have uniqueness, we still don’t know how to ﬁnd a Laurent series. For a Taylor series, we can just keep diﬀerentiating and then get the coeﬃcients. For Laurent series, the above integral is often almost impossible to evaluate. So the technique to compute a Laurent series is blind guesswork. Example. We know z3 3! deﬁnes a holomorphic function, with a radius of convergence of ∞. Now consider sin z = z − z5 5! − · · · + cosec z = 1 sin z, which is holomorphic except for z = kπ, with k ∈ Z. So cosec z has a Laurent series near z = 0. Using we get sin z = z 1 − + O(z4), z2 6 cosec z = 1 z 1 + z2 6 + O(z4). From this, we can read oﬀ that the Laurent series has cn = 0 for all n ≤ −2, c−1 = 1, c1 = 1 5. If we want, we can go further, but we already see that cosec has a simple pole at z = 0. By periodicity, cosec has a simple pole at all other singularities. Example. Consider instead sin 1 z = 1 z − 1 3!z3 + 1 5!z5 − · · ·. We see this is holomorphic on C∗, with cn = 0 for inﬁnitely many n < 0. So this has an isolated essential singularity. Example. Consider cosec 1 kπ for k ∈ N =
{1, 2, 3, · · · }. So it is not holomorphic at any punctured neighbourhood B(0, r) \ {0} of zero. So this has a non-isolated singularity at zero, and there is no Laurent series in a neighbourhood of zero.. This has singularities at z = 1 z 44 2 Contour integration IB Complex Analysis We’ve already done most of the theory. In the remaining of the course, we will use these techniques to do stuﬀ. We will spend most of our time trying to evaluate integrals, but before that, we will have a quick look on how we can use Laurent series to evalue some series. Example (Series summation). We claim that f (z) = ∞ n=−∞ 1 (z − n)2 is holomorphic on C \ Z, and moreover if we let f (z) = π2 sin2(πz), We will reserve the name f for the original series, and refer to the function z → π2 sin2(πz) as g instead, until we have proven that they are the same. Our strategy is as follows — we ﬁrst show that f (z) converges and is holomorphic, which is not hard, given the Weierstrass M -test and Morera’s theorem. To show that indeed we have f (z) = g(z), we ﬁrst show that they have equal principal part, so that f (z) − g(z) is entire. We then show it is zero by proving f − g is bounded, hence constant, and that f (z) − g(z) → 0 as z → ∞ (in some appropriate direction). For any ﬁxed w ∈ C \ Z, we can compare it with 1 n2 and apply the Weierstrass M -test. We pick r > 0 such that \|w − n\| > 2r for all n ∈ Z. Then for all z ∈ B(w; r), we have \|z − n\| ≥ max{r, n − \|w\| − r}. Hence 1 1 r2, By comparison to 1 n2, we know M -test, we know our series converges uniformly on B(w, r). 1 (n − \|w\| − r)2 \|z − n\|2
≤ min = Mn. n Mn converges. So by the Weierstrass By our results around Morera’s theorem, we see that f is a uniform limit of holomorphic functions N n=−N 1 (z−n)2, and hence holomorphic. Since w was arbitrary, we know f is holomorphic on C \ Z. Note that we do not say the sum converges uniformly on C \ Z. It’s just that for any point w ∈ C \ Z, there is a small neighbourhood of w on which the sum is uniformly convergent, and this is suﬃcient to apply the result of Morera’s. For the second part, note that f is periodic, since f (z + 1) = f (z). Also, at z2 + holomorphic stuﬀ near z = 0. So f has sin2(πz) also has a double pole at each 0, f has a double pole, since f (z) = 1 a double pole at each k ∈ Z. Note that k ∈ Z. 1 Now, consider the principal parts of our functions — at k ∈ Z, f (z) has principal part 1 (z−k)2. Looking at our previous Laurent series for cosec(z), if g(z) = π 2 sin πz, then limz→0 z2g(z) = 1. So g(z) must have the same principal part at 0 and hence at k for all k ∈ Z. 45 2 Contour integration IB Complex Analysis Thus h(z) = f (z) − g(z) is holomorphic on C \ Z. However, since its principal part vanishes at the integers, it has at worst a removable singularity. Removing the singularity, we know h(z) is entire. Since we want to prove f (z) = g(z), we need to show h(z) = 0. We ﬁrst show it is boundedness. We know f and g are both periodic with period 1. So it suﬃces to focus attention on the strip − 1 2 ≤ x = Re(z) ≤ 1 2. To show this is bounded on the rectangle, it suﬃces to show that h(x + iy) → 0 as y → ±∞, by continuity. To do so
, we show that f and g both vanish as y → ∞. So we set z = x + iy, with \|x\| ≤ 1 2. Then we have \|g(z)\| ≤ 4π2 \|eπy − e−πy\| → 0 as y → ∞. Exactly analogously, \|f (z)\| ≤ n∈Z 1 \|x + iy − n\|2 ≤ 1 y2 + 2 ∞ n=1 1 (n − 1 2 )2 + y2 → 0 as y → ∞. So h is bounded on the strip, and tends to 0 as y → ∞, and is hence constant by Liouville’s theorem. But if h → 0 as y → ∞, then the constant better be zero. So we get h(z) = 0. 46 3 Residue calculus IB Complex Analysis 3 Residue calculus 3.1 Winding numbers Recall that the type of the singularity of a point depends on the coeﬃcients in the Laurent series, and these coeﬃcients play an important role in determining the behaviour of the functions. Among all the inﬁnitely many coeﬃcients, it turns out the coeﬃcient of z−1 is the most important one, as we will soon see. We call this the residue of f. Deﬁnition (Residue). Let f : B(a, r) \ {a} → C be holomorphic, with Laurent series ∞ f (z) = cn(z − a)n. Then the residue of f at a is n=−∞ Res(f, a) = Resf (a) = c−1. Note that if ρ < r, then by deﬁnition of the Laurent coeﬃcients, we know ∂B(a,ρ) f (z) dz = 2πic−1. So we can alternatively write the residue as Resf (a) = 1 2πi ∂B(a,ρ) f (z) dz. This gives us a formulation of the residue without reference to the Laurent series. Deforming paths if necessary, it is not too far-fetching to imagine that for any simple curve γ around the singularity a, we have f (z) dz = 2πi Res
(f, a). γ Moreover, if the path actually encircles two singularities a and b, then deforming the path, we would expect to have γ f (z) dz = 2πi(Res(f, a) + Res(f, b)), and this generalizes to multiple singularities in the obvious way. If this were true, then it would be very helpful, since this turns integration into addition, which is (hopefully) much easier! Indeed, we will soon prove that this result holds. However, we ﬁrst get rid of the technical restriction that we only work with simple (i.e. non-self intersecting) curves. This is completely not needed. We are actually not really worried in the curve intersecting itself. The reason why we’ve always talked about simple closed curves is that we want to avoid the curve going around the same point many times. There is a simple workaround to this problem — we consider arbitrary curves, and then count how many times we are looping around the point. If we are looping around it twice, then we count its contribution twice! Indeed, suppose we have the following curve around a singularity: 47 3 Residue calculus IB Complex Analysis a We see that the curve loops around a twice. Also, by the additivity of the integral, we can break this curve into two closed contours. So we have 1 2πi γ f (z) dz = 2 Resf (a). So what we want to do now is to deﬁne properly what it means for a curve to loop around a point n times. This will be called the winding number. There are many ways we can deﬁne the winding number. The deﬁnition we will pick is based on the following observation — suppose, for convenience, that the point in question is the origin. As we move along a simple closed curve around 0, our argument will change. If we keep track of our argument continuously, then we will ﬁnd that when we return to starting point, the argument would have increased by 2π. If we have a curve that winds around the point twice, then our argument will increase by 4π. What we do is exactly the above — given a path, ﬁnd a continuous function that gives the “argument” of the path, and then deﬁne the winding number to be the di
ﬀerence between the argument at the start and end points, divided by 2π. For this to make sense, there are two important things to prove. First, we need to show that there is indeed a continuous “argument” function of the curve, in a sense made precise in the lemma below. Then we need to show the winding number is well-deﬁned, but that is easier. Lemma. Let γ : [a, b] → C be a continuous closed curve, and pick a point w ∈ C \ image(γ). Then there are continuous functions r : [a, b] → R > 0 and θ : [a, b] → R such that γ(t) = w + r(t)eiθ(t). Of course, at each point t, we can ﬁnd r and θ such that the above holds. The key point of the lemma is that we can do so continuously. Proof. Clearly r(t) = \|γ(t) − w\| exists and is continuous, since it is the composition of continuous functions. Note that this is never zero since γ(t) is never w. The actual content is in deﬁning θ. To deﬁne θ(t), we for simplicity assume w = 0. Furthermore, by considering r(t), which is continuous and well-deﬁned since r is never instead the function γ(t) zero, we can assume \|γ(t)\| = 1 for all t. Recall that the principal branch of log, and hence of the argument Im(log), takes values in (−π, π) and is deﬁned on C \ R≤0. 48 3 Residue calculus IB Complex Analysis If γ(t) always lied in, say, the right-hand half plane, we would have no problem deﬁning θ consistently, since we can just let θ(t) = arg(γ(t)) for arg the principal branch. There is nothing special about the right-hand half plane. Similarly, if γ lies in the region as shaded below: α i.e. we have γ(t) ∈ z : Re z eiα > 0 for a ﬁxed α, we can deﬁne θ(t) = α
+ arg γ(t) eiα. Since γ : [a, b] → C is continuous, it is uniformly continuous, and we can ﬁnd a subdivision a = a0 < a1 < · · · < am = b, √ such that if s, t ∈ [ai−1, ai], then \|γ(s) − γ(t)\| < belong to such a half-plane. So we deﬁne θj : [aj−1, aj] → R such that γ(t) = eiθj (t) 2, and hence γ(s) and γ(t) for t ∈ [aj−1, aj], and 1 ≤ j ≤ n − 1. On each region [aj−1, aj], this gives a continuous argument function. We cannot immediately extend this to the whole of [a, b], since it is entirely possible that θj(aj) = θj+1(aj). However, we do know that θj(aj) are both values of the argument of γ(aj). So they must diﬀer by an integer multiple of 2π, say 2nπ. 49 3 Residue calculus IB Complex Analysis Then we can just replace θj+1 by θj+1 − 2nπ, which is an equally valid argument function, and then the two functions will agree at aj. Hence, for j > 1, we can successively re-deﬁne θj such that the resulting map θ is continuous. Then we are done. We can thus use this to deﬁne the winding number. Deﬁnition (Winding number). Given a continuous path γ : [a, b] → C such that γ(a) = γ(b) and w ∈ image(γ), the winding number of γ about w is θ(b) − θ(a) 2π, where θ : [a, b] → R is a continuous function as above. This is denoted by I(γ, w) or nγ(W ). I and n stand for index and number respectively. Note that we always have I(γ, w) ∈ Z, since θ(b) and θ(a) are arguments of the same
number. More importantly, I(γ, w) is well-deﬁned — suppose γ(t) = r(t)eiθ1(t) = r(t)eiθ2(t) for continuous functions θ1, θ2 : [a, b] → R. Then θ1 − θ2 : [a, b] → R is continuous, but takes values in the discrete set 2πZ. So it must in fact be constant, and thus θ1(b) − θ1(a) = θ2(b) − θ2(a). So far, what we’ve done is something that is true for arbitrary continuous closed curve. However, if we focus on piecewise C 1-smooth closed path, then we get an alternative expression: Lemma. Suppose γ : [a, b] → C is a piecewise C 1-smooth closed path, and w ∈ image(γ). Then I(γ, w) = 1 2πi γ 1 z − w dz. Proof. Let γ(t) − w = r(t)eiθ(t), with now r and θ piecewise C 1-smooth. Then γ 1 z − w dz = b dt a b γ(t) γ(t) − w r(t) r(t) = [ln r(t) + iθ(t)]b a = i(θ(b) − θ(a)) = a + iθ(t) dt So done. = 2πiI(γ, w). In some books, this integral expression is taken as the deﬁnition of the winding number. While this is elegant in complex analysis, it is not clear a priori that this is an integer, and only works for piecewise C 1-smooth closed curves, not arbitrary continuous closed curves. On the other hand, what is evident from this expression is that I(γ, w) is continuous as a function of w ∈ C \ image(γ), since it is even holomorphic as a function of w. Since I(γ; w) is integer valued, I(γ) must be locally constant on path components of C \ image(γ). 50 3 Residue calculus IB Complex Analysis We can quickly verify that this
is a sensible deﬁnition, in that the winding number around a point “outside” the curve is zero. More precisely, since image(γ) is compact, all points of suﬃciently large modulus in C belong to one component of C \ image(γ). This is indeed the only path component of C \ image(γ) that is unbounded. To ﬁnd the winding number about a point in this unbounded component, note that I(γ; w) is consistent on this component, and so we can consider arbitrarily larger w. By the integral formula, \|I(γ, w)\| ≤ 1 2π length(γ) max z∈γ 1 \|w − z\| → 0 as w → ∞. So it does vanish outside the curve. Of course, inside the other path components, we can still have some interesting values of the winding number. 3.2 Homotopy of closed curves The last ingredient we need before we can get to the residue theorem is the idea of homotopy. Recall we had this weird, ugly deﬁnition of elementary deformation of curves — given φ, ψ : [a, b] → U, which are closed, we say ψ is an elementary deformation or convex deformation of φ if there exists a decomposition a = x0 < x1 < · · · < xn = b and convex open sets C1, · · ·, Cn ⊆ U such that for xi−1 ≤ t ≤ xi, we have φ(t) and ψ(t) in Ci. It was a rather unnatural deﬁnition, since we have to make reference to this arbitrarily constructed dissection of [a, b] and convex sets Ci. Moreover, this deﬁnition fails to be transitive (e.g. on R \ {0}, rotating a circle about the center by, say, π 10 is elementary, but rotating by π is not). Yet, this deﬁnition was cooked up just so that it immediately follows that elementary deformations preserve integrals of holomorphic functions around the loop. The idea now is to deﬁne a more general and natural notion of deforming a curve, known as “homotopy”. We will then show that each homotopy can be given by a sequence
of elementary deformations. So homotopies also preserve integrals of holomorphic functions. Deﬁnition (Homotopy of closed curves). Let U ⊆ C be a domain, and let φ : [a, b] → U and ψ : [a, b] → U be piecewise C 1-smooth closed paths. A homotopy from φ : ψ is a continuous map F : [0, 1] × [a, b] → U such that F (0, t) = φ(t), F (1, t) = ψ(t), and moreover, for all s ∈ [0, t], the map t → F (s, t) viewed as a map [a, b] → U is closed and piecewise C 1-smooth. We can imagine this as a process of “continuously deforming” the path φ to ψ, with a path F (s, · ) at each point in time s ∈ [0, 1]. Proposition. Let φ, ψ : [a, b] → U be homotopic (piecewise C 1) closed paths in a domain U. Then there exists some φ = φ0, φ1, · · ·, φN = ψ such that each φj is piecewise C 1 closed and φi+1 is obtained from φi by elementary deformation. Proof. This is an exercise in uniform continuity. We let F : [0, 1] × [a, b] → U be a homotopy from φ to ψ. Since image(F ) is compact and U is open, there 51 3 Residue calculus IB Complex Analysis is some ε > 0 such that B(F (s, t), ε) ⊆ U for all (s, t) ∈ [0, 1] × [a, b] (for each s, t, pick the maximum εs,t > 0 such that B(F (s, t), εs,t) ⊆ U. Then εs,t varies continuously with s, t, hence attains its minimum on the compact set [0, 1] × [a, b]. Then picking ε to be the minimum works). Since F is uniformly continuous, there is some δ such that (s,
t)−(s, t) < δ implies \|F (s, t) − F (s, t)\| < ε. Now we pick n ∈ N such that 1+(b−a) n < δ, and let j n xj = a + (b − a) n, t φi(t) = F i Cij = B F i, ε n, xj Then Cij is clearly convex. These deﬁnitions are cooked up precisely so that if s ∈ i−1 and t ∈ [xj−1, xj], then F (s, t) ∈ Cij. So the result follows. n, i n Corollary. Let U be a domain, f : U → C be holomorphic, and γ1, γ2 be homotopic piecewise C 1-smooth closed curves in U. Then γ1 f (z) dz = γ2 f (z) dz. This means the integral around any path depends only on the homotopy class of the path, and not the actual path itself. We can now use this to “upgrade” our Cauchy’s theorem to allow arbitrary simply connected domains. The theorem will become immediate if we adopt the following alternative deﬁnition of a simply connected domain: Deﬁnition (Simply connected domain). A domain U is simply connected if every C 1 smooth closed path is homotopic to a constant path. This is in fact equivalent to our earlier deﬁnition that every continuous map S1 → U can be extended to a continuous map D2 → U. This is almost immediately obvious, except that our old deﬁnition only required the map to be continuous, while the new deﬁnition only works with piecewise C 1 paths. We will need something that allows us to approximate any continuous curve with a piecewise C 1-smooth one, but we shall not do that here. Instead, we will just forget about the old deﬁnition and stick to the new one. Rewriting history, we get the following corollary: Corollary (Cauchy’s theorem for simply connected domains). Let U be a simply connected domain, and let f : U → C be holomorphic. If γ is any piecewise C 1-
smooth closed curve in U, then γ f (z) dz = 0. We will sometimes refer to this theorem as “simply-connected Cauchy”, but we are not in any way suggesting that Cauchy himself is simply connected. Proof. By deﬁnition of simply-connected, γ is homotopic to the constant path, and it is easy to see the integral along a constant path is zero. 52 3 Residue calculus IB Complex Analysis 3.3 Cauchy’s residue theorem We ﬁnally get to Cauchy’s residue theorem. This in some sense a mix of all the results we’ve previously had. Simply-connected Cauchy tells us the integral of a holomorphic f around a closed curve depends only on its homotopy class, i.e. we can deform curves by homotopy and this preserves the integral. This means the value of the integral really only depends on the “holes” enclosed by the curve. We also had the Cauchy integral formula. This says if f : B(a, r) → C is holomorphic, w ∈ B(a, ρ) and ρ < r, then f (w) = 1 2πi ∂B(a,ρ) f (z) z − w dz. Note that f (w) also happens to be the residue of the function f (z) z−w. So this really says if g has a simple pole at a inside the region bounded by a simple closed curve γ, then 1 2π γ g(z) dz = Res(g, a). The Cauchy’s residue theorem says the result holds for any type of singularities, and any number of singularities. Theorem (Cauchy’s residue theorem). Let U be a simply connected domain, and {z1, · · ·, zk} ⊆ U. Let f : U \ {z1, · · ·, zk} → C be holomorphic. Let γ : [a, b] → U be a piecewise C 1-smooth closed curve such that zi = image(γ) for all i. Then 1 2πi k f (z) dz = I(γ, zi) Res(f ; zi). γ j=1 The
Cauchy integral formula and simply-connected Cauchy are special cases of this. Proof. At each zi, f has a Laurent expansion f (z) = n∈Z n (z − zi)n, c(i) valid in some neighbourhood of zi. Let gi(z) be the principal part, namely gi(z) = −1 n=−∞ n (z − zi)n. c(i) From the proof of the Laurent series, we know gi(z) gives a holomorphic function on U \ {zi}. We now consider f −g1−g2−· · ·−gk, which is holomorphic on U \{z1, · · ·, zk}, and has a removable singularity at each zi. So (f − g1 − · · · − gk)(z) dz = 0, γ by simply-connected Cauchy. Hence we know γ f (z) dz = k j=1 γ 53 gj(z) dz. 3 Residue calculus IB Complex Analysis For each j, we use uniform convergence of the series compact subsets of U \ {zj}, and hence on γ, to write n≤−1 c(j) n (z − zj)n on γ gj(z) dz = c(j) n n≤−1 γ (z − zj)n dz. However, for n = −1, the function (z − zj)n has an antiderivative, and hence the integral around γ vanishes. So this is equal to c(j) −1 γ 1 z − zj dz. But c(j) deﬁnition of the winding number (up to a factor of 2πi). So we get −1 is by deﬁnition the residue of f at zj, and the integral is just the integral γ f (z) dz = 2πi k j=1 Res(f ; zj)I(γ, zj). So done. 3.4 Overview We’ve done most of the theory we need. In the remaining of the time, we are going to use these tools to do something useful. In particular, we will use the residue theorem heavily to compute integrals. But before that,
we shall stop and look at what we have done so far. Our ﬁrst real interesting result was Cauchy’s theorem for a triangle, which had a rather weird hypothesis — if f : U → C is holomorphic and ∆ ⊆ U is at triangle, then f (z) dz = 0. ∂∆ To prove this, we dissected our triangle into smaller and smaller triangles, and then the result followed how the numbers and bounds magically ﬁt in together. To accompany this, we had another theorem that used triangles. Suppose U is a star domain and f : U → C is continuous. Then if ∂∆ f (z) dz = 0 for all triangles, then there is a holomorphic F with F (z) = f (z). Here we deﬁned F by z F (z) = f (z) dz, z0 where z0 is the “center” of the star, and we integrate along straight lines. The triangle condition ensures this is well-deﬁned. These are the parts where we used some geometric insight — in the ﬁrst case we thought of subdividing, and in the second we decided to integrate along paths. 54 3 Residue calculus IB Complex Analysis These two awkward theorems about triangles ﬁt in perfectly into the convex Cauchy theorem, via the fundamental theorem of calculus. This tells us that if f : U → C is holomorphic and U is convex, then γ f (z) dz = 0 for all closed γ ⊆ U. We then noticed this allows us to deform paths nicely and still preserve the integral. We called these nice deformations elementary deformations, and then used it to obtain the Cauchy integral formula, namely f (w) = 1 2πi ∂B(a,ρ) f (z) z − w dz for f : B(a, r) → C, ρ < r and w ∈ B(a, ρ). This formula led us to some classical theorems like the Liouville theorem and the maximum principle. We also used the power series trick to prove Taylor’s theorem, saying any holomorphic function is locally equal to some power series, which we call the Taylor series. In particular, this shows that holomorphic functions are inﬁnitely
diﬀerentiable, since all power series are. We then notice that for U a convex domain, if f : U → C is continuous and γ f (z) dz = 0 for all curves γ, then f has an antiderivative. Since f is the derivative of its antiderivative (by deﬁnition), it is then (inﬁnitely) diﬀerentiable. So a function is holomorphic on a simply connected domain if and only if the integral along any closed curve vanishes. Since the latter property is easily shown to be conserved by uniform limits, we know the uniform limit of holomorphic functions is holomorphic. Then we ﬁgured out that we can use the same power series expansion trick to deal with functions with singularities. It’s just that we had to include negative powers of z. Adding in the ideas of winding numbers and homotopies, we got the residue theorem. We showed that if U is simply connected and f : U \ {z1, · · ·, zk} → C is holomorphic, then 1 2πi γ f (z) dz = Res(f, zi)I(γ, zi). This will further lead us to Rouch´e’s theorem and the argument principle, to be done later. Throughout the course, there weren’t too many ideas used. Everything was built upon the two “geometric” theorems of Cauchy’s theorem for triangles and the antiderivative theorem. Afterwards, we repeatedly used the idea of deforming 1 z−w, and that’s it. and cutting paths, as well as the power series expansion of 3.5 Applications of the residue theorem This section is more accurately described as “Integrals, integrals, integrals”. Our main objective is to evaluate real integrals, but to do so, we will pretend they are complex integrals, and apply the residue theorem. 55 3 Residue calculus IB Complex Analysis Before that, we ﬁrst come up with some tools to compute residues, since we will have to do that quite a lot. Lemma. Let f : U \ {a} → C be holomorphic with a pole at a, i.e f is meromorphic on U. (i) If the pole is simple,
then (ii) If near a, we can write Res(f, a) = lim z→a (z − a)f (z). f (z) = g(z) h(z), where g(a) = 0 and h has a simple zero at a, and g, h are holomorphic on B(a, ε) \ {a}, then (iii) If Res(f, a) = g(a) h(a). f (z) = g(z) (z − a)k near a, with g(a) = 0 and g is holomorphic, then Res(f, a) = g(k−1)(a) (k − 1)!. Proof. (i) By deﬁnition, if f has a simple pole at a, then f (z) = c−1 (z − a) + c0 + c1(z − a) + · · ·, and by deﬁnition c−1 = Res(f, a). Then the result is obvious. (ii) This is basically L’Hˆopital’s rule. By the previous part, we have Res(f ; a) = lim z→a (z − a) g(z) h(z) = g(a) lim z→a z − a h(z) − h(a) = g(a) h(a). (iii) We know the residue Res(f ; a) is the coeﬃcient of (z − a)k−1 in the Taylor series of g at a, which is exactly 1 (k−1)! g(k−1)(a). Example. We want to compute the integral ∞ 0 1 1 + x4 dx. We consider the following contour: 56 3 Residue calculus IB Complex Analysis × e3iπ/4 × eiπ/4 −R R × × We notice the two of the poles lie in the unbounded region. So I(γ, · ) = 0 for these. 1+x4 has poles at x4 = −1, as indicated in the diagram. Note that 1 We can write the integral as γR 1 1 + z4 dz = R −R 1 1 + x4 dx + π 0 iReiθ 1 + R4e4iθ
dθ. The ﬁrst term is something we care about, while the second is something we despise. So we might want to get rid of it. We notice the integrand of the second integral is O(R−3). Since we are integrating it over something of length R, the whole thing tends to 0 as R → ∞. We also know the left hand side is just γR 1 1 + z4 dz = 2πi(Res(f, eiπ/4) + Res(f, e3iπ/4)). So we just have to compute the residues. But our function is of the form given by part (ii) of the lemma above. So we know Res(f, eiπ/4) = 1 4z3 z=eiπ/4 = 1 4 e−3πi/4, and similarly at ei3π/4. On the other hand, as R → ∞, the ﬁrst integral on the right is ∞ −∞ 1+x4 dx, which is, by evenness, twice of what we want. So 1 ∞ 2 0 1 1 + x4 dx = ∞ −∞ 1 1 + x4 dx = − 2πi 4 (eiπ/4 + e3πi/4) = π √ 2. Hence our integral is ∞ 0 1 1 + x4 dx = π √ 2 2. When computing contour integrals, there are two things we have to decide. First, we need to pick a nice contour to integrate along. Secondly, as we will see in the next example, we have to decide what function to integrate. Example. Suppose we want to integrate R cos(x) 1 + x + x2 dx. 57 3 Residue calculus IB Complex Analysis We know cos, as a complex function, is everywhere holomorphic, and 1 + x + x2 have two simple zeroes, namely at the cube roots of unity. We pick the same contour, and write ω = e2πi/3. Then we have ×ω −R R Life would be good if cos were bounded, for the integrand would then be O(R−2), and the circular integral vanishes. Unfortunately, at, say, iR, cos(z) is large. So instead, we consider f (z) = eiz 1 +
z + z2. Now, again by the previous lemma, we get Res(f ; ω) = eiω 2ω + 1. On the semicircle, we have π 0 f (Reiθ)Reiθ dθ ≤ π 0 Re− sin θR \|R2e2iθ + Reiθ + 1\| dθ, which is O(R−1). So this vanishes as R → ∞. The remaining is not quite the integral we want, but we can just take the real part. We have R cos x 1 + x + x2 dx = Re R f (z) dz = Re lim R→∞ γR f (z) dz = Re(2πi Res(f, ω)) √ e− 3/2 cos = 2π √ 3 1 2. Another class of integrals that often come up are integrals of trigonometric functions, where we are integrating along the unit circle. Example. Consider the integral π/2 0 1 1 + sin2(t) dt. We use the expression of sin in terms of the exponential function, namely sin(t) = eit − e−it 2i. 58 3 Residue calculus IB Complex Analysis So if we are on the unit circle, and z = eit, then sin(t) = z − z−1 2. dz dt = ieit. dt = dz iz. Moreover, we can check So Hence we get π/2 0 1 1 + sin2(t) dt = = = 1 4 1 4 1 1 + sin2(t) dt 2π 0 1 1 + (z−z−1)2 −4 dz iz \|z\|=1 iz z4 − 6z2 + 1 dz. \|z\|=1 The base is a quadratic in z2, which we can solve. We ﬁnd the roots to be 1 ± and −1 ± √ 2. √ 2 × × × × The residues at the point want is √ 2 − 1 and − √ √ 16. So the integral we 2 − 2 + 1 give − i √ √ 2i − 2i + 16 16 = π √ 2 2. π/2 0 1 1 + sin2(t) =
2πi Most rational functions of trigonometric functions can be integrated around \|z\| = 1 in this way, using the fact that sin(kt) = eikt − e−ikt 2i = zk − z−k 2, cos(kt) = eikt + e−ikt 2 = zk + z−k 2. We now develop a few lemmas that help us evaluate the contributions of certain parts of contours, in order to simplify our work. 59 3 Residue calculus IB Complex Analysis Lemma. Let f : B(a, r) \ {a} → C be holomorphic, and suppose f has a simple pole at a. We let γε : [α, β] → C be given by t → a + εeit. γε β α a ε Then γε lim ε→0 f (z) dz = (β − α) · i · Res(f, a). Proof. We can write c z − a near a, where c = Res(f ; a), and g : B(a, δ) → C is holomorphic near a. We take ε < δ. Then f (z) = + g(z) γε g(z) dz ≤ (β − α) · ε sup z∈γε \|g(z)\|. But g is bounded on B(α, δ). So this vanishes as ε → 0. So the remaining integral is γε c z − a lim ε→0 dz = c lim ε→0 = c lim ε→0 γε β α 1 z − a dz 1 εeit · iεeit dt = i(β − α)c, as required. A lemma of a similar ﬂavor allows us to consider integrals on expanding semicircles. Lemma (Jordan’s lemma). Let f be holomorphic on a neighbourhood of inﬁnity in C, i.e. on {\|z\| > r} for some r > 0. Assume that zf (z) is bounded in this region. Then for α > 0, we have γR f (z)eiαz dz → 0 60 3 Residue calculus IB Complex Analysis as R → ∞, where γR(t)
= Reit for t ∈ [0, π] is the semicircle (which is not closed). γR −R R In previous cases, we had f (z) = O(R−2), and then we can bound the integral simply as O(R−1) → 0. In this case, we only require f (z) = O(R−1). The drawback is that the case f (z) dz need not work — it is possible that this does not vanish. However, if we have the extra help from eiαx, then we do get that the integral vanishes. γR Proof. By assumption, we have \|f (z)\| ≤ M \|z\| for large \|z\| and some constant M > 0. We also have \|eiαz\| = e−Rα sin t on γR. To avoid messing with sin t, we note that on (0, π decreasing, since 2 ], the function sin θ θ is d dθ sin θ θ = θ cos θ − sin θ θ2 ≤ 0. Then by consider the end points, we ﬁnd sin(t) ≥ 2t π for t ∈ [0, π 2 ]. This gives us the bound \|eiαz\| = e−Rα sin t ≤ e−Ra2t/π e−Ra2t/ So we get π/2 0 eiRαeit f (Reit)Reit dt ≤ = 2π e−2αRt/π · M dt 0 1 2R (1 − eαR) → 0 as R → ∞. The estimate for is analogous. π π/2 f (z)eiαz dz 61 3 Residue calculus IB Complex Analysis Example. We want to show ∞ 0 sin x x dx = π 2. Note that sin x x has a removable singularity at x = 0. So everything is ﬁne. Our ﬁrst thought might be to use our usual semi-circular contour that looks like this: γR −R R If we look at this and take the function sin z z, then we get no control at iR ∈ γR. So what we would like to do is to replace the sine with an exponential. If we let f (z)
= eiz z, then we now have the problem that f has a simple pole at 0. So we consider a modiﬁed contour γR,ε −R × ε −ε R Now if γR,ε denotes the modiﬁed contour, then the singularity of eiz z the contour, and Cauchy’s theorem says lies outside γR,ε f (z) dz = 0. Considering the R-semicircle γR, and using Jordan’s lemma with α = 1 and 1 z as the function, we know f (z) dz → 0 as R → ∞. γR Considering the ε-semicircle γε, and using the ﬁrst lemma, we get a contribution of −iπ, where the sign comes from the orientation. Rearranging, and using the fact that the function is even, we get the desired result. 62 3 Residue calculus IB Complex Analysis Example. Suppose we want to evaluate ∞ −∞ eax cosh x dx, where a ∈ (−1, 1) is a real constant. To do this, note that the function f (z) = eaz cosh z has simple poles where z = n + 1 above, then we would run into inﬁnitely many singularities, which is not fun. iπ for n ∈ Z. So if we did as we have done 2 Instead, we note that cos(x + iπ) = − cosh x. Consider a rectangular contour πi γ1 γ− vert ×πi 2 γ+ vert −R γ0 R We now enclose only one singularity, namely ρ = iπ 2, where Res(f, ρ) = eaρ cosh(ρ) = ieaπi/2. We ﬁrst want to see what happens at the edges. We have γ+ vert f (z) dz = π 0 ea(R+iy) cosh(R + iy) i dy. hence we can bound this as f (z) dz γ+ vert ≤ π 0 2eaR eR − e−R dy → 0 as R → ∞, since a < 1. We can do a similar bound for γvert−,
where we use the fact that a > −1. Thus, letting R → ∞, we get R eax cosh x dx + −∞ +∞ eaπieax cosh(x + iπ) dx = 2πi(−ieaπi/2). Using the fact that cosh(x + iπ) = − cos(x), we get R eax cosh x dx = 2πeaiπ/2 1 + eaπi = π sec πa 2. 63 3 Residue calculus IB Complex Analysis Example. We provide a(nother) proof that 1 n2 = π2 6. n≥1 Recall we just avoided having to encircle inﬁnitely poles by picking a rectangular contour. Here we do the opposite — we encircle inﬁnitely many poles, and then we can use this to evaluate the inﬁnite sum of residues using contour integrals., which is holomorphic on C except We consider the function f (z) = π cot(πz) for simple poles at Z \ {0}, and a triple pole at 0. We can check that at n ∈ Z \ {0}, we can write z2 f (z) = π cos(πz) z2 · 1 sin(πz), where the second term has a simple zero at n, and the ﬁrst is non-vanishing at n = 0. Then we have compute Res(f ; n) = π cos(πn) n2 · 1 π cos(πn) = 1 n2. Note that the reason why we have those funny π’s all around the place is so that we can get this nice expression for the residue. At z = 0, we get cot(z) = 1 − z2 2 + O(z4) z − −1 + O(z5) z3 3 = 1 z − z 3 + O(z2). So we get So the residue is − π2 π cot(πz) z2 π2 3z 3. Now we consider the following square contour: 1 z3 − + · · · = (N + 1 2 )i γN −(N + 1 2 ) N + 1 2 × × × × × × × × × ×
× −(N + 1 2 )i Since we don’t want the contour itself to pass through singularities, we make the square pass through ± N + 1. Then the residue theorem says 2 γN f (z) dz = 2πi 2 N n=1 1 n2 − π2 3. 64 3 Residue calculus IB Complex Analysis We can thus get the desired series if we can show that γN f (z) dz → 0 as n → ∞. We ﬁrst note that γN f (z) dz π cot πz z2 4(2N + 1) \| cot πz\| 4(2N + 1)π 2 N + 1 2 \| cot πz\|O(N −1). ≤ sup γN ≤ sup γN = sup γN So everything is good if we can show supγN \| cot πz\| is bounded as N → ∞. On the vertical sides, we have z = π N + 1 2 + iy, and thus \| cot(πz)\| = \| tan(iπy)\| = \| tanh(πy)\| ≤ 1, while on the horizontal sides, we have and \| cot(πz)\| ≤ eπ(N +1/2) + e−π(N +1/2) eπ(N +1/2) − e−π(N +1/2) = coth N + 1 2 π. While it is not clear at ﬁrst sight that this is bounded, we notice x → coth x is decreasing and positive for x ≥ 0. So we win. Example. Suppose we want to compute the integral ∞ 0 log x 1 + x2 dx. The point is that to deﬁne log z, we have to cut the plane to avoid multivaluedness. In this case, we might choose to cut it along iR ≤ 0, giving a branch of log, for which arg(z) ∈ − π. We need to avoid running through zero. So we might look at the following contour: 2, 3π 2 65 3 Residue calculus IB Complex Analysis ×i −R ε ε R On the large semicircular arc of radius R, the integrand \|f (z)\|\|dz
\| = O R · log R R2 = O log R R → 0 as R → ∞. On the small semicircular arc of radius ε, the integrand \|f (z)\|\|dz\| = O(ε log ε) → 0 as ε → 0. Hence, as ε → 0 and R → ∞, we are left with the integral along the negative real axis. Along the negative real axis, we have log z = log \|z\| + iπ. So the residue theorem says ∞ 0 log x 1 + x2 dx + 0 ∞ log \|z\| + iπ 1 + x2 (−dx) = 2πi Res(f ; i). We can compute the residue as Res(f, i) = log i 2i = 1 2 iπ 2i = π 4. So we ﬁnd ∞ log x 1 + x2 dx + iπ Taking the real part of this, we obtain 2 0 ∞ 0 1 1 + x2 dx = iπ2 2. ∞ 0 log x 1 + x2 dx = 0. In this case, we had a branch cut, and we managed to avoid it by going around our magic contour. Sometimes, it is helpful to run our integral along the branch cut. 66 3 Residue calculus IB Complex Analysis Example. We want to compute ∞ 0 √ x x2 + ax + b dx, where a, b ∈ R. To deﬁne along the real line, and consider the keyhole contour z, we need to pick a branch cut. We pick it to lie √ As usual this has a small circle of radius ε around the origin, and a large circle of radius R. Note that these both avoid the branch cut. Again, on the R circle, we have \|f (z)\|\|dz\| = O 1 √ R → 0 as R → ∞. On the ε-circle, we have \|f (z)\|\|dz\| = O(ε3/2) → 0 as ε → 0. √ z = e 1 √ Viewing by 2πi. So going in the wrong direction. Therefore the residue theorem says 2 log z, on the two pieces of the contour along R≥0, log z diﬀers z changes sign. This cancels with the sign change arising from 2πi
residues inside contour = 2 ∞ 0 √ x x2 + ax + b dx. What the residues are depends on what the quadratic actually is, but we will not go into details. 3.6 Rouch´es theorem We now want to move away from computing integrals, and look at a diﬀerent application — Rouch´es theorem. Recall one of the ﬁrst applications of complex analysis is to use Liouville’s theorem to prove the fundamental theorem of algebra, and show that every polynomial has a root. One might wonder — if we know a bit more about the polynomial, can we say a bit more about how the roots behave? 67 3 Residue calculus IB Complex Analysis To do this, recall we said that if f : B(a; r) → C is holomorphic, and f (a) = 0, then f has a zero of order k if, locally, f (z) = (z − a)kg(z), with g holomorphic and g(a) = 0. Analogously, if f : B(a, r) \ {a} → C is holomorphic, and f has at worst a pole at a, we can again write f (z) = (z − a)kg(z), where now k ∈ Z may be negative. Since we like numbers to be positive, we say the order of the zero/pole is \|k\|. It turns out we can use integrals to help count poles and zeroes. Theorem (Argument principle). Let U be a simply connected domain, and let f be meromorphic on U. Suppose in fact f has ﬁnitely many zeroes z1, · · ·, zk and ﬁnitely many poles w1, · · ·, w. Let γ be a piecewise-C 1 closed curve such that zi, wj ∈ image(γ) for all i, j. Then I(f ◦ γ, 0) = 1 2πi γ f (z) f (z) dz = k i=1 ord(f ; zi)Iγ(zi) − j=1 ord(f, wj)I(γ, wj). Note that the ﬁrst equality comes from the fact that I(f ◦ γ, 0) = 1 2π
i f ◦γ dw w = 1 2πi γ df f (z) = 1 2πi γ f (z) f (z) dz. In particular, if γ is a simple closed curve, then all the winding numbers of γ about points zi, wj lying in the region bound by γ are all +1 (with the right choice of orientation). Then number of zeroes − number of poles = 1 2π (change in argument of f along γ). Proof. By the residue theorem, we have 1 2πi γ f (z) f (z) dz = Res f f z∈U, z I(γ, z), where we sum over all zeroes and poles of z. Note that outside these zeroes and poles, the function f (z) f (z) is holomorphic. Now at each zi, if f (z) = (z − zj)kg(z), with g(zj) = 0, then by direct computation, we get f (z) f (z) Since at zj, g is holomorphic and non-zero, we know g(z) zj. So k z − zj g(z) g(z) = +. g(z) is holomorphic near Res f f, zj = k = ord(f, zj). 68 3 Residue calculus IB Complex Analysis Analogously, by the same proof, at the wi, we get Res f f, wj = − ord(f ; wj). So done. This might be the right place to put the following remark — all the time, we have assumed that a simple closed curve “bounds a region”, and then we talk about which poles or zeroes are bounded by the curve. While this seems obvious, it is not. This is given by the Jordan curve theorem, which is actually hard. Instead of resorting to this theorem, we can instead deﬁne what it means to bound a region in a more convenient way. One can say that for a domain U, a closed curve γ ⊆ U bounds a domain D ⊆ U if I(γ, z) = +1 0 z ∈ D z ∈ D, for a particular choice of orientation on γ. However, we shall not worry ourselves with this. The main application of the argument
principle is Rouch´es theorem. Corollary (Rouch´es theorem). Let U be a domain and γ a closed curve which bounds a domain in U (the key case is when U is simply connected and γ is a simple closed curve). Let f, g be holomorphic on U, and suppose \|f \| > \|g\| for all z ∈ image(γ). Then f and f + g have the same number of zeroes in the domain bound by γ, when counted with multiplicity. Proof. If \|f \| > \|g\| on γ, then f and f + g cannot have zeroes on the curve γ. We let h(z) = f (z) + g(z) f (z) = 1 + g(z) f (z). This is a natural thing to consider, since zeroes of f + g is zeroes of h, while poles of h are zeroes of f. Note that by assumption, for all z ∈ γ, we have h(z) ∈ B(1, 1) ⊆ {z : Re z > 0}. Therefore h◦γ is a closed curve in the half-plane {z : Re z > 0}. So I(h◦γ; 0) = 0. Then by the argument principle, h must have the same number of zeros as poles in D, when counted with multiplicity (note that the winding numbers are all +1). Thus, as the zeroes of h are the zeroes of f + g, and the poles of h are the poles of f, the result follows. Example. Consider the function z6 + 6z + 3. This has three roots (with multiplicity) in {1 < \|z\| < 2}. To show this, note that on \|z\| = 2, we have \|z\|4 = 16 > 6\|z\| + 3 ≥ \|6z + 3\|. So if we let f (z) = z4 and g(z) = 6z + 3, then f and f + g have the same number of roots in {\|z\| < 2}. Hence all four roots lie inside {\|z\| < 2}. 69 3 Residue calculus IB Complex Analysis On the other hand, on \|z\| = 1, we have \|6z\| = 6 > \|z4 + 3\|. So
6z and z6 + 6z + 3 have the same number of roots in {\|z\| < 1}. So there is exactly one root in there, and the remaining three must lie in {1 < \|z\| < 2} (the bounds above show that \|z\| cannot be exactly 1 or 2). So done. Example. Let P (x) = xn + an−1xn−1 + · · · + a1x + a0 ∈ Z[x], and suppose a0 = 0. If \|an−1\| > 1 + \|an−2\| + · · · + \|a1\| + \|a0\|, then P is irreducible over Z (and hence irreducible over Q, by Gauss’ lemma from IB Groups, Rings and Modules). To show this, we let f (z) = an−1zn−1, g(z) = zn + an−2zn−2 + · · · + a1z + a0. Then our hypothesis tells us \|f \| > \|g\| on \|z\| = 1. So f and P = f + g both have n − 1 roots in the open unit disc {\|z\| < 1}. Now if we could factor P (z) = Q(z)R(z), where Q, R ∈ Z[x], then at least one of Q, R must have all its roots inside the unit disk. Say all roots of Q are inside the unit disk. But we assumed a0 = 0. So 0 is not a root of P. Hence it is not a root of Q. But the product of the roots Q is a coeﬃcient of Q, hence an integer strictly between 0 and 1. This is a contradiction. The argument principle and Rouch´es theorem tell us how many roots we have got. However, we do not know if they are distinct or not. This information is given to us via the local degree theorem. Before we can state it, we have to deﬁne the local degree. Deﬁnition (Local degree). Let f : B(a, r) → C be holomorphic and nonconstant. Then the local degree of f at a, written deg(f, a) is the order of the zero of f (z) − f (a) at a. If we take the Taylor expansion of f
about a, then the local degree is the degree of the ﬁrst non-zero term after the constant term. Lemma. The local degree is given by where deg(f, a) = I(f ◦ γ, f (a)), γ(t) = a + reit, with 0 ≤ t ≤ 2π, for r > 0 suﬃciently small. Proof. Note that by the identity theorem, we know that, f (z) − f (a) has an isolated zero at a (since f is non-constant). So for suﬃciently small r, the function f (z) − f (a) does not vanish on B(a, r) \ {a}. If we use this r, then f ◦ γ never hits f (a), and the winding number is well-deﬁned. The result then follows directly from the argument principle. 70 3 Residue calculus IB Complex Analysis Proposition (Local degree theorem). Let f : B(a, r) → C be holomorphic and non-constant. Then for r > 0 suﬃciently small, there is ε > 0 such that for any w ∈ B(f (a), ε) \ {f (a)}, the equation f (z) = w has exactly deg(f, a) distinct solutions in B(a, r). Proof. We pick r > 0 such that f (z)−f (a) and f (z) don’t vanish on B(a, r)\{a}. We let γ(t) = a + reit. Then f (a) ∈ image(f ◦ γ). So there is some ε > 0 such that B(f (a), ε) ∩ image(f ◦ γ) = ∅. We now let w ∈ B(f (a), ε). Then the number of zeros of f (z) − w in B(a, r) is just I(f ◦ γ, w), by the argument principle. This is just equal to I(f ◦ γ, f (a)) = deg(f, a), by the invariance of I(Γ, ∗) as we move ∗ in a component C \ Γ. Now if w = f (
a), since f (z) = 0 on B(a, r) \ {a}, all roots of f (z) − w must be simple. So there are exactly deg(f ; a) distinct zeros. The local degree theorem says the equation f (z) = w has deg(f, a) roots for w suﬃciently close to f (a). In particular, we know there are some roots. So B(f (a), ε) is contained in the image of f. So we get the following result: Corollary (Open mapping theorem). Let U be a domain and f : U → C is holomorphic and non-constant, then f is an open map, i.e. for all open V ⊆ U, we get that f (V ) is open. Proof. This is an immediate consequence of the local degree theorem. It suﬃces to prove that for every z ∈ U and r > 0 suﬃciently small, we can ﬁnd ε > 0 such that B(f (a), ε) ⊆ f (B(a, r)). This is true by the local degree theorem. Recall that Liouville’s theorem says every holomorphic f : C → B(0, 1) is constant. However, for any other simply connected domain, we know there are some interesting functions we can write down. Corollary. Let U ⊆ C be a simply connected domain, and U = C. Then there is a non-constant holomorphic function U → B(0, 1). This is a weak form of the Riemann mapping theorem, which says that there is a conformal equivalence to B(0, 1). This just says there is a map that is not boring. Proof. We let q ∈ C \ U, and let φ(z) = z − q. So φ : U → C is non-vanishing. It is also clearly holomorphic and non-constant. By an exercise (possibly on the example sheet), there is a holomorphic function g : U → C such that φ(z) = eg(z) for all z. In particular, our function φ(z) = z − q : U → C∗ can be written as φ(z) = h(z)2, for some function h :
U → C∗ (by letting h(z) = e 1 2 g(z)). We let y ∈ h(U ), and then the open mapping theorem says there is some r > 0 with B(y, r) ⊆ h(U ). But notice φ is injective by observation, and that h(z1) = ±h(z2) implies φ(z1) = φ(z2). So we deduce that B(−y, r) ∩ h(U ) = ∅ (note that since y = 0, we have B(y, r) ∩ B(−y, r) = ∅ for suﬃciently small r). Now deﬁne f : z → r 2(h(z) + y). This is a holomorphic function f : U → B(0, 1), and is non-constant. This shows the amazing diﬀerence between C and C \ {0}. 71 and limits indexed on D are coproducts and products indexed on the set D. Coproducts are disjoint unions in S or U, wedges (or one-point unions) in T, free products in G, and direct sums in A b. Products are Cartesian products in all of these categories; more precisely, they are Cartesian products of underlying sets, with additional structure. If D is the category displayed schematically as e d / f or d / d′, where we have displayed all objects and all non-identity morphisms, then the colimits indexed on D are called pushouts or coequalizers, respectively. Similarly, if D is displayed schematically as e / d f or d / d′, / / $. THE VAN KAMPEN THEOREM 17 then the limits indexed on D are called pullbacks or equalizers, respectively. A given category may or may not have all colimits, and it may have some but not others. A category is said to be cocomplete if it has all colimits, complete if it has all limits. The categories S, U, T, G, and A b are complete and cocomplete. If a category has coproducts and coequalizers, then it is cocomplete, and similarly for completeness. The proof is a worthwhile exercise. 7. The van Kampen theorem The following is a modern
dress treatment of the van Kampen theorem. I should admit that, in lecture, it may make more sense not to introduce the fundamental groupoid and to go directly to the fundamental group statement. The direct proof is shorter, but not as conceptual. However, as far as I know, the deduction of the fundamental group version of the van Kampen theorem from the fundamental groupoid version does not appear in the literature in full generality. The proof well illustrates how to manipulate colimits formally. We have used the van Kampen theorem as an excuse to introduce some basic categorical language, and we shall use that language heavily in our treatment of covering spaces in the next chapter. Theorem (van Kampen). Let O = {U } be a cover of a space X by path connected open subsets such that the intersection of ﬁnitely many subsets in O is again in O. Regard O as a category whose morphisms are the inclusions of subsets and observe that the functor Π, restricted to the spaces and maps in O, gives a diagram Π\|O : O −→ G P of groupoids. The groupoid Π(X) is the colimit of this diagram. In symbols, Π(X) ∼= colimU∈O Π(U ). Proof. We must verify the universal property. For a groupoid C and a map η : Π\|O −→ C of O-shaped diagrams of groupoids, we must construct a map ˜η : Π(X) −→ C of groupoids that restricts to ηU on Π(U ) for each U ∈ O. On objects, that is on points of X, we must deﬁne ˜η(x) = ηU (x) for x ∈ U. This is independent of the choice of U since O is closed under ﬁnite intersections. If a path f : x → y lies entirely in a particular U, then we must deﬁne ˜η[f ] = η([f ]). Again, since O is closed under ﬁnite intersections, this speciﬁcation is independent of the choice of U if f lies entirely in more than one U. Any path f is the composite of ﬁnitely many paths fi, each of which does lie in a single U, and we must de�
��ne ˜η[f ] to be the composite of the ˜η[fi]. Clearly this speciﬁcation will give the required unique map ˜η, provided that ˜η so speciﬁed is in fact well deﬁned. Thus suppose that f is equivalent to g. The equivalence is given by a homotopy h : f ≃ g through paths x → y. We may subdivide the square I × I into subsquares, each of which is mapped into one of the U. We may choose the subdivision so that the resulting subdivision of I × {0} reﬁnes the subdivision used to decompose f as the composite of paths fi, and similarly for g and the resulting subdivision of I × {1}. We see that the relation [f ] = [g] in Π(X) is a consequence of a ﬁnite number of relations, each of which holds in one of the Π(U ). Therefore ˜η([f ]) = ˜η([g]). This veriﬁes the universal property and proves the theorem. The fundamental group version of the van Kampen theorem “follows formally.” That is, it is an essentially categorical consequence of the version just proved. Arguments like this are sometimes called proof by categorical nonsense. 18 CATEGORICAL LANGUAGE AND THE VAN KAMPEN THEOREM Theorem (van Kampen). Let X be path connected and choose a basepoint x ∈ X. Let O be a cover of X by path connected open subsets such that the intersection of ﬁnitely many subsets in O is again in O and x is in each U ∈ O. Regard O as a category whose morphisms are the inclusions of subsets and observe that the functor π1(−, x), restricted to the spaces and maps in O, gives a diagram π1\|O : O −→ G of groups. The group π1(X, x) is the colimit of this diagram. In symbols, π1(X, x) ∼= colimU∈O π1(U, x). We proceed in two steps. Lemma. The van Kampen theorem holds when the cover O is ﬁnite. Proof. This step is based
on the nonsense above about skeleta of categories. We must verify the universal property, this time in the category of groups. For a group G and a map η : π1\|O −→ G of O-shaped diagrams of groups, we must show that there is a unique homomorphism ˜η : π1(X, x) −→ G that restricts to ηU on π1(U, x). Remember that we think of a group as a groupoid with a single object and with the elements of the group as the morphisms. The inclusion of categories J : π1(X, x) −→ Π(X) is an equivalence. An inverse equivalence F : Π(X) −→ π1(X, x) is determined by a choice of path classes x −→ y for y ∈ X; we choose cx when y = x and so ensure that F ◦ J = Id. Because the cover O is ﬁnite and closed under ﬁnite intersections, we can choose our paths inductively so that the path x −→ y lies entirely in U whenever y is in U. This ensures that the chosen paths determine compatible inverse equivalences FU : Π(U ) −→ π1(U, x) to the inclusions JU : π1(U, x) −→ Π(U ). Thus the functors Π(U ) FU / / π1(U, x) ηU / G specify an O-shaped diagram of groupoids Π\|O −→ G. By the fundamental groupoid version of the van Kampen theorem, there is a unique map of groupoids that restricts to ηU ◦ FU on Π(U ) for each U. The composite ξ : Π(X) −→ G π1(X, x) J / Π(X) ξ / G is the required homomorphism ˜η. It restricts to ηU on π1(U, x) by a little “diagram chase” and the fact that FU ◦ JU = Id. It is unique because ξ is unique. In fact, if we are given ˜η : π1(X, x) −→ G that restricts to ηU on each π1(U, x
), then ˜η ◦ F : Π(X) −→ G restricts to ηU ◦ FU on each Π(U ); therefore ξ = ˜η ◦ F and thus ξ ◦ J = ˜η. Proof of the van Kampen theorem. We deduce the general case from the case just proved. Let F be the set of those ﬁnite subsets of the cover O that are closed under ﬁnite intersection. For S ∈ F, let US be the union of the U in S. Then S is a cover of US to which the lemma applies. Thus colimU∈S π1(U, x) ∼= π1(US, x). Regard F as a category with a morphism S −→ T whenever US ⊂ UT. We claim ﬁrst that colimS ∈F π1(US, x) ∼= π1(X, x). / / / 8. EXAMPLES OF THE VAN KAMPEN THEOREM 19 In fact, by the usual subdivision argument, any loop I −→ X and any equivalence h : I × I −→ X between loops has image in some US. This implies directly that π1(X, x), together with the homomorphisms π1(US, x) −→ π1(X, x), has the universal property that characterizes the claimed colimit. We claim next that colimU∈O π1(U, x) ∼= colimS ∈F π1(US, x), and this will complete the proof. Substituting in the colimit on the right, we have colimS ∈F π1(US, x) ∼= colimS ∈F colimU∈S π1(U, x). By a comparison of universal properties, this iterated colimit is isomorphic to the single colimit colim(U,S )∈(O,F ) π1(U, x). Here the indexing category (O, F ) has objects the pairs (U, S ) with U ∈ S ; there is a morphism (U, S ) −→ (V, T ) whenever both U ⊂ V and US ⊂ UT.
A moment’s reﬂection on the relevant universal properties should convince the reader of the claimed identiﬁcation of colimits: the system on the right diﬀers from the system on the left only in that the homomorphisms π1(U, x) −→ π1(V, x) occur many times in the system on the right, each appearance making the same contribution to the colimit. If we assume known a priori that colimits of groups exist, we can formalize this as follows. We have a functor O −→ F that sends U to the singleton set {U } and thus a functor O −→ (O, F ) that sends U to (U, {U }). The functor π1(−, x) : O −→ G factors through (O, F ), hence we have an induced map of colimits colimU∈O π1(U, x) −→ colim(U,S )∈(O,F ) π1(U, x). Projection to the ﬁrst coordinate gives a functor (O, F ) −→ O. Its composite with π1(−, x) : O −→ G deﬁnes the colimit on the right, hence we have an induced map of colimits colim(U,S )∈(O,F ) π1(U, x) −→ colimU∈O π1(U, x). These maps are inverse isomorphisms. 8. Examples of the van Kampen theorem So far, we have only computed the fundamental groups of the circle and of contractible spaces. The van Kampen theorem lets us extend these calculations. We now drop notation for the basepoint, writing π1(X) instead of π1(X, x). Proposition. Let X be the wedge of a set of path connected based spaces Xi, each of which contains a contractible neighborhood Vi of its basepoint. Then π1(X) is the coproduct (= free product) of the groups π1(Xi). Proof. Let Ui be the union of Xi and the Vj for j 6= i. We apply the van Kampen theorem with O taken to be the Ui and their ﬁnite intersections. Since any
intersection of two or more of the Ui is contractible, the intersections make no contribution to the colimit and the conclusion follows. Corollary. The fundamental group of a wedge of circles is a free group with one generator for each circle. 20 CATEGORICAL LANGUAGE AND THE VAN KAMPEN THEOREM Any compact surface is homeomorphic to a sphere, or to a connected sum of tori T 2 = S1 × S1, or to a connected sum of projective planes RP 2 = S2/Z2 (where we write Z2 = Z/2Z). We shall see shortly that π1(RP 2) = Z2. We also have the following observation, which is immediate from the universal property of products. Using this information, it is an exercise to compute the fundamental group of any compact surface from the van Kampen theorem. Lemma. For based spaces X and Y, π1(X × Y ) ∼= π1(X) × π1(Y ). We shall later use the following application of the van Kampen theorem to prove that any group is the fundamental group of some space. We need a deﬁnition. Definition. A space X is said to be simply connected if it is path connected and satisﬁes π1(X) = 0. Proposition. Let X = U ∪V, where U, V, and U ∩V are path connected open neighborhoods of the basepoint of X and V is simply connected. Then π1(U ) −→ π1(X) is an epimorphism whose kernel is the smallest normal subgroup of π1(U ) that contains the image of π1(U ∩ V ). Proof. Let N be the cited kernel and consider the diagram π1(U ∩ V ) π1(U )/N π1(U ) 8ppppppppppp &NNNNNNNNNNN *UUUUUUUUUUUUUUUUUUU %LLLLLLLLLL 4iiiiiiiiiiiiiiiii 9ssssssssss ξ /___ π1(X) π1(V ) = 0 The universal property gives rise to the map ξ, and ξ is an isomorphism since, by an easy algebraic inspection, π1(U )/N is the
pushout in the category of groups of the homomorphisms π1(U ∩ V ) −→ π1(U ) and π1(U ∩ V ) −→ 0. PROBLEMS (1) Compute the fundamental group of the two-holed torus (the compact surface of genus 2 obtained by sewing together two tori along the boundaries of an open disk removed from each). (2) The Klein bottle K is the quotient space of S1 × I obtained by identifying (z, 0) with (z−1, 1) for z ∈ S1. Compute π1(K). (3) ∗ Let X = {(p, q)\|p 6= −q} ⊂ Sn × Sn. Deﬁne a map f : Sn −→ X by f (p) = (p, p). Prove that f is a homotopy equivalence. (4) Let C be a category that has all coproducts and coequalizers. Prove that C is cocomplete (has all colimits). Deduce formally, by use of opposite categories, that a category that has all products and equalizers is complete. % * & 8 / 9 4 CHAPTER 3 Covering spaces We run through the theory of covering spaces and their relationship to fundamental groups and fundamental groupoids. This is standard material, some of the oldest in algebraic topology. However, I know of no published source for the use that we shall make of the orbit category O(π1(B, b)) in the classiﬁcation of coverings of a space B. This point of view gives us the opportunity to introduce some ideas that are central to equivariant algebraic topology, the study of spaces with group actions. In any case, this material is far too important to all branches of mathematics to omit. 1. The deﬁnition of covering spaces While the reader is free to think about locally contractible spaces, weaker conditions are appropriate for the full generality of the theory of covering spaces. A space X is said to be locally path connected if for any x ∈ X and any neighborhood U of x, there is a smaller neighborhood V of x each of whose points can be connected to x by a path in U. This is equivalent to the seemingly more stringent requirement that the topology of X have a basis consisting of path connected open sets.
In fact, if X is locally path connected and U is an open neighborhood of a point x, then the set V = {y \| y can be connected to x by a path in U } is a path connected open neighborhood of x that is contained in U. Observe that if X is connected and locally path connected, then it is path connected. Throughout this chapter, we assume that all given spaces are connected and locally path connected. Definition. A map p : E −→ B is a covering (or cover, or covering space) if it is surjective and if each point b ∈ B has an open neighborhood V such that each component of p−1(V ) is open in E and is mapped homeomorphically onto V by p. We say that a path connected open subset V with this property is a fundamental neighborhood of B. We call E the total space, B the base space, and Fb = p−1(b) a ﬁber of the covering p. Any homeomorphism is a cover. A product of covers is a cover. The projection R −→ S1 is a cover. Each fn : S1 −→ S1 is a cover. The projection Sn −→ RP n is a cover, where the real projective space RP n is obtained from Sn by identifying antipodal points. If f : A −→ B is a map (where A is connected and locally path connected) and D is a component of the pullback of f along p, then p : D −→ A is a cover. 21 22 COVERING SPACES 2. The unique path lifting property The following result is abstracted from what we saw in the case of the particular cover R −→ S1. It describes the behavior of p with respect to path classes and fundamental groups. Theorem (Unique path lifting). Let p : E −→ B be a covering, let b ∈ B, and let e, e′ ∈ Fb. (i) A path f : I −→ B with f (0) = b lifts uniquely to a path g : I −→ E such that g(0) = e and p ◦ g = f. (ii) Equivalent paths f ≃ f ′ : I −→ B that start at b lift to equivalent paths g ≃ g′ : I −→ E that start at e, hence g(1) = g′(1). (iii) p∗ :
π1(E, e) −→ π1(B, b) is a monomorphism. (iv) p∗(π1(E, e′)) is conjugate to p∗(π1(E, e)). (v) As e′ runs through Fb, the groups p∗(π1(E, e′)) run through all conjugates of p∗(π1(E, e)) in π1(B, b). Proof. For (i), subdivide I into subintervals each of which maps to a fundamental neighborhood under f, and lift f to g inductively by use of the prescribed homeomorphism property of fundamental neighborhoods. For (ii), let h : I × I −→ B be a homotopy f ≃ f ′ through paths b −→ b′. Subdivide the square into subsquares each of which maps to a fundamental neighborhood under f. Proceeding inductively, we see that h lifts uniquely to a homotopy H : I × I −→ E such that H(0, 0) = e and p ◦ H = h. By uniqueness, H is a homotopy g ≃ g′ through paths e −→ e′, where g(1) = e′ = g′(1). Parts (iii)–(v) are formal consequences of (i) and (ii), as we shall see in the next section. Definition. A covering p : E −→ B is regular if p∗(π1(E, e)) is a normal subgroup of π1(B, b). It is universal if E is simply connected. As we shall explain in §4, for a universal cover p : E −→ B, the elements of Fb are in bijective correspondence with the elements of π1(B, b). We illustrate the force of this statement. Example. For n ≥ 2, Sn is a universal cover of RP n. Therefore π1(RP n) has only two elements. There is a unique group with two elements, and this proves our earlier claim that π1(RP n) = Z2. 3. Coverings of groupoids Much of the theory of covering spaces can be recast conceptually in terms of fundamental groupoids. This point of view separates the essentials of the topology from the formalities and gives a convenient language in which
to describe the algebraic classiﬁcation of coverings. Definition. (i) Let C be a category and x be an object of C. The category x\C of objects under x has objects the maps f : x −→ y in C ; for objects f : x −→ y and g : x −→ z, the morphisms γ : f −→ g in x\C are the morphisms γ : y −→ z in C such that γ ◦ f = g : x −→ z. Composition and identity maps are given by composition and identity maps in C. When C is a groupoid, γ = g ◦ f −1, and the objects of x\C therefore determine the category. 3. COVERINGS OF GROUPOIDS 23 (ii) Let C be a small groupoid. Deﬁne the star of x, denoted St(x) or StC (x), to be the set of objects of x\C, that is, the set of morphisms of C with source x. Write C (x, x) = π(C, x) for the group of automorphisms of the object x. (iii) Let E and B be small connected groupoids. A covering p : E −→ B is a functor that is surjective on objects and restricts to a bijection p : St(e) −→ St(p(e)) for each object e of E. For an object b of B, let Fb denote the set of objects of E such that p(e) = b. Then p−1(St(b)) is the disjoint union over e ∈ Fb of St(e). Parts (i) and (ii) of the unique path lifting theorem can be restated as follows. Proposition. If p : E −→ B is a covering of spaces, then the induced functor Π(p) : Π(E) −→ Π(B) is a covering of groupoids. Parts (iii), (iv), and (v) of the unique path lifting theorem are categorical consequences that apply to any covering of groupoids, where they read as follows. Proposition. Let p : E −→ B be a covering of groupoids, let b be an object of B, and let e and e′ be objects of Fb. (i) p : π(E,
e) −→ π(B, b) is a monomorphism. (ii) p(π(E, e′)) is conjugate to p(π(E, e)). (iii) As e′ runs through Fb, the groups p(π(E, e′)) run through all conjugates of p(π(E, e)) in π(B, b). Proof. For (i), if g, g′ ∈ π(E, e) and p(g) = p(g′), then g = g′ by the injectivity of p on St(e). For (ii), there is a map g : e −→ e′ since E is connected. Conjugation by g gives a homomorphism π(E, e) −→ π(E, e′) that maps under p to conjugation of π(B, b) by its element p(g). For (iii), the surjectivity of p on St(e) gives that any f ∈ π(B, b) is of the form p(g) for some g ∈ St(e). If e′ is the target of g, then p(π(E, e′)) is the conjugate of p(π(E, e)) by f. The ﬁbers Fb of a covering of groupoids are related by translation functions. Definition. Let p : E −→ B be a covering of groupoids. Deﬁne the ﬁber translation functor T = T (p) : B −→ S as follows. For an object b of B, T (b) = Fb. For a morphism f : b −→ b′ of B, T (f ) : Fb −→ Fb′ is speciﬁed by T (f )(e) = e′, where e′ is the target of the unique g in St(e) such that p(g) = f. It is an exercise from the deﬁnition of a covering of a groupoid to verify that T is a well deﬁned functor. For a covering space p : E −→ B and a path f : b −→ b′, T (f ) : Fb −→ Fb′ is given by T (f )(e) = g(
1) where g is the path in E that starts at e and covers f. Proposition. Any two ﬁbers Fb and Fb′ of a covering of groupoids have the same cardinality. Therefore any two ﬁbers of a covering of spaces have the same cardinality. Proof. For f : b −→ b′, T (f ) : Fb −→ Fb′ is a bijection with inverse T (f −1). 24 COVERING SPACES 4. Group actions and orbit categories The classiﬁcation of coverings is best expressed in categorical language that involves actions of groups and groupoids on sets. A (left) action of a group G on a set S is a function G × S −→ S such that es = s (where e is the identity element) and (g′g)s = g′(gs) for all s ∈ S. The isotropy group Gs of a point s is the subgroup {g\|gs = s} of G. An action is free if gs = s implies g = e, that is, if Gs = e for every s ∈ S. The orbit generated by a point s is {gs\|g ∈ G}. An action is transitive if for every pair s, s′ of elements of S, there is an element g of G such that gs = s′. Equivalently, S consists of a single orbit. If H is a subgroup of G, the set G/H of cosets gH is a transitive G-set. When G acts transitively on a set S, we obtain an isomorphism of G-sets between S and the G-set G/Gs for any ﬁxed s ∈ S by sending gs to the coset gGs. The following lemma describes the group of automorphisms of a transitive G-set S. For a subgroup H of G, let N H denote the normalizer of H in G and deﬁne W H = N H/H. Such quotient groups W H are sometimes called Weyl groups. Lemma. Let G act transitively on a set S, choose s ∈ S, and let H = Gs. Then W H is isomorphic to the group AutG(S) of automorphisms of the G-set S. Proof. For n ∈ N H
with image ¯n ∈ W H, deﬁne an automorphism φ(¯n) of S by φ(¯n)(gs) = gns. For an automorphism φ of S, we have φ(s) = ns for some n ∈ G. For h ∈ H, hns = φ(hs) = φ(s) = ns, hence n−1hn ∈ Gs = H and n ∈ N H. Clearly φ = φ(¯n), and it is easy to check that this bijection between W H and AutG(S) is an isomorphism of groups. We shall also need to consider G-maps between diﬀerent G-sets G/H. Lemma. A G-map α : G/H −→ G/K has the form α(gH) = gγK, where the element γ ∈ G satisﬁes γ−1hγ ∈ K for all h ∈ H. Proof. If α(eH) = γK, then the relation γK = α(eH) = α(hH) = hα(eH) = hγK implies that γ−1hγ ∈ K for h ∈ H. Definition. The category O(G) of canonical orbits has objects the G-sets G/H and morphisms the G-maps of G-sets. The previous lemmas give some feeling for the structure of O(G) and lead to the following alternative description. Lemma. The category O(G) is isomorphic to the category G whose objects are the subgroups of G and whose morphisms are the distinct subconjugacy relations γ−1Hγ ⊂ K for γ ∈ G. If we regard G as a category with a single object, then a (left) action of G on a set S is the same thing as a covariant functor G −→ S. (A right action is the same thing as a contravariant functor.) If B is a small groupoid, it is therefore natural to think of a covariant functor T : B −→ S as a generalization of a group action. For each object b of B, T restricts to an action of π(B, b) on T (b). We say
that the functor T is transitive if this group action is transitive for each object b. If B is connected, this holds for all objects b if it holds for any one object b. 5. THE CLASSIFICATION OF COVERINGS OF GROUPOIDS 25 For example, for a covering of groupoids p : E −→ B, the ﬁber translation functor T restricts to give an action of π(B, b) on the set Fb. For e ∈ Fb, the isotropy group of e is precisely p(π(E, e)). That is, T (f )(e) = e if and only if the lift of f to an element of St(e) is an automorphism of e. Moreover, the action is transitive since there is an isomorphism in E connecting any two points of Fb. Therefore, as a π(B, b)-set, Fb ∼= π(B, b)/p(π(E, e)). Definition. A covering p : E −→ B of groupoids is regular if p(π(E, e)) is a normal subgroup of π(B, b). It is universal if p(π(E, e)) = {e}. Clearly a covering space is regular or universal if and only if its associated covering of fundamental groupoids is regular or universal. A covering of groupoids is universal if and only if π(B, b) acts freely on Fb, and then Fb is isomorphic to π(B, b) as a π(B, b)-set. Specializing to covering spaces, this sharpens our earlier claim that the elements of Fb and π1(B, b) are in bijective correspondence. 5. The classiﬁcation of coverings of groupoids Fix a small connected groupoid B throughout this section and the next. We explain the classiﬁcation of coverings of B. This gives an algebraic prototype for the classiﬁcation of coverings of spaces. We begin with a result that should be called the fundamental theorem of covering groupoid theory. We assume once and for all that all given groupoids are small and connected. Theorem. Let p : E −→ B be a covering of groupoids, let X be a groupoid, and let f : X −→ B be a functor. Choose
a base object x0 ∈ X, let b0 = f (x0), and choose e0 ∈ Fb0. Then there exists a functor g : X −→ E such that g(x0) = e0 and p ◦ g = f if and only if f (π(X, x0)) ⊂ p(π(E, e0)) in π(B, b0). When this condition holds, there is a unique such functor g. Proof. If g exists, its properties directly imply that im(f ) ⊂ im(p). For an object x of X and a map α : x0 −→ x in X, let ˜α be the unique element of St(e0) such that p(˜α) = f (α). If g exists, g(α) must be ˜α and therefore g(x) must be the target T (f (α))(e0) of ˜α. The inclusion f (π(X, x0)) ⊂ p(π(E, e0)) ensures that T (f (α))(e0) is independent of the choice of α, so that g so speciﬁed is a well deﬁned functor. In fact, given another map α′ : x0 −→ x, α−1 ◦ α′ is an element of π(X, x0). Therefore f (α)−1 ◦ f (α′) = f (α−1 ◦ α′) = p(β) for some β ∈ π(E, e0). Thus p(˜α ◦ β) = f (α) ◦ p(β) = f (α) ◦ f (α)−1 ◦ f (α′) = f (α′). This means that ˜α ◦ β is the unique element ˜α′ of St(e0) such that p(˜α′) = f (α′), and its target is the target of ˜α, as required. 26 COVERING SPACES Definition. A map g : E −→ E ′ of coverings of B is a functor g such that the following diagram of functors is commutative: E g / E ′ AAAAAAAA p }\|\|\|\|\|\|\|\| ′ p B. Let Cov(B
) denote the category of coverings of B; when B is understood, we write Cov(E, E ′) for the set of maps E −→ E ′ of coverings of B. Lemma. A map g : E −→ E ′ of coverings is itself a covering. Proof. The functor g is surjective on objects since, if e′ ∈ E ′ and we choose an object e ∈ E and a map f : g(e) −→ e′ in E ′, then e′ = g(T (p′(f ))(e)). The map g : StE (e) −→ StE ′(g(e)) is a bijection since its composite with the bijection p′ : StE ′ (g(e)) −→ StB(p′(g(e))) is the bijection p : StE (e) −→ StB(p(e)). The fundamental theorem immediately determines all maps of coverings of B in terms of group level data. Theorem. Let p : E −→ B and p′ : E ′ −→ B be coverings and choose base objects b ∈ B, e ∈ E, and e′ ∈ E ′ such that p(e) = b = p′(e′). There exists a map g : E −→ E ′ of coverings with g(e) = e′ if and only if p(π(E, e)) ⊂ p′(π(E ′, e′)), and there is then only one such g. In particular, two maps of covers g, g′ : E −→ E ′ coincide if g(e) = g′(e) for any one object e ∈ E. Moreover, g is an isomorphism if and only if the displayed inclusion of subgroups of π(B, b) is an equality. Therefore E and E ′ are isomorphic if and only if p(π(E, e)) and p′(π(E ′, e′)) are conjugate whenever p(e) = p′(e′). Corollary. If it exists, the universal cover of B is unique up to isomorphism and covers any other cover. That the universal cover does exist will be proved in the next section. It is useful to recast the previous theorem in terms
of actions on ﬁbers. Theorem. Let p : E −→ B and p′ : E ′ −→ B be coverings, choose a base object b ∈ B, and let G = π(B, b). If g : E −→ E ′ is a map of coverings, then g restricts to a map Fb −→ F ′ b of G-sets, and restriction to ﬁbers speciﬁes a bijection between Cov(E, E ′) and the set of G-maps Fb −→ F ′ b. Proof. Let e ∈ Fb and f ∈ π(B, b). By deﬁnition, f e is the target of the map ˜f ∈ StE (e) such that p( ˜f ) = f. Clearly g(f e) is the target of g( ˜f ) ∈ StE ′ (g(e)) and p′(g( ˜f )) = p( ˜f ) = f. Again by deﬁnition, this gives g(f e) = f g(e). The previous theorem shows that restriction to ﬁbers is an injection on Cov(E, E ′). To show b be a G-map. Choose e ∈ Fb and let e′ = α(e). surjectivity, let α : Fb −→ F ′ Since α is a G-map, the isotropy group p(π(E, e)) of e is contained in the isotropy group p′(π(E ′, e′)) of e′. Therefore the previous theorem ensures the existence of a covering map g that restricts to α on ﬁbers. Definition. Let Aut(E ) ⊂ Cov(E, E ) denote the group of automorphisms of a cover E. Note that, since it is possible to have conjugate subgroups H and H ′ of / } 6. THE CONSTRUCTION OF COVERINGS OF GROUPOIDS 27 a group G such that H is a proper subgroup of H ′, it is possible to have a map of covers g : E −→ E such that g is not an isomorphism. Corollary. Let p : E −→ B be a covering and choose objects b ∈ B and e �
� Fb. Write G = π(B, b) and H = p(π(E, e)). Then Aut(E ) is isomorphic to the group of automorphisms of the G-set Fb and therefore to the group W H. If p is regular, then Aut(E ) ∼= G/H. If p is universal, then Aut(E ) ∼= G. 6. The construction of coverings of groupoids We have given an algebraic classiﬁcation of all possible covers of B: there is at most one isomorphism class of covers corresponding to each conjugacy class of subgroups of π(B, b). We show that all of these possibilities are actually realized. Since this algebraic result is not needed in the proof of its topological analogue, we shall not give complete details. Theorem. Choose a base object b of B and let G = π(B, b). There is a functor E (−) : O(G) −→ Cov(B) that is an equivalence of categories. For each subgroup H of G, the covering p : E (G/H) −→ B has a canonical base object e in its ﬁber over b such that p(π(E (G/H), e)) = H. Moreover, Fb = G/H as a G-set and, for a G-map α : G/H −→ G/K in O(G), the restriction of E (α) : E (G/H) −→ E (G/K) to ﬁbers over b coincides with α. Proof. The idea is that, up to bijection, StE (G/H)(e) must be the same set for each H, but the nature of its points can diﬀer with H. At one extreme, E (G/G) = B, p = id, e = b, and the set of morphisms from b to any other object b′ is a copy of π(B, b). At the other extreme, E (G/e) is a universal cover of B and there is just one morphism from e to any other object e′. In general, the set of objects of E (G/H) is deﬁned to be StB(b)/H, the coset of the identity morphism being e. Here
G and hence its subgroup H act from the right on StB(b) by composition in B. We deﬁne p : E (G/H) −→ B on objects by letting p(f H) be the target of f, which is independent of the coset representative f. We deﬁne morphism sets by f ′ ◦ h ◦ f −1\|h ∈ H Again, this is independent of the choices of coset representatives f and f ′. Composition and identities are inherited from those of B, and p is given on morphisms by the displayed inclusions. It is easy to check that p : E (G/H) −→ B is a covering, and it is clear that p(π(E (G/H), e)) = H. ⊂ B(p(f H), p(f ′H)). E (G/H)(f H, f ′H) = This deﬁnes the object function of the functor E : O(G) −→ Cov(B). To deﬁne E on morphisms, consider α : G/H −→ G/K. If α(eH) = gK, then g−1Hg ⊂ K and α(f H) = f gK. The functor E (α) : E (G/H) −→ E (G/K) sends the object f H to the object α(f H) = f gK and sends the morphism f ′ ◦ h ◦ f −1 to the same morphism of B regarded as f ′g ◦ g−1hg ◦ g−1f −1. It is easily checked that each E (α) is a well deﬁned functor, and that E is functorial in α. To show that the functor E (−) is an equivalence of categories, it suﬃces to show that it maps the morphism set O(G)(G/H, G/K) bijectively onto the morphism set Cov(E (G/H), E (G/K)) and that every covering of B is isomorphic to one of the coverings E (G/H). These statements are immediate from the results of the previous section. 28 COVERING SPACES The following remarks place the orbit category O
(π(B, b)) in perspective by relating it to several other equivalent categories. Remark. Consider the category S B of functors T : B −→ S and natural transformations. Let G = π(B, b). Regarding G as a category with one object b, it is a skeleton of B, hence the inclusion G ⊂ B is an equivalence of categories. Therefore, restriction of functors T to G-sets T (b) gives an equivalence of categories from S B to the category of G-sets. This restricts to an equivalence between the respective subcategories of transitive objects. We have chosen to focus on transitive objects since we prefer to insist that coverings be connected. The inclusion of the orbit category O(G) in the category of transitive G-sets is an equivalence of categories because O(G) is a full subcategory that contains a skeleton. We could shrink O(G) to a skeleton by choosing one H in each conjugacy class of subgroups of G, but the resulting equivalent subcategory is a less natural mathematical object. 7. The classiﬁcation of coverings of spaces In this section and the next, we shall classify covering spaces and their maps by arguments precisely parallel to those for covering groupoids in the previous sections. In fact, applied to the associated coverings of fundamental groupoids, some of the algebraic results directly imply their topological analogues. We begin with the following result, which deserves to be called the fundamental theorem of covering space theory and has many other applications. It asserts that the fundamental group gives the only “obstruction” to solving a certain lifting problem. Recall our standing assumption that all given spaces are connected and locally path connected. Theorem. Let p : E −→ B be a covering and let f : X −→ B be a continuous map. Choose x ∈ X, let b = f (x), and choose e ∈ Fb. There exists a map g : X −→ E such that g(x) = e and p ◦ g = f if and only if f∗(π1(X, x)) ⊂ p∗(π1(E, e)) in π1(B, b). When this condition holds, there is a unique such map g. Proof. If g exists, its properties directly imply that im(f∗) ⊂ im(p∗
). Thus assume that im(f∗) ⊂ im(p∗). Applied to the covering Π(p) : Π(E) −→ Π(B), the analogue for groupoids gives a functor Π(X) −→ Π(E) that restricts on objects to the unique map g : X −→ E of sets such that g(x) = e and p ◦ g = f. We need only check that g is continuous, and this holds because p is a local homeomorphism. In detail, if y ∈ X and g(y) ∈ U, where U is an open subset of E, then there is a smaller open neighborhood U ′ of g(y) that p maps homeomorphically onto an open subset V of B. If W is any path connected neighborhood of y such that f (W ) ⊂ V, then g(W ) ⊂ U ′ by inspection of the deﬁnition of g. Definition. A map g : E −→ E′ of coverings over B is a map g such that the following diagram is commutative: E g AAAAAAAA p B. / E′ ~\|\|\|\|\|\|\|\| ′ p Let Cov(B) denote the category of coverings of the space B; when B is understood, we write Cov(E, E′) for the set of maps E −→ E′ of coverings of B. / ~ 7. THE CLASSIFICATION OF COVERINGS OF SPACES 29 Lemma. A map g : E −→ E′ of coverings is itself a covering. Proof. The map g is surjective by the algebraic analogue. The fundamental neighborhoods for g are the components of the inverse images in E′ of the neigh- borhoods of B which are fundamental for both p and p′. The following remarkable theorem is an immediate consequence of the funda- mental theorem of covering space theory. Theorem. Let p : E −→ B and p′ : E′ −→ B be coverings and choose b ∈ B, e ∈ E, and e′ ∈ E′ such that p(e) = b = p′(e′). There exists a map g : E −→ E′ of coverings with g(e) = e′ if and only if p∗(π1(E, e)) ⊂ p′
∗(π1(E′, e′)), and there is then only one such g. In particular, two maps of covers g, g′ : E −→ E′ coincide if g(e) = g′(e) for any one e ∈ E. Moreover, g is a homeomorphism if and only if the displayed inclusion of subgroups of π1(B, b) is an equality. Therefore E and E′ are homeomorphic if and only if p∗(π1(E, e)) and p′ ∗(π1(E′, e′)) are conjugate whenever p(e) = p′(e′). Corollary. If it exists, the universal cover of B is unique up to isomorphism and covers any other cover. Under a necessary additional hypothesis on B, we shall prove in the next section that the universal cover does exist. We hasten to add that the theorem above is atypical of algebraic topology. It is not usually the case that algebraic invariants like the fundamental group totally determine the existence and uniqueness of maps of topological spaces with prescribed properties. The following immediate implication of the theorem gives one explanation. Corollary. The fundamental groupoid functor induces a bijection Cov(E, E′) −→ Cov(Π(E), Π(E′)). Just as for groupoids, we can recast the theorem in terms of ﬁbers. In fact, via the previous corollary, the following result is immediate from its analogue for groupoids. Theorem. Let p : E −→ B and p′ : E′ −→ B be coverings, choose a basepoint b ∈ B, and let G = π1(B, b). If g : E −→ E′ is a map of coverings, then g restricts to a map Fb −→ F ′ b of G-sets, and restriction to ﬁbers speciﬁes a bijection between Cov(E, E′) and the set of G-maps Fb −→ F ′ b. Definition. Let Aut(E) ⊂ Cov(E, E) denote the group of automorphisms of a cover E. Again, just as for groupoids, it is possible to have a map of covers g : E −→ E such that g is not an isomorphism. Cor
ollary. Let p : E −→ B be a covering and choose b ∈ B and e ∈ Fb. Write G = π1(B, b) and H = p∗(π1(E, e)). Then Aut(E) is isomorphic to the group of automorphisms of the G-set Fb and therefore to the group W H. If p is regular, then Aut(E) ∼= G/H. If p is universal, then Aut(E) ∼= G. 30 COVERING SPACES 8. The construction of coverings of spaces We have now given an algebraic classiﬁcation of all possible covers of B: there is at most one isomorphism class of covers corresponding to each conjugacy class of subgroups of π1(B, b). We show here that all of these possibilities are actually realized. We shall ﬁrst construct universal covers and then show that the existence of universal covers implies the existence of all other possible covers. Again, while it suﬃces to think in terms of locally contractible spaces, appropriate generality demands a weaker hypothesis. We say that a space B is semi-locally simply connected if every point b ∈ B has a neighborhood U such that π1(U, b) −→ π1(B, b) is the trivial homomorphism. Theorem. If B is connected, locally path connected, and semi-locally simply connected, then B has a universal cover. Proof. Fix a basepoint b ∈ B. We turn the properties of paths that must hold in a universal cover into a construction. Deﬁne E to be the set of equivalence classes of paths f in B that start at b and deﬁne p : E −→ B by p[f ] = f (1). Of course, the equivalence relation is homotopy through paths from b to a given endpoint, so that p is well deﬁned. Thus, as a set, E is just StΠ(B)(b), exactly as in the construction of the universal cover of Π(B). The topology of B has a basis consisting of path connected open subsets U such that π1(U, u) −→ π1(B, u) is trivial for all u ∈ U. Since every loop in U is equivalent in
B to the trivial loop, any two paths u −→ u′ in such a U are equivalent in B. We shall topologize E so that p is a cover with these U as fundamental neighborhoods. For a path f in B that starts at b and ends in U, deﬁne a subset U [f ] of E by U [f ] = {[g] \| [g] = [c · f ] for some c : I −→ U }. The set of all such U [f ] is a basis for a topology on E since if U [f ] and U ′[f ′] are two such sets and [g] is in their intersection, then W [g] ⊂ U [f ] ∩ U ′[f ′] for any open set W of B such that p[g] ∈ W ⊂ U ∩ U ′. For u ∈ U, there is a unique [g] in each U [f ] such that p[g] = u. Thus p maps U [f ] homeomorphically onto U and, if we choose a basepoint u in U, then p−1(U ) is the disjoint union of those U [f ] such that f ends at u. It only remains to show that E is connected, locally path connected, and simply connected, and the second of these is clear. Give E the basepoint e = [cb]. For [f ] ∈ E, deﬁne a path ˜f : I −→ E by ˜f (s) = [fs], where fs(t) = f (st); ˜f is continuous since each ˜f −1(U [g]) is open by the deﬁnition of U [g] and the continuity of f. Since ˜f starts at e and ends at [f ], E is path connected. Since fs(1) = f (s), p ◦ ˜f = f. Thus, by deﬁnition, T [f ](e) = [ ˜f (1)] = [f ]. Restricting attention to loops f, we see that T [f ](e) = e if and only if [f ] = e as an element of π1(B, b). Thus the action of π1(B, b) on
Fb is free and the isotropy group p∗(π1(E, e)) is trivial. We shall construct general covers by passage to orbit spaces from the universal cover, and we need some preliminaries. 8. THE CONSTRUCTION OF COVERINGS OF SPACES 31 Definition. A G-space X is a space X that is a G-set with continuous action map G × X −→ X. Deﬁne the orbit space X/G to be the set of orbits {Gx\|x ∈ X} with its topology as a quotient space of X. The deﬁnition makes sense for general topological groups G. However, our interest here is in discrete groups G, for which the continuity condition just means that action by each element of G is a homeomorphism. The functoriality on O(G) of our construction of general covers will be immediate from the following observation. Lemma. Let X be a G-space. Then passage to orbit spaces deﬁnes a functor X/(−) : O(G) −→ U. Proof. The functor sends G/H to X/H and sends a map α : G/H −→ G/K to the map X/H −→ X/K that sends the coset Hx to the coset Kγ−1x, where α is given by the subconjugacy relation γ−1Hγ ⊂ K. The starting point of the construction of general covers is the following descrip- tion of regular covers and in particular of the universal cover. Proposition. Let p : E −→ B be a cover such that Aut(E) acts transitively on Fb. Then the cover p is regular and E/ Aut(E) is homeomorphic to B. Proof. For any points e, e′ ∈ Fb, there exists g ∈ Aut(E) such that g(e) = e′ and thus p∗(π1(E, e)) = p∗(π1(E, e′)). Therefore all conjugates of p∗(π1(E, e)) are equal to p∗(π1(E, e)) and p∗(π1(E, e)) is a normal subgroup of π1(B, b). The homeomorphism is clear since, locally, both p
and passage to orbits identify the diﬀerent components of the inverse images of fundamental neighborhoods. Theorem. Choose a basepoint b ∈ B and let G = π1(B, b). There is a functor E(−) : O(G) −→ Cov(B) that is an equivalence of categories. For each subgroup H of G, the covering p : E(G/H) −→ B has a canonical basepoint e in its ﬁber over b such that p∗(π1(E(G/H), e)) = H. Moreover, Fb ∼= G/H as a G-set and, for a G-map α : G/H −→ G/K in O(G), the restriction of E(α) : E(G/H) −→ E(G/K) to ﬁbers over b coincides with α. Proof. Let p : E −→ B be the universal cover of B and ﬁx e ∈ E such that p(e) = b. We have the isomorphism Aut(E) ∼= π1(B, b) given by mapping g : E −→ E to the path class [f ] ∈ G such that g(e) = T (f )(e), where T (f )(e) is the endpoint of the path ˜f that starts at e and lifts f. We identify subgroups of G with subgroups of Aut(E) via this isomorphism. We deﬁne E(G/H) to be the orbit space E/H and we let q : E −→ E/H be the quotient map. We may identify B with E/Aut(E), and inclusion of orbits speciﬁes a map p′ : E/H −→ B such that p′ ◦ q = p : E −→ B. If U ⊂ B is a fundamental neighborhood for p and V is a component of p−1(U ) ⊂ E, then p−1(U ) = g∈Aut(E) gV. Passage to orbits over H simply identiﬁes some of these components, and we see ` immediately that both p′ and q are covers. If e′ = q(e), then p′ ∗ maps π1(E/H
, e′) isomorphically onto H since, by construction, the isotropy group of e′ under the action of π1(B, b) is precisely H. Rewriting p′ = p and e′ = e generically, this gives 32 COVERING SPACES the stated properties of the coverings E(G/H). The functoriality on O(G) follows directly from the previous lemma. The functor E(−) is an equivalence of categories since the results of the previous section imply that it maps the morphism set O(G)(G/H, G/K) bijectively onto the morphism set Cov(E(G/H), E(G/K)) and that every covering of B is isomorphic to one of the coverings E(G/H). The classiﬁcation theorems for coverings of spaces and coverings of groupoids are nicely related. In fact, the following diagram of functors commutes up to natural isomorphism: O(π1(B, b)) E(−) xqqqqqqqqqq E (−) 'OOOOOOOOOOO Cov(B) Π / Cov(Π(B)). Corollary. Π : Cov(B) −→ Cov(Π(B)) is an equivalence of categories. PROBLEMS In the following two problems, let G be a connected and locally path connected topological group with identity element e, let p : H −→ G be a covering, and ﬁx f ∈ H such that p(f ) = e. Prove the following. (Hint: Make repeated use of the fundamental theorem for covering spaces.) (1) (a) H has a unique continuous product H × H −→ H with identity element f such that p is a homomorphism. (b) H is a topological group under this product, and H is Abelian if G is. (2) (a) The kernel K of p is a discrete normal subgroup of H. (b) In general, any discrete normal subgroup K of a connected topolog- ical group H is contained in the center of H. (c) For k ∈ K, deﬁne t(k) : H −→ H by t(k)(h) = kh. Then k −→ t(k) speciﬁes an
isomorphism between K and the group Aut(H). Let X and Y be connected, locally path connected, and Hausdorﬀ. A map f : X −→ Y is said to be a local homeomorphism if every point of X has an open neighborhood that maps homeomorphically onto an open set in Y. 3. Give an example of a surjective local homeomorphism that is not a cov- ering. 4. * Let f : X −→ Y be a local homeomorphism, where X is compact. Prove that f is a (surjective!) covering with ﬁnite ﬁbers. Let X be a G-space, where G is a (discrete) group. For a subgroup H of G, deﬁne X H = {x\|hx = x for all h ∈ H} ⊂ X; X H is the H-ﬁxed point subspace of X. Topologize the set of functions G/H −→ X as the product of copies of X indexed on the elements of G/H, and give the set of G-maps G/H −→ X the subspace topology. 5. Show that the space of G-maps G/H −→ X is naturally homeomorphic to X H. In particular, O(G/H, G/K) ∼= (G/K)H. x'/ 8. THE CONSTRUCTION OF COVERINGS OF SPACES 33 6. Let X be a G-space. Show that passage to ﬁxed point spaces, G/H 7−→ X H, is the object function of a contravariant functor X (−) : O(G) −→ U. CHAPTER 4 Graphs We deﬁne graphs, describe their homotopy types, and use them to show that a subgroup of a free group is free and that any group is the fundamental group of some space. 1. The deﬁnition of graphs We give the deﬁnition in a form that will later make it clear that a graph is exactly a one-dimensional CW complex. Note that the zero-sphere S0 is a discrete space with two points. We think of S0 as the boundary of I and so label the points 0 and 1. Definition. A graph X is a space that is obtained from a
(discrete) set X 0 of points, called vertices, and a (discrete) set J of functions j : S0 −→ X 0 as the quotient space of the disjoint union X 0∐(J ×I) that is obtained by identifying (j, 0) with j(0) and (j, 1) with j(1). The images of the intervals {j} × I are called edges. A graph is ﬁnite if it has only ﬁnitely many vertices and edges or, equivalently, if it is a compact space. A graph is locally ﬁnite if each vertex is a boundary point of only ﬁnitely many edges or, equivalently, if it is a locally compact space. A subgraph A of X is a graph A ⊂ X with A0 ⊂ X 0. That is, A is the union of some of the vertices and edges of X. Observe that a graph is a locally contractible space: any neighborhood of any point contains a contractible neighborhood of that point. Therefore a connected graph has all possible covers. 2. Edge paths and trees An oriented edge k : I −→ X in a graph X is the traversal of an edge in either the forward or backward direction. An edge path is a ﬁnite composite of oriented edges kn with kn+1(0) = kn(1). Such a path is reduced if it is never the case that kn+1 is kn with the opposite orientation. An edge path is closed if it starts and ends at the same vertex (and is thus a loop). Definition. A tree is a connected graph with no closed reduced edge paths. A subspace A of a space X is a deformation retract if there is a homotopy h : X × I −→ X such that h(x, 0) = x, h(a, t) = a, and h(x, 1) ∈ A for all x ∈ X, a ∈ A, and t ∈ I. Such a homotopy is called a deformation of X onto A. Lemma. Any vertex v0 of a tree T is a deformation retract of T. Proof. This is true by induction on the number of edges when T is ﬁnite since we can prune the last branch. For the general case, observe that each vertex v lies in some
ﬁnite connected subtree T (v) that also contains v0. Choose an edge path 35 36 GRAPHS a(v) : I −→ T (v) connecting v to v0. For an edge j from v to v′, T (v) ∪ T (v′) ∪ j is a ﬁnite connected subtree of T. On the square j × I, we deﬁne h : j × I −→ T (v) ∪ T (v′) ∪ j by requiring h = a(v) on {v} × I, h = a(v′) on {v′} × I, h(x, 0) = x and h(x, 1) = v0 for all x ∈ j, and extending over the interior of the square by use of the simple connectivity of T (v) ∪ T (v′) ∪ j. As j runs over the edges, these homotopies glue together to specify a deformation h of T onto v0. A subtree of a graph X is maximal if it is contained in no strictly larger tree. Lemma. If a tree T is a subgraph of a graph X, then T is contained in a If X is connected, then a tree in X is maximal if and only if it maximal tree. contains all vertices of X. Proof. Since the union of an increasing family of trees in X is a tree, the ﬁrst statement holds by Zorn’s lemma. If X is connected, then a tree containing all vertices is maximal since addition of an edge would result in a subgraph that contains a closed reduced edge path and, conversely, a tree T that does not contain all vertices is not maximal since a vertex not in T can be connected to a vertex in T by a reduced edge path consisting of edges not in T. 3. The homotopy types of graphs Graph theory is a branch of combinatorics. The homotopy theory of graphs is essentially trivial, by the following result. Theorem. Let X be a connected graph with maximal tree T. Then the quotient space X/T is the wedge of one circle for each edge of X not in T, and the quotient map q : X −→ X/T is a homotopy equivalence. Proof. The ﬁrst clause is evident. The
second is a direct consequence of a later result (that will be left as an exercise): for a suitably nice inclusion, called a “coﬁbration,” of a contractible space T in a space X, the quotient map X −→ X/T is a homotopy equivalence. A direct proof in the present situation is longer and uglier. With the notation in our proof that a vertex v0 is a deformation retract of T via a deformation h, deﬁne a loop bj = a(v′) · j · a(v)−1 at v0 for each edge j : v −→ v′ not in T. The bj together specify a map b from X/T ∼= j S1 to X. The composite q ◦ b : X/T −→ X/T is the wedge over j of copies of the loop v0 : S1 −→ S1 and is therefore homotopic to the identity. To prove that cv0 · id · c−1 b ◦ q is homotopic to the identity, observe that h is a homotopy id ≃ b ◦ q on T. This homotopy extends to a homotopy H : id ≃ b ◦ q on all of X. To see this, we need only construct H on j × I for an edge j : v −→ v′ not in T. The following schematic description of the prescribed behavior on the boundary of the square makes it clear W 5. APPLICATIONS TO GROUPS 37 that H exists: −1 a(v) j a(v) cv j a(v ′ ) ****************** cv′ a(v ′ ) 4. Covers of graphs and Euler characteristics Deﬁne the Euler characteristic χ(X) of a ﬁnite graph X to be V − E, where V is the number of vertices of X and E is the number of edges. By induction on the number of edges, χ(T ) = 1 for any ﬁnite tree. The determination of the homotopy types of graphs has the following immediate implication. Corollary. If X is a connected graph, then π1(X) is a free group with one generator for each edge not in a given maximal tree. If X is ﬁnite, then �
�1(X) is free on 1 − χ(X) generators; in particular, χ(X) ≤ 1, with equality if and only if X is a tree. Theorem. If B is a connected graph with vertex set B0 and p : E −→ B is a covering, then E is a connected graph with vertex set E0 = p−1(B0) and with one edge for each edge j of B and point e ∈ Fj(0). Therefore, if B is ﬁnite and p is a ﬁnite cover whose ﬁbers have cardinality n, then E is ﬁnite and χ(E) = nχ(B). Proof. Regard an edge j of B as a path I −→ B and let k(e) : I −→ E be the unique path such that p ◦ k = j and k(e)(0) = e, where e ∈ Fj(0). We claim that E is a graph with E0 as vertex set and the k(e) as edges. An easy path lifting argument shows that each point of E − E0 is an interior point of exactly one edge, hence we have a continuous bijection from the graph E0 ∐ (K × I)/(∼) to E, where K is the evident set of “attaching maps” S0 −→ E0 for the speciﬁed edges. This map is a homeomorphism since it is a local homeomorphism over B. 5. Applications to groups The following purely algebraic result is most simply proved by topology. Theorem. A subgroup H of a free group G is free. If G is free on k generators and H has ﬁnite index n in G, then H is free on 1 − n + nk generators. Proof. Realize G as π1(B), where B is the wedge of one circle for each generator of G in a given free basis. Construct a covering p : E −→ B such that p∗(π1(E)) = H. Since E is a graph, H must be free. If G has k generators, then If [G : H] = n, then Fb has cardinality n and χ(E) = nχ(B). χ(B) = 1 − k. Therefore 1
− χ(E) = 1 − n + nk. We can extend the idea to realize any group as the fundamental group of some connected space. Theorem. For any group G, there is a connected space X such that π1(X) is isomorphic to G. 38 GRAPHS Proof. We may write G = F/N for some free group F and normal subgroup N. As above, we may realize the inclusion of N in F by passage to fundamental groups from a cover p : E −→ B. Deﬁne the (unreduced) cone on E to be CE = (E × I)/(E × {1}) and deﬁne X = B ∪p CE/(∼), where (e, 0) ∼ p(e). Let U and V be the images in X of B ∐ (E × [0, 3/4)) and E × (1/4, 1], respectively, and choose a basepoint in E × {1/2}. Since U and U ∩ V are homotopy equivalent to B and E via evident deformations and V is contractible, a consequence of the van Kampen theorem gives the conclusion. The space X constructed in the proof is called the “homotopy coﬁber” of the map p. It is an important general construction to which we shall return shortly. PROBLEMS (1) Let F be a free group on two generators a and b. How many subgroups of F have index 2? Specify generators for each of these subgroups. (2) Prove that a non-trivial normal subgroup N with inﬁnite index in a free group F cannot be ﬁnitely generated. (3) * Essay: Describe a necessary and suﬃcient condition for a graph to be embeddable in the plane. CHAPTER 5 Compactly generated spaces We brieﬂy describe the category of spaces in which algebraic topologists customarily work. The ordinary category of spaces allows pathology that obstructs a clean development of the foundations. The homotopy and homology groups of spaces are supported on compact subspaces, and it turns out that if one assumes a separation property that is a little weaker than the Hausdorﬀ property, then one can reﬁne the point-set topology of spaces to
eliminate such pathology without changing these invariants. We shall leave the proofs to the reader, but the wise reader will simply take our word for it, at least on a ﬁrst reading: we do not want to overemphasize this material, the importance of which can only become apparent in retrospect. 1. The deﬁnition of compactly generated spaces We shall understand compact spaces to be both compact and Hausdorﬀ, following Bourbaki. A space X is said to be “weak Hausdorﬀ” if g(K) is closed in X for every map g : K −→ X from a compact space K into X. When this holds, the image g(K) is Hausdorﬀ and is therefore a compact subspace of X. This separation property lies between T1 (points are closed) and Hausdorﬀ, but it is not much weaker than the latter. A subspace A of X is said to be “compactly closed” if g−1(A) is closed in K for any map g : K −→ X from a compact space K into X. When X is weak Hausdorﬀ, this holds if and only if the intersection of A with each compact subset of X is closed. A space X is a “k-space” if every compactly closed subspace is closed. A space X is “compactly generated” if it is a weak Hausdorﬀ k-space. For example, any locally compact space and any weak Hausdorﬀ space that satisﬁes the ﬁrst axiom of countability (every point has a countable neighborhood basis) is compactly generated. We have expressed the deﬁnition in a form that should make the following statement clear. Lemma. If X is a compactly generated space and Y is any space, then a function f : X −→ Y is continuous if and only if its restriction to each compact subspace K of X is continuous. We can make a space X into a k-space by giving it a new topology in which a space is closed if and only if it is compactly closed in the original topology. We call the resulting space kX. Clearly the identity function kX −→ X is continuous. If X is weak Hausdorﬀ, then
so is kX, hence kX is compactly generated. Moreover, X and kX then have exactly the same compact subsets. Write X ×c Y for the product of X and Y with its usual topology and write X × Y = k(X ×c Y ). If X and Y are weak Hausdorﬀ, then X × Y = kX × kY. If X is locally compact and Y is compactly generated, then X × Y = X ×c Y. 39 40 COMPACTLY GENERATED SPACES By deﬁnition, a space X is Hausdorﬀ if the diagonal subspace ∆X = {(x, x)} is closed in X ×c X. The weak Hausdorﬀ property admits a similar characterization. Lemma. If X is a k-space, then X is weak Hausdorﬀ if and only if ∆X is closed in X × X. 2. The category of compactly generated spaces One major source of point-set level pathology can be passage to quotient spaces. Use of compactly generated topologies alleviates this. Proposition. If X is compactly generated and π : X −→ Y is a quotient map, then Y is compactly generated if and only if (π × π)−1(∆Y ) is closed in X × X. The interpretation is that a quotient space of a compactly generated space by a “closed equivalence relation” is compactly generated. We are particularly interested in the following consequence. Proposition. If X and Y are compactly generated spaces, A is a closed subspace of X, and f : A −→ Y is any continuous map, then the pushout Y ∪f X is compactly generated. Another source of pathology is passage to colimits over sequences of maps Xi −→ Xi+1. When the given maps are inclusions, the colimit is the union of the sets Xi with the “topology of the union;” a set is closed if and only if its intersection with each Xi is closed. Proposition. If {Xi} is a sequence of compactly generated spaces and inclu- sions Xi −→ Xi+1 with closed images, then colim Xi is compactly generated. We now adopt a more categorical point of view. We redeﬁne U to be the category of compactly
generated spaces and continuous maps, and we redeﬁne T to be its subcategory of based spaces and based maps. Let wU be the category of weak Hausdorﬀ spaces. We have the functor k : wU −→ U, and we have the forgetful functor j : U −→ wU, which embeds U as a full subcategory of wU. Clearly U (X, kY ) ∼= wU (jX, Y ) for X ∈ U and Y ∈ wU since the identity map kY −→ Y is continuous and continuity of maps deﬁned on compactly generated spaces is compactly determined. Thus k is right adjoint to j. We can construct colimits and limits of spaces by performing these constructions on sets: they inherit topologies that give them the universal properties of colimits and limits in the classical category of spaces. Limits of weak Hausdorﬀ spaces are weak Hausdorﬀ, but limits of k-spaces need not be k-spaces. We construct limits of compactly generated spaces by applying the functor k to their limits as spaces. It is a categorical fact that functors which are right adjoints preserve limits, so this does give categorical limits in U. This is how we deﬁned X × Y, for example. Point-set level colimits of weak Hausdorﬀ spaces need not be weak Hausdorﬀ. However, if a point-set level colimit of compactly generated spaces is weak Hausdorﬀ, then it is a k-space and therefore compactly generated. We shall only be interested in colimits in those cases where this holds. The propositions above give examples. In such cases, these constructions give categorical colimits in U. 2. THE CATEGORY OF COMPACTLY GENERATED SPACES 41 From here on, we agree that all given spaces are to be compactly generated, and we agree to redeﬁne any construction on spaces by applying the functor k to it. For example, for spaces X and Y in U, we understand the function space Map(X, Y ) = Y X to mean the set of continuous maps from X to Y with the kiﬁcation of the standard compact-open topology; the latter topology has as basis the
ﬁnite intersections of the subsets of the form {f \|f (K) ⊂ U } for some compact subset K of X and open subset U of Y. This leads to the following adjointness homeomorphism, which holds without restriction when we work in the category of compactly generated spaces. Proposition. For spaces X, Y, and Z in U, the canonical bijection is a homeomorphism. Z (X×Y ) ∼= (Z Y )X Observe in particular that a homotopy X × I −→ Y can equally well be viewed as a map X −→ Y I. These adjoint, or “dual,” points of view will play an important role in the next two chapters. (1) PROBLEMS (a) Any subspace of a weak Hausdorﬀ space is weak Hausdorﬀ. (b) Any closed subspace of a k-space is a k-space. (c) An open subset U of a compactly generated space X is compactly generated if each point has an open neighborhood in X with closure contained in U. (2) * A Tychonoﬀ (or completely regular) space X is a T1-space (points are closed) such that for each point x ∈ X and each closed subset A such that x /∈ A, there is a function f : X −→ I such that f (x) = 0 and f (a) = 1 if a ∈ A. Prove the following (e.g., Kelley, General Topology). (a) A space is Tychonoﬀ if and only if it can be embedded in a cube (a product of copies of I). (b) There are Tychonoﬀ spaces that are not k-spaces, but every cube is a compact Hausdorﬀ space. (3) Brief essay: In view of Problems 1 and 2, what should we mean by a “subspace” of a compactly generated space. (We do not want to restrict the allowable set of subsets.) CHAPTER 6 Coﬁbrations Exact sequences that feature in the study of homotopy, homology, and cohomology groups all can be derived homotopically from the theory of coﬁber and ﬁber sequences that we present in this and the
following two chapters. Abstractions of these ideas are at the heart of modern axiomatic treatments of homotopical algebra and of the foundations of algebraic K-theory. The theories of coﬁber and ﬁber sequences illustrate an important, but informal, duality theory, known as Eckmann-Hilton duality. It is based on the adjunction between Cartesian products and function spaces. Our standing hypothesis that all spaces in sight are compactly generated allows the theory to be developed without further restrictions on the given spaces. We discuss “coﬁbrations” here and the “dual” notion of “ﬁbrations” in the next chapter. 1. The deﬁnition of coﬁbrations Definition. A map i : A −→ X is a coﬁbration if it satisﬁes the homotopy extension property (HEP). This means that if h ◦ i0 = f ◦ i in the diagram A i X i0 i0 A × I h {xxxxxxxxx cF F ˜h F F i×id F X × I, Y f?~~~~~~~~ then there exists ˜h that makes the diagram commute. Here i0(x) = (x, 0). We do not require ˜h to be unique, and it usually isn’t. Using our alternative way of writing homotopies, we see that the “test diagram” displayed in the deﬁnition can be rewritten in the equivalent form A i } X h / / Y I >} p0 } ˜h } / Y, f where p0(ξ) = ξ(0). Pushouts of coﬁbrations are coﬁbrations, in the sense of the following result. We generally write B ∪g X for the pushout of a given coﬁbration i : A −→ X and a map g : A −→ B. 43 / / { / /? c > / 44 COFIBRATIONS Lemma. If i : A −→ X is a coﬁbration and g : A −→ B is any map, then the induced map B −→ B ∪g X is a coﬁbration. Proof. Notice that (B ∪g X)×I ∼=
(B ×I)∪g×id (X ×I) and consider a typical test diagram for the HEP. The proof is a formal chase of the following diagram: A g #GGGGGGGGGG B i pushout ;wwwwwwwww X Y f ;wwwwwwwww B ∪g X A × I g×id wooooooooooo / B × I h ˜h pushout xrrrrrrrrrrr eK gNNNNNNNNNNN / (B ∪g X) × I ¯h i×id / X × I. i0 i0 We ﬁrst use that A −→ X is a coﬁbration to obtain a homotopy ¯h : X ×I −→ Y and then use the right-hand pushout to see that ¯h and h induce the required homotopy ˜h. 2. Mapping cylinders and coﬁbrations Although the HEP is expressed in terms of general test diagrams, there is a certain universal test diagram. Namely, we can let Y in our original test diagram be the “mapping cylinder” M i ≡ X ∪i (A × I), which is the pushout of i and i0. Indeed, suppose that we can construct a map r that makes the following diagram commute: A i X i0 M i =\|\|\|\|\|\|\|\| i0 A × I {vvvvvvvvv cH H r H H i×id H X × I. By the universal property of pushouts, the given maps f and h in our original test diagram induce a map M i −→ Y, and its composite with r gives a homotopy ˜h that makes the test diagram commute. A map r that makes the previous diagram commute satisﬁes r ◦ j = id, where j : M i −→ X × I is the map that restricts to i0 on X and to i × id on A × I. As a matter of point-set topology, left as an exercise, it follows that a coﬁbration is an inclusion with closed image. A CRITERION FOR A MAP TO BE A COFIBRATION 45 3. Replacing maps by coﬁbrations We can use the mapping cylinder construction to decompose an arbitrary map f : X −→ Y as the composite of a coﬁbration and a homotopy equivalence
. That is, up to homotopy, any map can be replaced by a coﬁbration. To see this, recall that M f = Y ∪f (X × I) and observe that f coincides with the composite X j −→ M f r−→ Y, where j(x) = (x, 1) and where r(y) = y on Y and r(x, s) = f (x) on X × I. If i : Y −→ M f is the inclusion, then r ◦ i = id and id ≃ i ◦ r. In fact, we can deﬁne a deformation h : M f × I −→ M f of M f onto i(Y ) by setting h(y, t) = y and h((x, s), t) = (x, (1 − t)s). It is not hard to check directly that j : X −→ M f satisﬁes the HEP, and this will also follow from the general criterion for a map to be a coﬁbration to which we turn next. 4. A criterion for a map to be a coﬁbration We want a criterion that allows us to recognize coﬁbrations when we see them. We shall often consider pairs (X, A) consisting of a space X and a subspace A. Coﬁbration pairs will be those pairs that “behave homologically” just like the associated quotient spaces X/A. Definition. A pair (X, A) is an NDR-pair (= neighborhood deformation retract pair) if there is a map u : X −→ I such that u−1(0) = A and a homotopy h : X × I −→ X such that h0 = id, h(a, t) = a for a ∈ A and t ∈ I, and h(x, 1) ∈ A if u(x) < 1; (X, A) is a DR-pair if u(x) < 1 for all x ∈ X, in which case A is a deformation retract of X. Lemma. If (h, u) and (j, v) represent (X, A) and (Y, B) as NDR-pairs, then (k, w) represents the “product pair�
� (X × Y, X × B ∪ A × Y ) as an NDR-pair, where w(x, y) = min(u(x), v(y)) and k(x, y, t) = (h(x, t), j(y, tu(x)/v(y))) if v(y) ≥ u(x) (h(x, tv(y)/u(x)), j(y, t)) if u(x) ≥ v(y). ( If (X, A) or (Y, B) is a DR-pair, then so is (X × Y, X × B ∪ A × Y ). Proof. If v(y) = 0 and v(y) ≥ u(x), then u(x) = 0 and both y ∈ B and x ∈ A; therefore we can and must understand k(x, y, t) to be (x, y). It is easy to check from this and the symmetric observation that k is a well deﬁned continuous homotopy as desired. Theorem. Let A be a closed subspace of X. Then the following are equivalent: (i) (X, A) is an NDR-pair. (ii) (X × I, X × {0} ∪ A × I) is a DR-pair. (iii) X × {0} ∪ A × I is a retract of X × I. (iv) The inclusion i : A −→ X is a coﬁbration. Proof. The lemma gives that (i) implies (ii), (ii) trivially implies (iii), and we have already seen that (iii) and (iv) are equivalent. Assume given a retraction 46 COFIBRATIONS r : X × I −→ X × {0} ∪ A × I. Let π1 : X × I −→ X and π2 : X × I −→ I be the projections and deﬁne u : X −→ I by u(x) = sup{t − π2r(x, t)\|t ∈ I} and h : X × I −→ X by h(x, t) = π1r(x, t). Then (h, u) represents (X, A) as an NDR-pair. Here u−1
(0) = A since u(x) = 0 implies that r(x, t) ∈ A × I for t > 0 and thus also for t = 0 since A × I is closed in X × I. 5. Coﬁber homotopy equivalence It is often important to work in the category of spaces under a given space A, and we shall later need a basic result about homotopy equivalences in this category. We shall also need a generalization concerning homotopy equivalences of pairs. The reader is warned that the results of this section, although easy enough to understand, have fairly lengthy and unilluminating proofs. A space under A is a map i : A −→ X. A map of spaces under A is a commutative diagram i ~~~~~~~~ X A f j @@@@@@@ / Y A homotopy between maps under A is a homotopy that at each time t is a map under A. We then write h : f ≃ f ′ rel A and have h(i(a), t) = j(a) for all a ∈ A and t ∈ I. There results a notion of a homotopy equivalence under A. Such an equivalence is called a “coﬁber homotopy equivalence.” The name is suggested by the following result, whose proof illustrates a more substantial use of the HEP than we have seen before. Proposition. Let i : A −→ X and j : A −→ Y be coﬁbrations and let f : X −→ Y be a map such that f ◦i = j. Suppose that f is a homotopy equivalence. Then f is a coﬁber homotopy equivalence. Proof. It suﬃces to ﬁnd a map g : Y −→ X under A and a homotopy g ◦ f ≃ id rel A. Indeed, g will then be a homotopy equivalence, and we can repeat the argument to obtain f ′ : X −→ Y such that f ′ ◦ g ≃ id rel A; it will follow formally that f ′ ≃ f rel A. By hypothesis, there is a map g′′ : Y −→ X that is a homotopy inverse to f. Since g′′ ◦ f ≃ id, g′′ ◦ j ≃ i.
Since j satisﬁes the HEP, it follows directly that g′′ is homotopic to a map g′ such that g′ ◦ j = i. It suﬃces to prove that g′ ◦ f : X −→ X has a left homotopy inverse e : X −→ X under A, since g = e ◦ g′ will then satisfy g ◦ f ≃ id rel A. Replacing our original map f with g′ ◦ f, we see that it suﬃces to obtain a left homotopy inverse under A to a map f : X −→ X such that f ◦ i = i and f ≃ id. Choose a homotopy h : f ≃ id. Since h0 ◦ i = f ◦ i = i and h1 = id, we can apply the HEP to h ◦ (i × id) : A × I −→ X and the identity map of X to obtain a homotopy k : id ≃ k1 ≡ e such that k ◦ (i × id) = h ◦ (i × id). Certainly e ◦ i = i. Now apply the HEP to the following ~ / 5. COFIBER HOMOTOPY EQUIVALENCE 47 diagram ytttttttttt X eJ i×id i×id × id J ;xxxxxxxx. i0 i0 Here J is the homotopy e ◦ f ≃ id speciﬁed by J(x, s) = k(f (x), 1 − 2s) if s ≤ 1/2 if 1/2 ≤ s. h(x, 2s − 1) ( The homotopy between homotopies K is speciﬁed by K(a, s, t) = k(i(a), 1 − 2s(1 − t)) h(i(a), 1 − 2(1 − s)(1 − t)) if s ≤ 1/2 if s ≥ 1/2. ( Traversal of L around the three faces of I × I other than that speciﬁed by J gives a homotopy e ◦ f = J0 = L0,0 ≃ L0,1 ≃ L1,1 ≃ L1,0 = J1 = id rel A. The proposition applies to the following
previously encountered situation. Example. Let i : A −→ X be a coﬁbration. We then have the commutative diagram j ~}}}}}}}} M i A r i @@@@@@@ / X, where j(a) = (a, 1). The obvious homotopy inverse ι : X −→ M i has ι(x) = (x, 0) and is thus very far from being a map under A. The proposition ensures that ι is homotopic to a map under A that is homotopy inverse to r under A. The following generalization asserts that, for inclusions that are coﬁbrations, a pair of homotopy equivalences is a homotopy equivalence of pairs. It is often used implicitly in setting up homology and cohomology theories on pairs of spaces. Proposition. Assume given a commutative diagram A i X d f B j / Y in which i and j are coﬁbrations and d and f are homotopy equivalences. Then (f, d) : (X, A) −→ (Y, B) is a homotopy equivalence of pairs. / / y ; / e ~ / / / / 48 COFIBRATIONS Proof. The statement means that there are homotopy inverses e of d and g of f such that g ◦ j = i ◦ e together with homotopies H : g ◦ f ≃ id and K : f ◦ g ≃ id that extend homotopies h : e ◦ d ≃ id and k : d ◦ e ≃ id. Choose any homotopy inverse e to d, together with homotopies h : e ◦ d ≃ id and ℓ : d ◦ e ≃ id. By HEP for j, there is a homotopy inverse g′ for f such that g′ ◦ j = i ◦ e. Then, by HEP for i, there is a homotopy m of g′ ◦ f such that m ◦ (i × id) = i ◦ h. Let φ = m1. Then φ ◦ i = i and φ is a coﬁber homotopy equivalence by the previous result. Let ψ : X −→ X be a homotopy inverse under i and let
n : ψ ◦ φ ≃ id be a homotopy under i. Deﬁne g = ψ ◦ g′. Clearly g ◦ j = i ◦ e. Using that the pairs (I × I, I × {0}) and (I × I, I × {0} ∪ ∂I × I) are homeomorphic, we can construct a homotopy between homotopies Λ by applying HEP to the diagram (A × I × 0) ∪ (A × ∂I × I) ⊂ / A × I × I i×id i×id Γ ztttttttttt dJJJJJJJJJJ Λ X 6lllllllllllllll γ (X × I × 0) ∪ (X × ∂I × I) ⊂ / X × I × I. γ(x, s, 0) = if s ≤ 1/2 ψ(m(x, 2s)) if s ≥ 1/2, n(x, 2s − 1) γ(x, 0, t) = (g ◦ f )(x) = (ψ ◦ g′ ◦ f )(x), γ(x, 1, t) = x, i(h(a, 2s/(1 + t))) i(a) if 2s ≤ 1 + t if 2s ≥ 1 + t Γ(a, s, t) = Here and while Deﬁne H(x, s) = Λ(x, s, 1). Then H : g ◦ f ≃ id and H ◦ (i × id) = i ◦ h. Application of this argument with d and f replaced by e and g gives a left homotopy inverse f ′ to g and a homotopy L : f ′◦g ≃ id such that f ′◦i = j ◦d and L◦(j ×id) = j ◦ℓ. Adding homotopies by concentrating them on successive fractions of the unit interval and letting the negative of a homotopy be obtained by reversal of direction, deﬁne k = (−ℓ)(de × id) + dh(e × id) + ℓ and Then K : f ◦ g ≃ id and K
◦ (j × id) = j ◦ k. K = (−L)(f g × id) + f ′H(g × id) + L. PROBLEMS (1) Show that a coﬁbration i : A −→ X is an inclusion with closed image. (2) Let i : A −→ X be a coﬁbration, where A is a contractible space. Prove that the quotient map X −→ X/A is a homotopy equivalence. / z 6 / d CHAPTER 7 Fibrations We “dualize” the deﬁnitions and theory of the previous chapter to the study of ﬁbrations, which are “up to homotopy” generalizations of covering spaces. 1. The deﬁnition of ﬁbrations Definition. A surjective map p : E −→ B is a ﬁbration if it satisﬁes the covering homotopy property (CHP). This means that if h ◦ i0 = p ◦ f in the diagram Y i0 x Y × I f x ˜h x x h E <x p B, then there exists ˜h that makes the diagram commute. This notion of a ﬁbration is due to Hurewicz. There is a more general notion of a Serre ﬁbration, in which the test spaces Y are restricted to be cubes I n. Serre ﬁbrations are more appropriate for many purposes, but we shall make no use of them. The test diagram in the deﬁnition can be rewritten in the equivalent form _???????? f E p B p0 Y p0 EI =\| pI \| ˜h \| \|!BBBBBBBB h BI. Here p0(β) = β(0) for β ∈ BI. With this formulation, we can “dualize” the proof that pushouts of coﬁbrations are coﬁbrations to show that pullbacks of ﬁbrations are ﬁbrations. We often write A ×g E for the pullback of a given ﬁbration p : E −→ B and a map g : A −→ B. Lemma. If p : E −→ B is a ﬁ
bration and g : A −→ B is any map, then the induced map A ×g E −→ A is a ﬁbration. 2. Path lifting functions and ﬁbrations Although the CHP is expressed in terms of general test diagrams, there is a certain universal test diagram. Namely, we can let Y in our original test diagram be the “mapping path space” N p ≡ E ×p BI = {(e, β)\|β(0) = p(e)} ⊂ E × BI. 49 / / / / < o o = _! o o 50 FIBRATIONS That is, N p is the pullback of p and p0 in the second form of the test diagram and, with Y = N p, f and h in that diagram are the evident projections. A map s : N p −→ EI such that k ◦ s = id, where k : EI −→ N p has coordinates p0 and pI, is called a path lifting function. Thus s(e, β)(0) = e and p ◦ s(e, β) = β. Given a general test diagram, there results a map g : Y −→ N p determined by f and h, and we can take ˜h = s ◦ g. In general, path lifting functions are not unique. In fact, we have already studied the special kinds of ﬁbrations for which they are unique. Lemma. If p : E −→ B is a covering, then p is a ﬁbration with a unique path lifting function s. Proof. The unique lifts of paths with a given initial point specify s. Fibrations and coﬁbrations are related by the following useful observation. Lemma. If i : A −→ X is a coﬁbration and B is a space, then the induced map is a ﬁbration. p = Bi : BX −→ BA Proof. It is an easy matter to check that we have a homeomorphism BMi = BX×{0}∪A×I ∼= BX ×p (BA)I = N p. If r : X × I −→ M i is a retraction, then Br : N p ∼= BMi −→ BX×I ∼= (BX )I is a path lifting function. 3. Replacing maps
by ﬁbrations We can use the mapping path space construction to decompose an arbitrary map f : X −→ Y as the composite of a homotopy equivalence and a ﬁbration. That is, up to homotopy, any map can be replaced by a ﬁbration. To see this, recall that N f = X ×f Y I and observe that f coincides with the composite X ν−→ N f ρ −→ Y, where ν(x) = (x, cf (x)) and ρ(x, χ) = χ(1). Let π : N f −→ X be the projection. Then π ◦ ν = id and id ≃ ν ◦ π since we can deﬁne a deformation h : N f × I −→ N f of N f onto ν(X) by setting h(x, χ)(t) = (x, χt), where χt(s) = χ((1 − t)s). We check directly that ρ : N f −→ Y satisﬁes the CHP. Consider a test diagram A i0 x A × I g x ˜h x h ;x N f ρ Y. / / / / ; 4. A CRITERION FOR A MAP TO BE A FIBRATION 51 We are given g and h such that h ◦ i0 = ρ ◦ g and must construct ˜h that makes the diagram commute. We write g(a) = (g1(a), g2(a)) and set ˜h(a, t) = (g1(a), j(a, t)), where j(a, t)(s) = if 0 ≤ s ≤ 1/(1 + t) g2(a)(s + st) h(a, s + ts − 1) if 1/(1 + t) ≤ s ≤ 1. ( 4. A criterion for a map to be a ﬁbration Again, we want a criterion that allows us to recognize ﬁbrations when we see them. Here the idea of duality fails, and we instead think of ﬁbrations as generalizations of coverings. When restricted to the spaces U in a well chosen open cover O of the base space B, a covering is homeomorphic to the
projection U × F −→ U, where F is a ﬁxed discrete set. The obvious generalization of this is the notion of a bundle. A map p : E −→ B is a bundle if, when restricted to the spaces U in a well chosen open cover O of B, there are homeomorphisms φ : U × F −→ p−1(U ) such that p ◦ φ = π1, where F is a ﬁxed topological space. We require of a “well chosen” open cover that it be numerable. This means that there are continuous maps λU : B −→ I such that λ−1 U (0, 1] = U and that the cover is locally ﬁnite, in the sense that each b ∈ B has a neighborhood that intersects only ﬁnitely many U ∈ O. Any open cover of a paracompact space has a numerable reﬁnement. With this proviso on the open covers allowed in the deﬁnition of a bundle, the following result shows in particular that every bundle is a ﬁbration. Theorem. Let p : E −→ B be a map and let O be a numerable open cover of B. Then p is a ﬁbration if and only if p : p−1(U ) −→ U is a ﬁbration for every U ∈ O. Proof. Since pullbacks of ﬁbrations are ﬁbrations, necessity is obvious. Thus assume that p\|p−1(U ) is a ﬁbration for each U ∈ O. We shall construct a path lifting function for B by patching together path lifting functions for the p\|p−1(U ), but we ﬁrst set up the scaﬀolding of the patching argument. Choose maps λU : B −→ I U (0, 1] = U. For a ﬁnite ordered subset T = {U1,..., Un} of sets in O, such that λ−1 deﬁne c(T ) = n and deﬁne λT : BI −→ I by λT (β) = inf{(λUi ◦ β)(t)\|(i − 1)/n
≤ t ≤ i/n, 1 ≤ i ≤ n}. Let WT = λ−1 T (0, 1]. Equivalently, WT = {β\|β(t) ∈ Ui if t ∈ [(i − 1)/n, i/n]} ⊂ BI. The set {WT } is an open cover of BI, but it need not be locally ﬁnite. However, {WT \|c(T ) < n} is locally ﬁnite for each ﬁxed n. If c(T ) = n, deﬁne γT : BI −→ I by γT (β) = max{0, λT (β) − n c(S)<n λS(β)}, and deﬁne P VT = {β\|γT (β) > 0} ⊂ WT. Then {VT } is a locally ﬁnite open cover of BI. We choose a total ordering of the set of all ﬁnite ordered subsets T of O. 52 FIBRATIONS With this scaﬀolding in place, choose path lifting functions sU : p−1(U ) ×p U I −→ p−1(U )I for U ∈ O, so that (p ◦ sU )(e, β) = β and sU (e, β)(0) = e. For a given T = {U1,..., Un}, consider paths β ∈ VT. For 0 ≤ u < v ≤ 1, let β[u, v] be the If u ∈ [(i − 1)/n, i/n] and v ∈ [(j − 1)/n, j/n], where restriction of β to [u, v]. 0 ≤ i ≤ j ≤ n, and if e ∈ p−1(β(u)), deﬁne sT (e, β[u, v]) : [u, v] −→ E to be the path that starts at e and covers β[u, v] that is obtained by applying sUi to lift over [u, i/n] (or over [u, v] if i = j), using sUi+1, starting at the point where the ﬁrst lifted path ends, to lift over [i/
n, (i + 1)/n] and so on inductively, ending with use of sUj to lift over [(j − 1)/n, v]. (Technically, since we are lifting over partial intervals and the sU lift paths deﬁned on I to paths deﬁned on I, this involves a rescaling: we must shrink I linearly onto our subinterval, then apply the relevant part of β, next lift the resulting path, and ﬁnally apply the result to the linear expansion of our subinterval onto I.) For a point (e, β) in N p, deﬁne s(e, β) to be the concatenation of the paths sTj (ej−1, β[uj−1, uj]), 1 ≤ j ≤ q, where the Ti, in order, run through j i=1 γTi(β) for 1 ≤ j ≤ q, the set of all T such that β ∈ VT, where u0 = 0 and uj = and where e0 = e and ej is the endpoint of sTj (ej−1, β[uj−1, uj]) for 1 ≤ j < q. Certainly s(e, β) = e and (p ◦ s)(e, β) = β. It is not hard to check that s is well deﬁned and continuous, hence it is a path lifting function for p. P 5. Fiber homotopy equivalence It is often important to study ﬁbrations over a given base space B, working in the category of spaces over B. A space over B is a map p : E −→ B. A map of spaces over B is a commutative diagram D @@@@@@@ p f B E ~~~~~~~ q A homotopy between maps over B is a homotopy that at each time t is a map over B. There results a notion of a homotopy equivalence over B. Such an equivalence is called a “ﬁber homotopy equivalence.” The name is suggested by the following result, whose proof is precisely dual to the corresponding result for coﬁbrations and is left as an exercise. Proposition. Let p : D −→ B and q : E −→ B be ﬁbrations and let f : D −→ E be
a map such that q ◦ f = p. Suppose that f is a homotopy equivalence. Then f is a ﬁber homotopy equivalence. Example. Let p : E −→ B be a ﬁbration. We then have the commutative diagram E ν / N p???????????????? p p ~\|\|\|\|\|\|\|\| ρ B where ν(e) = (e, cp(e)) and ρ(e, χ) = χ(1). The obvious homotopy inverse π : N p −→ E is not a map over B, but the proposition ensures that it is homotopic to a map over B that is homotopy inverse to ν over B. / / / ~ 6. CHANGE OF FIBER 53 The result generalizes as follows, the proof again being dual to the proof of the corresponding result for coﬁbrations. Proposition. Assume given a commutative diagram D p A f d E q / B in which p and q are ﬁbrations and d and f are homotopy equivalences. Then (f, d) : p −→ q is a homotopy equivalence of ﬁbrations. The statement means that there are homotopy inverses e of d and g of f such that p ◦ g = e ◦ q together with homotopies H : g ◦ f ≃ id and K : f ◦ g ≃ id that cover homotopies h : e ◦ d ≃ id and k : d ◦ e ≃ id. 6. Change of ﬁber Translation of ﬁbers along paths in the base space played a fundamental role in our study of covering spaces. Fibrations admit an up to homotopy version of that theory that well illustrates the use of the CHP and will be used later. Let p : E −→ B be a ﬁbration with ﬁber Fb over b ∈ B and let ib : Fb −→ E be the inclusion. For a path β : I −→ B from b to b′, the CHP gives a lift ˜β in the diagram Fb × {0} ib ˜β 6mmmmmmmm / E p Fb × I π2 / I / B. β At time t, ˜β maps Fb to the �
��ber Fβ(t). In particular, at t = 1, this gives a map τ [β] ≡ [ ˜β1] : Fb −→ Fb′, which we call the translation of ﬁbers along the path class [β]. We claim that, as indicated by our choice of notation, the homotopy class of the map ˜β1 is independent of the choice of β in its path class. Thus suppose that β and β′ are equivalent paths from b to b′, let h : I × I −→ B be a homotopy β ≃ β′ through paths from b to b′, and let ˜β′ : Fb × I −→ E cover β′π2. Observe that if J 2 = I × ∂I ∪ {0} × I ⊂ I 2, then the pairs (I 2, J 2) and (I ×I, I ×{0}) are homeomorphic. Deﬁne f : Fb ×J 2 −→ E to be ˜β on Fb × I × {0}, ˜β′ on Fb × I × {1}, and ib ◦ π1 on Fb × {0} × I. Then another application of the CHP gives a lift ˜h in the diagram Fb × J 2 Fb × I 2 f / E p 6mmmmmmmm ˜h π2 / I 2 / B / / 54 FIBRATIONS Thus ˜h : ˜β ≃ ˜β′ through maps Fb × I −→ E, each of which starts at the inclusion of Fb in E. At time t = 1, this gives a homotopy ˜β1 ≃ ˜β′ 1. Thus τ [β] = [ ˜β1] is a well deﬁned homotopy class of maps Fb −→ Fb′. We think of τ [β] as a map in the homotopy category hU. It is clear that, in the homotopy category, τ [cb] = [id] and τ [γ · β] = τ [γ] ◦ τ [β] if γ(0) = β(1). It follows that τ [β] is an isomorphism with inverse τ [β−1]. This can be stated formally as follows. Theorem. L
ifting of equivalence classes of paths in B to homotopy classes of maps of ﬁbers speciﬁes a functor λ : Π(B) −→ hU. Therefore, if B is path connected, then any two ﬁbers of B are homotopy equivalent. Just as the fundamental group π1(B, b) of the base space of a covering acts on the ﬁber Fb, so the fundamental group π1(B, b) of the base space of a ﬁbration acts “up to homotopy” on the ﬁber, in a sense made precise by the following corollary. For a space X, let π0(X) denote the set of path components of X. The set of homotopy equivalences of X is denoted Aut(X) and is topologized as a subspace of the function space of maps X −→ X. The composite of homotopy equivalences is a homotopy equivalence, and composition deﬁnes a continuous product on Aut(X). With this product, Aut(X) is a “topological monoid,” namely a space with a continuous and associative multiplication with a two-sided identity element, but it is not a group. However, the path components of Aut(X) are the homotopy classes of homotopy equivalences of X, and these do form a group under composition. Corollary. Lifting of equivalence classes of loops speciﬁes a homomorphism π1(B, b) −→ π0(Aut(Fb)). We have the following naturality statement with respect to maps of ﬁbrations. Theorem. Let p and q be ﬁbrations in the commutative diagram D q A g f E p / B. For a path α : I −→ A from a to a′, the following diagram commutes in hU : Fa g Ff (a) τ [α] τ [f ◦α] Fa′ g / / Ff (a′). If, further, h : f ≃ f ′ and H : g ≃ g′ in the commutative diagram D × I H q×id E p A × I h / B, / / / / / / / / then the following diagram
in hU also commutes, where h(a)(t) = h(a, t): 6. CHANGE OF FIBER 55 Fa g \|zzzzzzzz ′ g "FFFFFFFF Ff (a) τ [h(a)] / Ff ′(a). Proof. Let ˜α : Fa × I −→ D lift α and ˜β : Ff (a) × I −→ E lift f ◦ α. Deﬁne j : Fa × J 2 −→ E to be g ◦ ˜α on Fa × I × {0}, ˜β ◦ (g × id) on Fa × I × {1}, and g ◦ π1 on Fa × {0} × I. Deﬁne k : I 2 −→ B to be the constant homotopy which at each time t is f ◦ α. Another application of the CHP gives a lift ˜k in the diagram: Fa × J 2 Fa × I 2 j / E p 6mmmmmmmm ˜k π2 / I 2 / B. k Here ˜k is a homotopy g ◦ ˜α ≃ ˜β ◦ (g × id) through homotopies starting at g ◦ π1 : Fa × I −→ E. This gives the diagram claimed in the ﬁrst statement. For the second statement, deﬁne α : I −→ A × I by α(t) = (a, t), so that h(a) = h ◦ α. Deﬁne ˜α : Fa −→ Fa × I by ˜α(f ) = (f, t). Then ˜α lifts α and τ [α] = [id] : Fa = Fa × {0} −→ Fa × {1} = Fa. We conclude that the second statement is a special case of the ﬁrst. (1) Prove the proposition stated in §5. PROBLEM \| " / / 6 / / CHAPTER 8 Based coﬁber and ﬁber sequences We use coﬁbrations and ﬁbrations in the category T of based spaces to generate two “exact sequences of spaces” from a given map of based spaces. We shall write ∗ generically for the basepoints of
based spaces. Much that we do for coﬁbrations can be done equally well in the unbased context of the previous chapter. However, the dual theory of ﬁbration sequences only makes sense in the based context. 1. Based homotopy classes of maps For based spaces X and Y, we let [X, Y ] denote the set of based homotopy classes of based maps X −→ Y. This set has a natural basepoint, namely the homotopy class of the constant map from X to the basepoint of Y. The appropriate analogue of the Cartesian product in the category of based spaces is the “smash product” X ∧ Y deﬁned by X ∧ Y = X × Y /X ∨ Y. Here X ∨ Y is viewed as the subspace of X × Y consisting of those pairs (x, y) such that either x is the basepoint of X or y is the basepoint of Y. The appropriate based analogue of the function space is the subspace F (X, Y ) of Y X consisting of the based maps, with the constant based map as basepoint. With these deﬁnitions, we have a natural homeomorphism of based spaces F (X ∧ Y, Z) ∼= F (X, F (Y, Z)) for based spaces X and Y. Recall that π0(X) denotes the set of path components of X. When X is based, so is this set, and we sometimes denote it by π0(X, ∗). Observe that [X, Y ] may be identiﬁed with π0(F (X, Y )). 2. Cones, suspensions, paths, loops Let X be a based space. We deﬁne the cone on X to be CX = X ∧ I, where I is given the basepoint 1. That is, CX = X × I/({∗} × I ∪ X × {1}). We view S1 as I/∂I, denote its basepoint by 1, and deﬁne the suspension of X to be ΣX = X ∧ S1. That is, ΣX = X × S1/({∗} × S1 ∪ X × {1}). These are sometimes called the reduced cone and
suspension, to distinguish them from the unreduced constructions, in which the line {∗} × I through the basepoint of X is not identiﬁed to a point. We shall make use of both constructions in our work, but we shall not distinguish them notationally. 57 58 BASED COFIBER AND FIBER SEQUENCES Dually, we deﬁne the path space of X to be P X = F (I, X), where I is given the basepoint 0. Thus the points of P X are the paths in X that start at the basepoint. We deﬁne the loop space of X to be ΩX = F (S1, X). Its points are the loops at the basepoint. We have the adjunction Passing to π0, this gives that F (ΣX, Y ) ∼= F (X, ΩY ). [ΣX, Y ] ∼= [X, ΩY ]. Composition of loops deﬁnes a multiplication on this set. Explicitly, for f, g : ΣX −→ Y, we write (g + f )(x ∧ t) = (g(x) · f (x))(t) = if 0 ≤ t ≤ 1/2 f (x ∧ 2t) g(x ∧ (2t − 1)) if 1/2 ≤ t ≤ 1. ( Lemma. [ΣX, Y ] is a group and [Σ2X, Y ] is an Abelian group. Proof. The ﬁrst statement is proved just as for the fundamental group. For the second, think of maps f, g : Σ2X −→ Y as maps S2 −→ F (X, Y ) and think of S2 as the quotient I 2/∂I 2. Then a homotopy between g + f and f + g can be pictured schematically as follows. Based coﬁbrations The deﬁnition of a coﬁbration has an evident based variant, in which all given and constructed maps in our test diagrams are required to be based. A based map i : A −→ X that is a coﬁbration in the unbased sense is necessarily a coﬁbration in the based sense since the
basepoint of X must lie in A. We say that X is “nondegenerately based,” or “well pointed,” if the inclusion of its basepoint is a coﬁbration in the unbased sense. If A and X are nondegenerately based and i : A −→ X is a based coﬁbration, then i is necessarily an unbased coﬁbration. We refer to based coﬁbrations simply as coﬁbrations in the rest of this chapter. Write Y+ for the union of a space Y and a disjoint basepoint and observe that we can identify X ∧ Y+ with X × Y /{∗} × Y. The space X ∧ I+ is called the reduced cylinder on X, and a based homotopy X × I −→ Y is the same thing as a based map X ∧ I+ −→ Y. We change notations and write M f for the based mapping cylinder Y ∪f (X ∧ I+) of a based map f. As in the unbased case, we conclude that a based map i : A −→ X is a coﬁbration if and only if M i is a retract of X ∧ I+. / / / / / / 4. COFIBER SEQUENCES 59 4. Coﬁber sequences For a based map f : X −→ Y, deﬁne the “homotopy coﬁber” Cf to be Cf = Y ∪f CX = M f /j(X), where j : X −→ M f sends x to (x, 1). As in the unbased case, our original map f is the composite of the coﬁbration j and the evident retraction r : M f −→ Y. Thus Cf is constructed by ﬁrst replacing f by the coﬁbration j and then taking the associated quotient space. Let i : Y −→ Cf be the inclusion. It is a coﬁbration since it is the pushout of f and the coﬁbration X −→ CX that sends x to (x, 0). Let be the quotient map. The sequence π : Cf −→ Cf /Y ∼= ΣX X f −→
Y i−→ Cf π−→ ΣX −Σf −−−→ ΣY −Σi−−−→ ΣCf −Σπ−−−→ Σ2X Σ2f −−→ Σ2Y −→ · · · is called the coﬁber sequence generated by the map f ; here (−Σf )(x ∧ t) = f (x) ∧ (1 − t). These “long exact sequences of based spaces” give rise to long exact sequences of pointed sets, where a sequence S′ f −→ S g −→ S′′ of pointed sets is said to be exact if g(s) = ∗ if and only if s = f (s′) for some s′. Theorem. For any based space Z, the induced sequence · · · −→ [ΣCf, Z] −→ [ΣY, Z] −→ [ΣX, Z] −→ [Cf, Z] −→ [Y, Z] −→ [X, Z] is an exact sequence of pointed sets, or of groups to the left of [ΣX, Z], or of Abelian groups to the left of [Σ2X, Z]. Exactness is clear at the ﬁrst stage, where we are considering the composite of f : X −→ Y and the inclusion i of Y in the coﬁber Cf. To see this, consider the diagram f X / Y g Z. i / Cf = Y ∪f CX xq q q q q q ˜g=g∪h Here h : g ◦ f ≃ c∗, and we view h as a map CX −→ Z. Thus we check exactness by using any given homotopy to extend g over the coﬁber. We emphasize that this applies to any composite pair of maps of the form (f, i), where i is the inclusion of the target of f in the coﬁber of f. We claim that, up to homotopy equivalence, each consecutive pair of maps in our coﬁber sequence is the composite of a map and the inclusion of its target in its coﬁber. This will imply the theorem. We observe that, for any map f, interchange of the cone and suspension coordinate gives
a homeomorphism ΣCf ∼= C(Σf ) / / x 60 BASED COFIBER AND FIBER SEQUENCES such that the following diagram commutes: ΣX Σf / ΣY Σi(f ) / ΣCf Σπ(f ) ∼= Σ2X τ ΣX / ΣY Σf i(Σf ) / C(Σf ) π(Σf ) / Σ2X. Here τ : Σ2X −→ Σ2X is the homeomorphism obtained by interchanging the two suspension coordinates; we shall see later, and leave as an exercise here, that τ is homotopic to − id. We have written i(f ), π(f ), etc., to indicate the maps to which the generic constructions i and π are applied. Using this inductively, we see that we need only verify our claim for the two pairs of maps (i(f ), π(f )) and (π(f ), −Σf ). The following two lemmas will imply the claim in these two cases. More precisely, they will imply the claim directly for the ﬁrst pair and will imply that the second pair is equivalent to a pair of the same form as the ﬁrst pair. Lemma. If i : A −→ X is a coﬁbration, then the quotient map ψ : Ci −→ Ci/CA ∼= X/A is a based homotopy equivalence. Proof. Since i is a coﬁbration, there is a retraction r : X ∧ I+ −→ M i = X ∪i (A ∧ I+). We embed X as X ×{1} in the source and collapse out A×{1} from the target. The resulting composite X −→ Ci maps A to {∗} and so induces a map φ : X/A −→ Ci. The map r restricts to the identity on A ∧ I+, and if we collapse out A ∧ I+ from its source and target, then r becomes a homotopy id ≃ ψ ◦ φ. The map r on X ∧ I+ glues together with the map h : CA ∧ I+ −→ CA speciﬁed by h(
a, s, t) = (a, max(s, t)) to give a homotopy Ci ∧ I+ −→ Ci from the identity to φ ◦ ψ. Lemma. The left triangle commutes and the right triangle commutes up to homotopy in the diagram X f / Y i(f ) / Cf π(f ) −Σf "EEEEEEEE i(i(f )) ΣX ψ / ΣY <xxxxxxxxx π(i(f )) Ci(f ) / · · · Proof. Observe that Ci(f ) is obtained by gluing the cones CX and CY along their bases via the map f : X −→ Y. The left triangle commutes since collapsing out CY from Ci(f ) is the same as collapsing out Y from Cf. A homotopy h : Ci(f ) ∧ I+ −→ ΣY from π to (−Σf ) ◦ ψ is given by and h(x, s, t) = (f (x), t − st) on CX h(y, s, t) = (y, s + t − st) on CY. FIBER SEQUENCES 61 5. Based ﬁbrations Similarly, the deﬁnition of a ﬁbration has an evident based variant, in which all given and constructed maps in our test diagrams are required to be based. A based ﬁbration p : E −→ B is necessarily a ﬁbration in the unbased sense, as we see by restricting to spaces of the form Y+ in test diagrams and noting that Y+ ∧I+ ∼= (Y ×I)+. Less obviously, if p is a based map that is an unbased ﬁbration, then it satisﬁes the based CHP for test diagrams in which Y is nondegenerately based. We refer to based ﬁbrations simply as ﬁbrations in the rest of this chapter. Observe that a based homotopy X ∧I+ −→ Y is the same thing as a based map X −→ F (I+, Y ). Here F (I+, Y ) is the same space as Y I, but given a basepoint determined by the basepoint of Y. Therefore the based version of the mapping path space N f of a
based map f : X −→ Y is the same space as the unbased version, but given a basepoint determined by the given basepoints of X and Y. However, because path spaces are always deﬁned with I having basepoint 0 rather than 1, we ﬁnd it convenient to redeﬁne N f correspondingly, setting N f = {(x, χ)\|χ(1) = f (x)} ⊂ X × Y I. As in the unbased case, we easily check that a based map p : E −→ B is a ﬁbration if and only if there is a based path lifting function s : N p −→ F (I+, E). 6. Fiber sequences For a based map f : X −→ Y, deﬁne the “homotopy ﬁber” F f to be F f = X ×f P Y = {(x, χ)\|f (x) = χ(1)} ⊂ X × P Y. Equivalently, F f is the pullback displayed in the diagram F f π X P Y p1 / Y, f where π(x, χ) = x. As a pullback of a ﬁbration, π is a ﬁbration. If ρ : N f −→ Y is deﬁned by ρ(x, χ) = χ(0), then f = ρ ◦ ν, where ν(x) = (x, cf (x)), and F f is the ﬁber ρ−1(∗). Thus the homotopy ﬁber F f is constructed by ﬁrst replacing f by the ﬁbration ρ and then taking the actual ﬁber. Let ι : ΩY −→ F f be the inclusion speciﬁed by ι(χ) = (∗, χ). The sequence · · · −→ Ω2X Ω2f −−→ Ω2Y −Ωι−−−→ ΩF f −Ωπ−−−→ ΩX −Ωf −−−→ ΩY ι−→ F f π−→ X
f −→ Y is called the ﬁber sequence generated by the map f ; here (−Ωf )(ζ)(t) = (f ◦ ζ)(1 − t) for ζ ∈ ΩX. These “long exact sequences of based spaces” also give rise to long exact se- quences of pointed sets, this time covariantly. / / / 62 BASED COFIBER AND FIBER SEQUENCES Theorem. For any based space Z, the induced sequence · · · −→ [Z, ΩF f ] −→ [Z, ΩX] −→ [Z, ΩY ] −→ [Z, F f ] −→ [Z, X] −→ [Z, Y ] is an exact sequence of pointed sets, or of groups to the left of [Z, ΩY ], or of Abelian groups to the left of [Z, Ω2Y ]. Exactness is clear at the ﬁrst stage. To see this, consider the diagram Z g ˜g=g×h xq f P Y / X π / Y f Here h : c∗ ≃ f ◦ g, and we view h as a map Z −→ P Y. Thus we check exactness by using any given homotopy to lift g to the ﬁber. We claim that, up to homotopy equivalence, each consecutive pair of maps in our ﬁber sequence is the composite of a map and the projection from its ﬁber onto its source. This will imply the theorem. We observe that, for any map f, interchange of coordinates gives a homeomorphism such that the following diagram commutes: ΩF f ∼= F (Ωf ) Ω2Y τ Ωι(f ) ΩF f Ωπ(f ) / ΩX Ωf / ΩY ∼= Ω2Y ι(Ωf ) / F (Ωf ) π(Ωf ) / ΩX Ωf / ΩY. Here τ is obtained by interchanging the loop coordinates and is homotopic to − id. We have written ι(f ), π(f ), etc., to indicate the maps
to which the generic constructions ι and π are applied. Using this inductively, we see that we need only verify our claim for the two pairs of maps (ι(f ), π(f )) and (−Ωf, ι(f )). The following two lemmas will imply the claim in these two cases. More precisely, they will imply the claim directly for the ﬁrst pair and will imply that the second pair is equivalent to a pair of the same form as the ﬁrst pair. The proofs of the lemmas are left as exercises. Lemma. If p : E −→ B is a ﬁbration, then the inclusion φ : p−1(∗) −→ F p speciﬁed by φ(e) = (e, c∗) is a based homotopy equivalence. Lemma. The right triangle commutes and the left triangle commutes up to homotopy in the diagram · · · / ΩX ι(f ) / F f π(f ) / X f / Y. −Ωf #GGGGGGGGG ι(π(f )) ΩY φ F π(f ) ;xxxxxxxx π(π(f )). CONNECTIONS BETWEEN COFIBER AND FIBER SEQUENCES 63 7. Connections between coﬁber and ﬁber sequences It is often useful to know that coﬁber sequences and ﬁber sequences can be connected to one another. The adjunction between loops and suspension has “unit” and “counit” maps η : X −→ ΩΣX and ε : ΣΩX −→ X. Explicitly, η(x)(t) = x ∧ t and ε(χ ∧ t) = χ(t) for x ∈ X, χ ∈ ΩX, and t ∈ S1. For a map f : X −→ Y, we deﬁne η : F f −→ ΩCf and ε : ΣF f −→ Cf by η(x, γ)(t) = ε(x, γ, t) = γ(2t) (x, 2t

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