Split (1)

train · 21.9k rows

text stringlengths 270 6.81k
→Vn,k such that pf = 11, is exactly a set of k − 1 orthonormal tangent vector ﬁelds v1, ···, vk−1 on S n−1 since f assigns to each x ∈ S n−1 an orthonormal k frame (x, v1(x), ···, vk−1(x)). We described a cell structure on Vn,k at the end of §3.D, and we claim that the (n − 1) skeleton of this cell structure is RPn−1/RPn−k−1 if 2k − 1 ≤ n. The cells of Vn,k were products ei1 × ··· × eim with n > i1 > ··· > im ≥ n − k, so the products with a single factor account for all of the (2n − 2k) skeleton, hence they account for all of the (n − 1) skeleton if n − 1 ≤ 2n − 2k, that is, if 2k − 1 ≤ n. The cells that are products with a single factor are the homeomorphic images of cells of RPn−1 under a map RPn−1→SO(n)→SO(n)/SO(n − k) = Vn,k. This map collapses RPn−k−1 to a point, so we get the desired conclusion that RPn−1/RPn−k−1 is the (n − 1) skeleton of Vn,k if 2k − 1 ≤ n. Now suppose we have f : S n−1→Vn,k with pf = 11. In particular, f ∗ is surjective on H n−1(−; Z2). If we deform f to a cellular map, with image in the (n − 1) skeleton, then by the preceding paragraph this will give a map g : S n−1→RPn−1/RPn−k−1 if 2k − 1 ≤ n, and this map will still induce a surjection on H n−1(−; Z2), hence an ≡ 1 mod 2, then by the isomorphism. If the number k happens to be such that earlier formula (∗) the operation n−k k−1 Sqk−1 : H n−k(RPn−1/RPn−k−1; Z2
)→H n−1(RPn−1/RPn−k−1; Z2) will be nonzero, contradicting the existence of the map g since obviously the operation Sqk−1 : H n−k(S n−1; Z2)→H n−1(S n−1; Z2) is zero. In order to guarantee that n−k k−1 ≡ 1 mod 2, write n = 2r (2s + 1) and choose k = 2r + 1. Assume for the moment that s ≥ 1. Then, and in view of the rule for computing binomial coeﬃcients in Z2, this is nonzero since the dyadic expansion of 2r +1s − 1 ends with a string of 1 ’s including a 1 in the single digit where the expansion of 2r is nonzero. Note that the earlier condition 2k − 1 ≤ n is satisﬁed since it becomes 2r +1 + 1 ≤ 2r +1s + 2r and we assume s ≥ 1. = n−k k−1 2r +1s−1 2r Summarizing, we have shown that for n = 2r (2s + 1), the sphere S n−1 cannot have 2r orthonormal tangent vector ﬁelds if s ≥ 1. This is also trivially true for s = 0 since S n−1 cannot have n orthonormal tangent vector ﬁelds. It is easy to see that this result is best possible when r ≤ 3 by explicitly constructing 2r − 1 orthonormal tangent vector ﬁelds on S n−1 when n = 2r m. When r = 1, view S n−1 as the unit sphere in Cm, and then x ֏ ix deﬁnes a tangent vector ﬁeld since the unit complex numbers 1 and i are orthogonal and multiplication by a unit complex number is an isometry of C, so x and ix are orthogonal in each coordinate of Cm, hence are orthogonal. When r = 2 the same construction works with H in place of C, using the maps x֏ix, x֏jx, and x֏kx to deﬁne three orthonormal tangent vector ﬁelds
on the unit sphere in Hm. When r = 3 we can follow the same Steenrod Squares and Powers Section 4.L 495 procedure with the octonions, constructing seven orthonormal tangent vector ﬁelds to the unit sphere in Om via an orthonormal basis 1, i, j, k, ··· for O. The upper bound of 2r − 1 for the number of orthonormal vector ﬁelds on S n−1 is not best possible in the remaining case n ≡ 0 mod 16. The optimal upper bound is obtained instead using K–theory; see [VBKT] or [Husemoller 1966]. The construction of the requisite number of vector ﬁelds is again algebraic, this time using Cliﬀord algebras. e Σ Σ Σ Example 4L.6: A Map of mod p Hopf Invariant One. Let us describe a construction for a map f : S 2p→S 3 such that in the mapping cone Cf = S 3 ∪f e2p+1, the ﬁrst Steenrod power P 1 : H 3(Cf ; Zp)→H 2p+1(Cf ; Zp) is nonzero, hence f is nonzero in π s 2p−3. The construction starts with the fact that a generator of H 2(K(Zp, 1); Zp) has nontrivial p th power, so the operation P 1 : H 2(K(Zp, 1); Zp)→H 2p(K(Zp, 1); Zp) is nontrivial K(Zp, 1), and we showed in by property (5). This remains true after we suspend to K(Zp, 1) has the homotopy type of a wedge sum of CW comProposition 4I.3 that H∗(Xi; Z) consisting only of a Zp in each dimension plexes Xi, 1 ≤ i ≤ p − 1, with congruent to 2i mod 2p − 2. We are interested here in the space X = X1, which has nontrivial Zp cohomology in dimensions 2, 3, 2p, 2p + 1, ···. Since X is, up to homoK(Zp, 1), the operation P
1 : H 3(X; Zp)→H 2p+1(X; Zp) is topy, a wedge summand of nontrivial. Since X is simply-connected, the construction in §4.C shows that we may take X to have (2p + 1) skeleton of the form S 2 ∪ e3 ∪ e2p ∪ e2p+1. In fact, using the notion of homology decomposition in §4.H, we can take this skeleton to be the reduced mapping cone Cg of a map of Moore spaces g : M(Zp, 2p − 1)→M(Zp, 2). It follows that the quotient Cg/S 2 is the reduced mapping cone of the composition g-----→ M(Zp, 2)→M(Zp, 2)/S 2 = S 3. The restriction h \|\|S 2p−1 repreh : M(Zp, 2p − 1) sents an element of π2p−1(S 3) that is either trivial or has order p since this restriction extends over the 2p cell of M(Zp, 2p − 1) which is attached by a map S 2p−1→S 2p−1 of degree p. In fact, h \|\| S 2p−1 is nullhomotopic since, as we will see in [SSAT] using the Serre spectral sequence, πi(S 3) contains no elements of order p if i ≤ 2p − 1. This implies that the space Ch = Cg/S 2 is homotopy equivalent to a CW complex Y obtained from S 3 ∨ S 2p by attaching a cell e2p+1. The quotient Y /S 2p then has the form S 3 ∪ e2p+1, so it is the mapping cone of a map f : S 2p→S 3. By construction there is a map Cg→Cf inducing an isomorphism on Zp cohomology in dimensions 3 and 2p + 1, so the operation P 1 is nontrivial in H ∗(Cf ; Zp) since this was true for Cg, the (2p + 1) skeleton of X. Example 4L.7: Moore Spaces. Let us use the operation Sq2 to show that for n ≥ 2, the identity map of M(Z2
, n) has order 4 in the group of basepoint-preserving homotopy classes of maps M(Z2, n)→M(Z2, n), with addition deﬁned via the suspension structure on M(Z2, n) = M(Z2, n − 1). According to Proposition 4H.2, this group is the middle term of a short exact sequence, the remaining terms of which contain only Σ 496 Chapter 4 Homotopy Theory elements of order 2. Hence if the identity map of M(Z2, n) has order 4, this short exact sequence cannot split. Σ In view of the short exact sequence just referred to, it will suﬃce to show that twice the identity map of M(Z2, n) is not nullhomotopic. If twice the identity were nullhomotopic, then the mapping cone C of this map would have the homotopy type M(Z2, n). This would force Sq2 : H n(C; Z2)→H n+2(C; Z2) to be trivial of M(Z2, n) ∨ since the source and target groups would come from diﬀerent wedge summands. However, we will now show that this Sq2 operation is nontrivial. Twice the identity map of M(Z2, n) can be regarded as the smash product of the degree 2 map S 1→S 1, z ֏ z2, with the identity map of M(Z2, n − 1). If we smash the coﬁbration sequence S 1→S 1→RP2 for this degree 2 map with M(Z2, n − 1) we get the coﬁber sequence M(Z2, n)→M(Z2, n)→C, in view of the identity (X/A) ∧ Y = (X ∧ Y )/(A ∧ Y ). This means we can view C as RP2 ∧ M(Z2, n − 1). The Cartan formula translated to cross products gives Sq2(α× β) = Sq0α× Sq2β+Sq1α× Sq1β+Sq2α× Sq0β. This holds for smash products as well as ordinary products, by naturality. Taking α to
be a generator of H 1(RP2; Z2) and β a generator of H n−1(M(Z2, n − 1); Z2), we have Sq2α = 0 = Sq2β, but Sq1α and Sq1β are nonzero since Sq1 is the Bockstein. By the K¨unneth formula, Sq1α× Sq1β then generates H n+2(RP2 ∧ M(Z2, n − 1); Z2) and we are done. Adem Relations and the Steenrod Algebra When Steenrod squares or powers are composed, the compositions satisfy certain relations, unfortunately rather complicated, known as Adem relations: SqaSqb = P aP b = P aβP b = j X j X j X b−j−1 Sqa+b−jSqj a−2j (−1)a+j (p−1)(b−j)−1 a−pj if a < 2b P a+b−jP j if a < pb (p−1)(b−j) a−pj (−1)a+j βP a+b−jP j (p−1)(b−j)−1 a−pj−1 (−1)a+j − j X By convention, the binomial coeﬃcient or if m < n. Also m 0 = 1 for m ≥ 0. P a+b−jβP j if a ≤ pb is taken to be zero if m or n is negative m n For example, taking a = 1 in the Adem relation for the Steenrod squares we have Sq1Sqb = (b − 1)Sqb+1, so Sq1Sq2i = Sq2i+1 and Sq1Sq2i+1 = 0. The relations Sq1Sq2i = Sq2i+1 and Sq1 = β explain the earlier comment that Sq2i is the analog of P i for p = 2. The Steenrod algebra A 2 is deﬁned to be the algebra over Z2 that is the quotient of the algebra of polynomials in the noncommuting variables Sq1, Sq2, ··· by the twosided ideal generated by the Adem
relations, that is, by the polynomials given by the diﬀerences between the left and right sides of the Adem relations. In similar fashion, A p for odd p is deﬁned to be the algebra over Zp formed by polynomials in the noncommuting variables β, P 1, P 2, ··· modulo the Adem relations and the relation Steenrod Squares and Powers Section 4.L 497 β2 = 0. Thus for every space X, H ∗(X; Zp) is a module over A p, for all primes p. The Steenrod algebra is a graded algebra, the elements of degree k being those that map H n(X; Zp) to H n+k(X; Zp) for all n. The next proposition implies that A, while A p for p odd is generated by β and the elements P pk 2 is generated as an algebra by the elements Sq2k. Proposition 4L.8. There is a relation Sqi = 0<j<i ajSqi−jSqj with coeﬃcients aj ∈ Z2 whenever i is not a power of 2. Similarly, if i is not a power of p there is a relation P i = 0<j<i ajP i−jP j with aj ∈ Zp. P P Proof: The argument is the same for p = 2 and p odd, so we describe the latter case. The idea is to write i as the sum a + b of integers a > 0 and b > 0 with a < pb, such that the coeﬃcient of the j = 0 term in the Adem relation for P aP b is nonzero. Then one can solve this relation for P a+b = P i. (p−1)b−1 a Let the p adic representation of i be i = i0 + i1p + ··· + ikpk with ik ≠ 0. Let b = pk and a = i − pk, so b > 0 and a > 0 if i is not a power of p. The is nonzero in Zp. The p adic expansion of (p − 1)b − 1 = claim is that (pk+1 − 1) − pk is (p − 1) + (p − 1)
p + ··· + (p − 2)pk, and the p adic expansion of (p−1)b−1 a is i0 + i1p + ··· + (ik − 1)pk. Hence and in each factor a of the latter product the numerator is nonzero in Zp so the product is nonzero in Zp. When p = 2 the last factor is omitted, and the product is still nonzero in Z2. ⊔⊓ p−2 ik−1 p−1 i0 ··· ≡ element a of a graded algebra such as A This proposition says that most of the Sqi ’s and P i ’s are decomposable, where an p is decomposable if it can be expressed in i aibi with each ai and bi having lower degree than a. The operation Sq2k ) = the form is indecomposable since for α a generator of H 1(RP∞; Z2) we saw that Sq2k α2k+1 ) = 0 for 0 < i < 2k. Similarly P pk α ∈ H 2(CP∞; Zp) is a generator then P pk and also β(αpk is indecomposable since if ) = 0 for 0 < i < pk but Sqi(α2k but P i(αpk ) = αpk+1 ) = 0. (αpk (α2k P Here is an application of the preceding proposition: Theorem 4L.9. Suppose H ∗(X; Zp) is the polynomial algebra Zp[α] on a generator α of dimension n, possibly truncated by the relation αm = 0 for m > p. Then if p = 2, n must be a power of 2, and if p is an odd prime, n must be of the form pkℓ where ℓ is an even divisor of 2(p − 1). As we mentioned in §3.2, there is a stronger theorem that n must be 1, 2, 4, or 8 when p = 2, and n must be an even divisor of 2(p − 1) when p is an odd prime. We also gave examples showing the necessity of the hypothesis m > p in the case of a truncated polynomial algebra. Proof: In the case p
= 2, Sqn(α) = α2 ≠ 0. If n is not a power of 2 then Sqn decomposes into compositions Sqn−jSqj with 0 < j < n. Such compositions must be zero since they pass through the group H n+j(X; Z2) which is zero for 0 < j < n. 498 Chapter 4 Homotopy Theory For odd p, the fact that α2 is nonzero implies that n is even, say n = 2k. Then must be P k(α) = αp ≠ 0. Since P k can be expressed in terms of P pi nonzero in H ∗(X; Zp). This implies that 2pi(p − 1), the amount by which P pi raises dimension, must be a multiple of n since H ∗(X; Zp) is concentrated in dimensions that are multiples of n. Since n divides 2pi(p − 1), it must be a power of p times a divisor of 2(p − 1), and this divisor must be even since n is even and p is odd. ⊔⊓ ’s, some P pi Corollary 4L.10. If H ∗(X; Z) is a polynomial algebra Z[α], possibly truncated by αm = 0 with m > 3, then \|α\| = 2 or 4. Proof: Passing from Z to Z2 coeﬃcients, the theorem implies that \|α\| is a power of 2, and taking Z3 coeﬃcients we see that \|α\| is a power of 3 times a divisor of ⊔⊓ 2(3 − 1) = 4. In particular, the octonionic projective plane OP2, constructed in Example 4.47 by attaching a 16 cell to S 8 via the Hopf map S 15→S 8, does not generalize to an octonionic projective n space OPn with n ≥ 3. In a similar vein, decomposability implies that if an element of π s ∗ is detected by a Sqi or P i then i must be a power of 2 for Sqi and a power of p for P i. For if Sqi is decomposable, then the map Sqi : H n(Cf : Z2)→H n+
i(Cf ; Z2) must be trivial since it is a sum of compositions that pass through trivial cohomology groups, and similarly for P i. Interestingly enough, the Adem relations can also be used in a positive way to ∗, as the proof of the following result will show. detect elements of π s Proposition 4L.11. If η ∈ π s nonzero in π s σ ∈ π s 1 is represented by the Hopf map S 3→S 2, then η2 is 3 and 2. Similarly, the other two Hopf maps represent elements ν ∈ π s 6 and π s 7 whose squares are nontrivial in π s 14. Proof: Let η : S n+1→S n be a suspension of the Hopf map, with mapping cone Cη obtained from S n by attaching a cell en+2 via η. If we assume the composition η)η is nullhomotopic, then we can deﬁne a map f : S n+3→Cη in the following ( way. Decompose S n+3 as the union of two cones CS n+2. On one of these cones Σ let f be a nullhomotopy of ( η)η. On the other cone let f be the composition CS n+2→CS n+1→Cη where the ﬁrst map is obtained by coning Σ map is a characteristic map for the cell en+2. η and the second Σ Steenrod Squares and Powers Section 4.L 499 We use the map f to attach a cell en+4 to Cη, forming a space X. This has Cη as its (n + 2) skeleton, so Sq2 : H n(X; Z2)→H n+2(X; Z2) is an isomorphism. The map Sq2 : H n+2(X; Z2)→H n+4(X; Z2) is also an isomorphism since the quotient map X→X/S n induces an isomorphism on cohomology groups above dimension n and 2η. Thus the composition X/S n is homotopy equivalent to the mapping cone of Sq2Sq2 : H n(X; Z2)→H n
+4(X; Z2) is an isomorphism. But this is impossible in view of the Adem relation Sq2Sq2 = Sq3Sq1, since Sq1 is trivial on H n(X; Z2). Σ The same argument shows that ν 2 and σ 2 are nontrivial using the relations ⊔⊓ Sq4Sq4 = Sq7Sq1 + Sq6Sq2 and Sq8Sq8 = Sq15Sq1 + Sq14Sq2 + Sq12Sq4. This line of reasoning does not work for odd primes and the element α ∈ π s 2p−3 detected by P 1 since the Adem relation for P 1P 1 is P 1P 1 = 2P 2, which is not helpful. And in fact α2 = 0 by the commutativity property of the product in π s ∗. When dealing with A 2 it is often convenient to abbreviate notation by writing a monomial Sqi1 Sqi2 ··· as SqI where I is the ﬁnite sequence of nonnegative integers i1, i2, ···. Call SqI admissible if no Adem relation can be applied to it, that is, if ij ≥ 2ij+1 for all j. The Adem relations imply that every monomial SqI can be written as a sum of admissible monomials. For if SqI is not admissible, it contains a pair SqaSqb to which an Adem relation can be applied, yielding a sum of terms SqJ for which J > I with respect to the lexicographic ordering on ﬁnite sequences of integers. These SqJ ’s have the same degree i1 + ··· + ik as SqI, and since the number of monomials SqI of a ﬁxed degree is ﬁnite, successive applications of the Adem relations eventually reduce any SqI to a sum of admissible monomials. For odd p, elements of A p are linear combinations of monomials βε1 P i1 βε2 P i2 ··· with each εj = 0 or 1. Such a monomial is admissible if ij ≥ εj+1 + pij+1 for
all j, which again means that no Adem relation can be applied to the monomial. As with A 2, the Adem relations suﬃce to reduce every monomial to a linear combination of ad- missible monomials, by the same argument as before but now using the lexicographic ordering on tuples (ε1 + pi1, ε2 + pi2, ···). Deﬁne the excess of the admissible monomial SqI to be j(ij − 2ij+1), the amount by which SqI exceeds being admissible. For odd p one might expect the exP cess of an admissible monomial βε1 P i1 βε2 P i2 ··· to be deﬁned as j(ij −pij+1 −εj+1), j(2ij − 2pij+1 − εj+1), for reasons which will become but instead it is deﬁned to be P clear below. P As we explained at the beginning of this section, cohomology operations corre- spond to elements in the cohomology of Eilenberg–MacLane spaces. Here is a rather important theorem which will be proved in [SSAT] since the proof makes heavy use of spectral sequences: 500 Chapter 4 Homotopy Theory Theorem. For each prime p, H ∗(K(Zp, n); Zp) is the free commutative algebra on (ιn) where ιn ∈ H n(K(Zp, n); Zp) is a generator and ranges over the generators all admissible monomials of excess less than n. Θ Θ Here ‘free commutative algebra’ means ‘polynomial algebra’ when p = 2 and ‘polynomial algebra on even-dimensional generators tensor exterior algebra on odd- dimensional generators’ when p is odd. We will say something about the rationale behind the ‘excess less than n ’ condition in a moment. Specializing the theorem to the ﬁrst two cases n = 1, 2, we have the following cohomology algebras: K(Z2, 1) : Z2[ι] K(Zp, 1) : K(Z2, 2) : Z2[ι, Sq1ι, Sq2S
q1ι, Sq4Sq2Sq1ι, ···] K(Zp, 2) : Zp[ι, βP 1βι, βP pP 1βι, βP p2 [ι]⊗Zp[βι] Λ Zp P pP 1βι, ···] ⊗ Zp [βι, P 1βι, P pP 1βι, P p2 P pP 1βι, ···] The theorem implies that the admissible monomials in A p are linearly indepenΛ dent, hence form a basis for A p as a vector space over Zp. For if some linear combination of admissible monomials were zero, then it would be zero when applied to the class ιn, but if we choose n larger than the excess of each monomial in the linear combination, this would contradict the freeness of the algebra H ∗(K(Zp, n); Zp). Even though the multiplicative structure of the Steenrod algebra is rather complicated, the Adem relations provide a way of performing calculations algorithmically by sys- tematically reducing all products to sums of admissible monomials. A proof of the linear independence of admissible monomials using more elementary techniques can be found in [Steenrod & Epstein 1962]. Another consequence of the theorem is that all cohomology operations with Zp coeﬃcients are polynomials in the Sqi ’s when p = 2 and polynomials in the P i ’s and β when p is odd, in view of Proposition 4L.1. We can also conclude that A p consists precisely of all the Zp cohomology operations that are stable, commuting with K(Zp, n)→K(Zp, n + 1) that pulls ιn+1 back to suspension. For consider the map the suspension of ιn. This map induces an isomorphism on homotopy groups πi for i ≤ 2n and a surjection for i = 2n + 1 by Corollary 4.24, hence the same is true for H ∗(K(Zp, n); Zp) homology and cohomology. Letting n go to inﬁnity, the limit lim ←-then exists in a strong sense. On
the one hand, this limit is exactly the stable opera- Σ tions by Proposition 4L.1 and the deﬁnition of a stable operation. On the other hand, the preceding theorem implies that this limit is A p since it says that all elements of H ∗(K(Zp, n); Zp) below dimension 2n are uniquely expressible as sums of admissible monomials applied to ιn. Now let us explain why the condition ‘excess less than n ’ in the theorem is natural. For a monomial SqI = Sqi1 Sqi2 ··· the deﬁnition of the excess e(I) can be rewritten as e Steenrod Squares and Powers Section 4.L 501 2k an equation i1 = e(I) + i2 + i3 + ···. Thus if e(I) > n, we have i1 > \|Sqi2 Sqi3 ··· (ιn)\|, hence SqI (ιn) = 0. And if e(I) = n then SqI (ιn) = (Sqi2 Sqi3 ··· (ιn))2 and either Sqi2 Sqi3 ··· has excess less than n or it has excess equal to n and we can repeat the process to write Sqi2 Sqi3 ··· (ιn) = (Sqi3 ··· (ιn))2, and so on, until we obtain an equation SqI (ιn) = with e(J) < n, so that SqI (ιn) is already in the algebra generated by the elements SqJ (ιn) with e(J) < n. The situation for odd p is similar. For an admissible monomial P I = βε1 P i1 βε2 P i2 ··· the deﬁnition of excess gives 2i1 = e(I) + ε2 + 2(p − 1)i2 + ···, so if e(I) > n we must have P I (ιn) = 0, and if e(I) = n then either P I(ιn) is a power P J (ιn) with e(J) < n, or, if P I begins with = 0 by the formula β(xm) = mxm−1β(x
), which is β, then P I (ιn) = β valid when \|x\| is even, as we may assume is the case here, otherwise (P J (ιn))pk = 0 by commutativity of cup product. SqJ (ιn) (P J (ιn))pk pk There is another set of relations among Steenrod squares equivalent to the Adem relations and somewhat easier to remember: k j Sq2n−k+j−1Sqn−j = 0 j X When k = 0 this is simply the relation Sq2n−1Sqn = 0, and the cases k > 0 are obtained from this via Pascal’s triangle. For example, from Sq7Sq4 = 0 we obtain the following table of relations: Sq7Sq4 Sq6Sq4 + Sq7Sq3 + Sq4Sq4 + Sq5Sq3 + Sq6Sq2 + Sq7Sq1 + Sq7Sq2 Sq5Sq4 = 0 Sq7Sq0 = 0 = 0 Sq6Sq0 Sq3Sq4 Sq2Sq4 + Sq3Sq3 + Sq3Sq2 Sq1Sq4 + + Sq5Sq0 Sq0Sq4 + Sq1Sq3 + Sq2Sq2 + Sq3Sq1 + Sq4Sq0 = 0 = 0 = 0 = 0 = 0 These relations are not in simplest possible form. For example, Sq5Sq3 = 0 in the fourth row and Sq3Sq2 = 0 in the seventh row, instances of Sq2n−1Sqn = 0. For P pn−k+j−1P n−j = 0 derived from Steenrod powers there are similar relations the basic relation P pn−1P n = 0. We leave it to the interested reader to show that these relations follow from the Adem relations. P k j j Constructing the Squares and Powers Now we turn to the construction of the Steenrod squares and powers, and the proof of their basic properties including the Adem relations. As will be seen, this all hinges on the fact that cohomology is maps into Eilenberg–MacLane spaces. The
case p = 2 is in some ways simpler than the case p odd, so in the ﬁrst part of the development we will specialize p to 2 whenever there is a signiﬁcant diﬀerence between the two cases. 502 Chapter 4 Homotopy Theory Before giving the construction in detail, let us describe the idea in the case p = 2. The cup product square α2 of an element α ∈ H n(X; Z2) can be viewed as a composition X→X × X→K(Z2, 2n), with the ﬁrst map the diagonal map and the second map representing the cross product α× α. Since we have Z2 coeﬃcients, cup product and cross product are strictly commutative, so if T : X × X→X × X is the map T (x1, x2) = (x2, x1) transposing the two factors, then T ∗(α× α) = α× α. Thinking of α× α as a map X × X→K(Z2, 2n), this says there is a homotopy ft from α× α to (α× α)T. If we follow the homotopy ft by the homotopy ftT, we obtain a homotopy from α× α to (α× α)T and then to (α× α)T 2 = α× α, in other words a loop of maps X × X→K(Z2, 2n). We can view this loop as a map S 1 × X × X→K(Z2, 2n). As we will see, if the homotopy ft is chosen appropriately, the loop of maps will be nullhomotopic, extending to a map D2 × X × X→K(Z2, 2n). Regarding D2 as the upper hemisphere of S 2, this gives half of a map S 2 × X × X→K(Z2, 2n), and once again we obtain the other half by composition with T. This process can in fact be repeated inﬁnitely often to yield a map S ∞ × X × X→K(Z2, 2n) with the property that each pair of points (s, x1, x2) and (−s, x2, x1) is sent to the same point
in K(Z2, 2n). This means that when we compose with the diagonal map S ∞ × X→S ∞ × X × X, (s, x)֏ (s, x, x), there is an induced quotient map RP∞ × X→K(Z2, 2n) extending α2 : X→K(Z2, 2n). This extended map represents a class in H 2n(RP∞ × X; Z2). By the K¨unneth formula and the fact that H ∗(RP∞; Z2) is the polynomial ring Z2[ω], this cohomology class i ωn−i × ai with ai ∈ H n+i(X; Z2). in H 2n(RP∞ × X; Z2) can be written in the form Then we deﬁne Sqi(α) = ai. P The construction of the map S ∞× X × X→K(Z2, 2n) will proceed cell by cell, so it will be convenient to eliminate any unnecessary cells. This is done by replacing X × X by the smash product X ∧X and factoring out a cross-sectional slice S ∞ in S ∞ × X ∧X. A further simpliﬁcation will be to use naturality to reduce to the case X = K(Z2, n). Now we begin the actual construction. For a space X with basepoint x0, let X ∧p denote the smash product X ∧···∧X of p copies of X. There is a map T : X ∧p→X ∧p, T (x1, ···, xp) = (x2, ···, xp, x1), permuting the factors cyclically. Note that when p = 2 this is just the transposition (x1, x2)֏(x2, x1). The map T generates an action of Zp on X ∧p. There is also the standard action of Zp on S ∞ viewed as the union of the unit spheres S 2n−1 in Cn, a generator of Zp rotating each C factor through an angle 2π /p, with quotient space an inﬁnite-dimensional lens space L∞,
or RP∞ when p = 2. On the product S ∞ × X ∧p there is then the diagonal action g(s, x) = (g(s), g(x)) for X denote the orbit space (S ∞ × X ∧p)/Zp of this diagonal action. This g ∈ Zp. Let X ∧p described in §3.G. The projection is the same as the Borel construction S ∞ × Zp S ∞ × X ∧p→S ∞ induces a projection π : X→L∞ with π −1(z) = X ∧p for all z ∈ L∞ X→L∞ is in fact a ﬁber bundle, since the action of Zp on S ∞ is free. This projection though we shall not need this fact and so we leave the proof as an exercise. The Zp action on X ∧p ﬁxes the basepoint x0 ∈ X ∧p, so the inclusion S ∞ × {x0} ֓ S ∞× X ∧p Γ Γ Γ Steenrod Squares and Powers Section 4.L 503 X. The composition L∞ ֓ induces an inclusion L∞ ֓ ﬁber bundle terminology this subspace L∞ ⊂ denote the quotient that the ﬁbers X ∧p in Γ the section L∞ in a single point. X→L∞ is the identity, so in X X/L∞ obtained by collapsing the section L∞ to a point. Note X since each ﬁber meets X are still embedded in the quotient X is a section of the bundle. Let Λ Γ Γ Γ Γ Λ If we replace S ∞ by S 1 in these deﬁnitions, we get subspaces 1X ⊂ X and 1X ⊂ Λ X. All these spaces have natural CW structures if X is a CW complex having x0 as a 0 cell. To see this, let L∞ be given its standard CW structure with one cell in Λ each dimension. This lifts to a CW structure on S ∞ with p cells in each dimension, and then T freely permutes the product cells of S ∞ × X ∧p so there is an induced X is a
subcomplex, so the quotient quotient CW structure on X. The section L∞ ⊂ Γ Γ X inherits a CW structure from Γ X. In particular, note that if the n skeleton of X X is S pn with its usual Γ is S n with its usual CW structure, then the pn skeleton of Λ CW structure. Γ, 1,, and 1 are functors: A map f : (X, x0)→(Y, y0) Λ We remark also that X→ f : induces maps Y, etc., in the evident way. Γ Λ Λ Γ Γ Γ Γ For brevity we write H ∗(−; Zp) simply as H ∗(−). For n > 0 let Kn denote a CW complex K(Zp, n) with (n−1) skeleton a point and n skeleton S n. Let ι ∈ H n(Kn) be the canonical fundamental class described in the discussion following Theorem 4.57. It will be notationally convenient to regard an element α ∈ H n(X) also as a map α : X→Kn such that α∗(ι) = α. Here we are assuming X is a CW complex. H ∗(X)⊗p→ H ∗(X)⊗p→ From §3.2 we have a reduced p fold cross product H ∗(X ∧p) where H ∗(X) with itself. This cross prodH ∗(X)⊗p denotes the p fold tensor product of H ∗(X ∧p) is an isomorphism since we are using Zp coeﬃcients. uct map e With this isomorphism in mind, we will use the notation α1 ⊗ ··· ⊗ αp rather than e H ∗(X ∧p). In particular, for each element α1 × ··· × αp for p fold cross products in H pn(X ∧p). Our ﬁrst α ∈ H n(X), n > 0, we have its p fold cross product α⊗p ∈ X) restricting to α⊗p in each ﬁber task will be to construct an element λ(α) ∈ H pn(
e X ∧p ⊂ Kn). X. By naturality it will suﬃce to construct λ(ι) ∈ H pn( e e e e e Λ The key point in the construction of λ(ι) is the fact that T ∗(ι⊗p) = ι⊗p. In terms Λ n →Kpn, this says the composition ι⊗p T is homotopic to ι⊗p, preserving of maps K∧p basepoints. Such a homotopy can be constructed as follows. The pn skeleton of K∧p n is (S n)∧p = S pn, with T permuting the factors cyclically. Thinking of S n as (S 1)∧n, the permutation T is a product of (p − 1)n2 transpositions of adjacent factors, so T has degree (−1)(p−1)n2 on S pn. If p is odd, this degree is +1, so the restriction of T to this skeleton is homotopic to the identity, hence ι⊗p T is homotopic to ι⊗p on this skeleton. This conclusion also holds when p = 2, signs being irrelevant in this case since we are dealing with maps S 2n→K2n and π2n(K2n) = Z2. Having a homotopy ι⊗p T ≃ ι⊗p on the pn skeleton, there are no obstructions to extending the homotopy over all higher-dimensional cells ei × (0, 1) since πi(Kpn) = 0 for i > pn. Λ 504 Chapter 4 Homotopy Theory n →Kpn deﬁnes a map The homotopy ι⊗p T ≃ ι⊗p : K∧p 1X is the quotient of I × X ∧p under the identiﬁcations (0, x) ∼ (1, T (x)). The homoΓ 1Kn→Kpn passes down to a quotient map topy is basepoint-preserving, so the map 1Kn→Kpn. Since Kn is obtained from S n by attaching cells of dimension greater λ1 : 1
Kn by attaching cells of dimension greater than than n, Λ Kn→Kpn since pn + 1. There are then no obstructions to extending λ1 to a map λ : Λ πi(Kpn) = 0 for i > pn. 1Kn→Kpn since Kn is obtained from Λ Γ Γ The map λ gives the desired element λ(ι) ∈ H pn( to each ﬁber K∧p up to homotopy since the restriction map H pn( pn skeleton of Kn) since the restriction of λ n is homotopic to ι⊗p. Note that this property determines λ uniquely n ) is injective, the n. We shall have occasion to use this Kn being contained in K∧p Kn)→H pn(K∧p Λ Λ argument again in the proof, so we refer to it as ‘the uniqueness argument’. For any α ∈ H n(X) let λ(α) be the composition Λ Λ X α restricts to α⊗p in each ﬁber. Λ Kn Λ Λ α-----→ λ-----→ Kpn. This restricts to α⊗p in each ﬁber X ∧p since Now we are ready to deﬁne some cohomology operations. There is an incluX as the quotient of the diagonal embedding S ∞ × X ֓ S ∞× X ∧p, X, we get a map X)→H ∗(L∞ × X) ≈ H ∗(L∞) ⊗ H ∗(X). For each Γ sion L∞× X ֓ (s, x) ֏ (s, x, ···, x). Composing with the quotient map ∇ : L∞ × X→ α ∈ H n(X) the element ∇∗(λ(α)) ∈ H pn(L∞ × X) may be written in the form X inducing ∇∗ : H ∗( X→ Λ Λ Γ Λ Λ ∇∗(λ(α)) = ω(p−1)n−i ⊗ θi(α) i where ω
j is a generator of H j(L∞) and θi(α) ∈ H n+i(X). Thus θi increases dimension by i. When p = 2 there is no ambiguity about ωj. For odd p we choose ω1 to be the class dual to the 1 cell of L∞ in its standard cell structure, then we take ω2 to j j 2 and ω2j+1 = ω1ω be the Bockstein βω1 and we set ω2j = ω 2. X It is clear that θi is a cohomology operation since θi(α) = α∗(θi(ι)). Note that θi = 0 for i < 0 since H n+i(Kn) = 0 for i < 0 except for i = −n, and in this special case θi = 0 since ∇ : L∞× X→ X sends L∞ × {x0} to a point. For p = 2 we set Sqi(α) = θi(α). For odd p we will show that θi = 0 unless i = 2k(p − 1) or 2k(p − 1) + 1. The operation P k will be deﬁned to be a certain constant times θ2k(p−1), and θ2k(p−1)+1 will be a constant times βP k, for β the mod p Bockstein. Λ Theorem 4L.12. The operations Sqi satisfy the properties (1)–(7). Proof: We have already observed that the θi ’s are cohomology operations, so property (1) holds. The basic property that λ(α) restricts to α⊗p in each ﬁber implies that θ(p−1)n(α) = αp since ω0 = 1. This gives the ﬁrst half of property (5) for Sqi. The second half follows from the fact that θi = 0 for i > (p − 1)n since the factor ω(p−1)n−i vanishes in this case. Next we turn to the Cartan formula. For any prime p we will show that λ(α`β) = (−1)p(
p−1)mn/2λ(α) ` λ(β) for m = \|α\| and n = \|β\|. This implies (3) when p = 2 Steenrod Squares and Powers Section 4.L 505 since if we let ω = ω1, hence ωj = ωj, then Sqi(α ` β) ⊗ ωn+m−i = ∇∗ = ∇∗ i X λ(α ` β) = ∇∗ λ(α) ` λ(β) λ(α) ` ∇∗ λ(β) X To show that λ(α ` β) = (−1)p(p−1)mn/2λ(α) ` λ(β) we use the following diagram: = = j X i X Sqj (α) ⊗ ωn−j ` k Sqk(β) ⊗ ωm−k j+k=i X Sqj (α) ` Sqk(β) ⊗ ωn+m−i ∆ Here is a generic symbol for diagonal maps x ֏ (x, x). These relate cross prod∗(ϕ ⊗ ψ) = ϕ ` ψ. The two unlabeled vertical maps are uct to cup product via induced by (s, x1, y1, ···, xp, yp) ֏ (s, x1, ···, xp, s, y1, ···, yp). The composition X→Kpm+pn going across the top of the diagram is λ(α ` β) since the composition X→Kpm+pn is λ(α) ⊗ λ(β) so X→ (α ` β). The composition Km+n is Λ X→Kpm+pn across the bottom of the diagram is λ(α) ` λ(β). The the composition Λ Λ Λ triangle on the left, the square, and the upper triangle on the right obviously commute X ∧ Λ ∆ Λ from the deﬁnitions. It remains to see that the third triangle commutes up to the sign (−1)p(p−1)mn/2. Since (Km ∧
Kn)∧p includes the (pm + pn) skeleton of (Km ∧ Kn), restriction to this ﬁber is injective on H pm+pn. On this ﬁber the two routes around the triangle give (ιm ⊗ ιn)⊗p and ι⊗p n. These diﬀer by a permutation that is the product of (p − 1) + (p − 2) + ··· + 1 = p(p − 1)/2 transpositions of adjacent factors. Since ιm and ιn have dimensions m and n, this permutation introduces a sign (−1)p(p−1)mn/2 by the commutativity property of cup product. This ﬁnishes the veriﬁcation of the Cartan formula when p = 2. m ⊗ ι⊗p Λ Λ Before proceeding further we need to make an explicit calculation to show that Sq0 is the identity on H 1(S 1). Viewing S 1 as the one-point compactiﬁcation of R, with the point at inﬁnity as the basepoint, the 2 sphere S 1 ∧S 1 becomes the one-point compactiﬁcation of R2. The map T : S 1 ∧ S 1→S 1 ∧ S 1 then corresponds to reﬂecting R2 across the line x = y, so after a rotation of coordinates this becomes reﬂection of S 2 across the equator. Hence 1S 1 is obtained from the shell I × S 2 by identifying its inner and outer boundary spheres via a reﬂection across the equator. The diagonal RP1 × S 1 ⊂ 1S 1 is a torus, obtained from the equatorial annulus I × S 1 ⊂ I × S 2 by identifying the two ends via the identity map since the equator is ﬁxed by the reﬂection. This RP1 × S 1 represents the 1S 1; Z2) as the ﬁber sphere S 1 ∧ S 1 since the upper half of the same element of H2( shell is a 3 cell whose mod 2 boundary in Γ 1S 1 is the union of these two surfaces. Γ Γ Γ 506 Chapter 4
Homotopy Theory For a generator α ∈ H 1(S 1), consider the element ∇∗(λ(α)) in H 2(RP∞ × S 1) ≈ Hom(H2(RP∞ × S 1; Z2), Z2). A basis for H2(RP∞ × S 1; Z2) is represented by RP2 × {x0} and RP1 × S 1. A cocycle representing ∇∗(λ(α)) takes the value 0 on RP2 × {x0} since RP∞ × {x0} collapses to a point in S 1). On RP1 × S 1, ∇∗(λ(α)) takes the value 1 since when λ(α) is pulled back to S 1 it takes the same Λ value on the homologous cycles RP1 × S 1 and S 1 ∧ S 1, namely 1 by the deﬁning property of λ(α) since α ⊗ α ∈ H 2(S 1 ∧ S 1) is a generator. Thus ∇∗(λ(α)) = ω1 ⊗ α and hence Sq0(α) = α by the deﬁnition of Sq0. S 1 and λ(α) lies in H 2( Λ Γ We use this calculation to prove that Sqi commutes with the suspension σ, where σ is deﬁned by σ (α) = ε ⊗ α ∈ H ∗(S 1 ∧ X) for ε a generator of H 1(S 1) and α ∈ H ∗(X). We have just seen that Sq0(ε) = ε. By (5), Sq1(ε) = ε2 = 0 and Sqi(ε) = 0 for i > 1. The Cartan formula then gives Sqi(σ (α)) = Sqi(ε ⊗ α) = j Sqj (ε) ⊗ Sqi−j(α) = ε ⊗ Sqi(α) = σ (Sqi(α)). From this it follows that Sq0 is the identity on H n(S n) for all n > 0. Since S n is P the n skeleton of Kn, this implies that S
q0 is the identity on the fundamental class ιn, hence Sq0 is the identity on all positive-dimensional classes. Property (7) is proved similarly: Sq1 coincides with the Bockstein β on the generator ω ∈ H 1(RP2) since both equal ω2. Hence Sq1 = β on the iterated suspensions of ω, and the n fold suspension of RP2 is the (n + 2) skeleton of Kn+1. Finally we have the additivity property (2). This holds in fact for any cohomology operation that commutes with suspension. For such operations, it suﬃces to prove additivity in spaces that are suspensions. Consider a composition X c-----→ X ∨ X α∨β------------→ Kn θ-----→ Km where c is the map that collapses an equatorial copy of X in Σ Σ Σ X to a point. The composition of the ﬁrst two maps is α + β, as in Lemma 4.60. Composing with the third map then gives θ(α + β). On the other hand, if we ﬁrst compose the second and third maps we get θ(α) ∨ θ(β), and then composing with the ﬁrst map gives Σ θ(α) + θ(β). The two ways of composing are equal, so θ(α + β) = θ(α) + θ(β). ⊔⊓ Theorem 4L.13. The Adem relations hold for Steenrod squares. Proof: The idea is to imitate the construction of Adem relations will come from the symmetry of Zp × Zp interchanging the factors. X using Zp × Zp in place of Zp. The The group Zp × Zp acts on S ∞ × S ∞ via (g, h)(s, t) = (g(s), h(t)), with quotient, obtained by writing points of X ∧p2 L∞× L∞. There is also an action of Zp × Zp on X ∧p2 as p2 tuples (xij) with subscripts i and j varying from 1 to p, and then letting the ﬁrst Zp act on the ﬁrst subscript and the second
Zp act on the second. Factoring out the diagonal action of Zp × Zp on S ∞ × S ∞ × X ∧p2 2X. This projects to L∞× L∞ with a section, and collapsing the section gives 2X. The ﬁbers since the action of Zp × Zp on S ∞× S ∞ is of the projection 2X→L∞× L∞ are X ∧p2 gives a quotient space Γ Λ Λ Λ Steenrod Squares and Powers Section 4.L 507 ( Λ Λ Λ 2X with 2X from the product S ∞ × S ∞ × X p2 free. We could also obtain by ﬁrst collapsing the subspace of points having at least one X coordinate equal to the basepoint x0 and then factoring out the Zp × Zp action. It will be useful to compare X). The latter space is the quotient of S ∞ × (S ∞ × X p)p in which one ﬁrst identiﬁes all points having at least one X coordiΛ nate equal to x0 and then one factors out by an action of the wreath product Zp ≀ Zp, p→Zp ≀ Zp→Zp→0 the group of order pp+1 deﬁned by a split exact sequence 0→Zp with conjugation by the quotient group Zp given by cyclic permutations of the p Zp factors of Zp In the coordinates (s, t1, x11, ···, x1p, ···, tp, xp1, ···, xpp) the p. ith factor Zp of Zp p acts in the block (ti, xi1, ···, xip), and the quotient Zp acts by cyclic permutation of the index i and by rotation in the s coordinate. There 2X→ X) induced by (s, t, x11, ···, x1p, ···, xp1, ···, xpp) ֏ is a natural map (s, t, x11, ···, x1p, ···, t, xp1, ···, xpp). In 2X one is factoring out by the action of
Zp × Zp. This is the subgroup of Zp ≀ Zp obtained by restricting the action of the quoΛ tient Zp on Zp p, where this action becomes trivial so that one has the direct product Zp × Zp. p to the diagonal subgroup Zp ⊂ Zp Λ Λ Λ ( 2Kn→Kp2n restricting to ι⊗p2 Λ in a ﬁber, then extends this over the part of Since it suﬃces to prove that the Adem relations hold on the class ι ∈ H n(Kn), in each ﬁber. we take X = Kn. There is a map λ2 : This is constructed by the same method used to construct λ. One starts with a map representing ι⊗p2 2Kn projecting to the 1 skeleton of L∞× L∞, and ﬁnally one extends inductively over higher-dimensional cells of 2Kn using the fact that Kp2n is an Eilenberg–MacLane space. The map λ2 ﬁts into the diagram at the right, where ∇2 is induced by the map (s, t, x)֏ (s, t, x, ···, x) and the unlabeled map is the one deﬁned Λ Λ 2 λ∗ The element ∇∗ r,s ωr ⊗ ωs ⊗ ϕr s, and we above. It is clear that the square commutes. Commutativity of the triangle up to homotopy follows from the fact that λ2 is uniquely determined, up to homotopy, by its restrictions to ﬁbers. 2 (ι) may be written in the form claim that the elements ϕr s satisfy the symmetry relation ϕr s = (−1)r s+p(p−1)n/2ϕsr. To verify this we use the commutative diagram at the right where the map τ on the left switches the two L∞ factors and the τ on the right is induced by switching the two S ∞ factors of S ∞ × S ∞ × K∧p2 and permuting the Kn factors of the smash product by interchanging the two subscripts in p2 tuples (xij).
This permutation is a product of p(p − 1)/2 transpositions, one for each pair is sent to (−1)p(p−1)n/2ι⊗p2 (i, j) with 1 ≤ i < j ≤ p, so in a ﬁber the class ι⊗p2. 2 (ι) = (−1)p(p−1)n/2λ∗ By the uniqueness property of λ2 this means that τ ∗λ∗ 2 (ι). Commutativity of the square then gives P n (−1)p(p−1)n/2∇∗ 2 λ∗ 2 (ι) = ∇∗ 2 τ ∗λ∗ 2 (ι) = τ ∗∇∗ 2 λ∗ 2 (ι) = (−1)r s ωs ⊗ ωr ⊗ ϕr s r,s X 508 Chapter 4 Homotopy Theory where the last equality follows from the commutativity property of cross products. The symmetry relation ϕr s = (−1)r s+p(p−1)n/2ϕsr follows by interchanging the indices r and s in the last summation. If we compute ∇∗ 2 λ∗ ing the map λ2, we obtain 2 (ι) using the lower route across the earlier diagram contain- ∇∗ 2 λ∗ 2 (ι) = = i X i,j X ω(p−1)pn−i ⊗ θi ω(p−1)pn−i ⊗ θi j ω(p−1)n−j ⊗ θj(ι) X ω(p−1)n−j ⊗ θj(ι) Now we specialize to p = 2, so θi = Sqi for all i. The Cartan formula converts the i,j,k ω2n−i ⊗ Sqk(ωn−j ) ⊗ Sqi−kSqj (ι). Plugging in the last summation above into value for Sqk(ωn−j ) computed in the discussion preceding Example 4L.3, we obtain ω2n−i ⊗ ωn−j
+k ⊗ Sqi−kSqj (ι). To write this summation more symmetri- cally with respect to the two ω terms, let n − j + k = 2n − ℓ. Then we get P n−j k i,j,k P n−j n+j−ℓ ω2n−i ⊗ ω2n−ℓ ⊗ Sqi+ℓ−n−jSqj(ι) i,j,ℓ X In view of the symmetry property of ϕr s, which becomes ϕr s = ϕsr for p = 2, switching i and ℓ in this formula leaves it unchanged. Hence we get the relation (∗) n−j n+j−ℓ Sqi+ℓ−n−jSqj (ι) = n−j n+j−i j X j X Sqi+ℓ−n−jSqj(ι) This holds for all n, i, and ℓ, and the idea is to choose these numbers so that the left side of this equation has only one nonzero term. Given integers r and s, let n−j n = 2r − 1 + s and ℓ = n + s, so that. If r is suﬃciently large, n+j−ℓ this will be 0 unless j = s. This is because the dyadic expansion of 2r − 1 consists entirely of 1 ’s, so the expansion of 2r − 1 − (j − s) will have 0 ’s in the positions where = 0 to the expansion of j − s has 1 ’s, hence these positions contribute factors of 0 1. Thus with n and ℓ chosen as above, the relation (∗) becomes 2r −1−(j−s) j−s = 2r −1−(j−s) j−s SqiSqs (ι) = 2r −1+s−j 2r −1+s+j−i Sqi+s−j Sqj (ι) = j X j X where the latter equality comes from the general relation 2r +s−j−1 i−2j Sqi+s−jSqj (ι) = x y x x−
y. = s−j−1 i−2j The ﬁnal step is to show that 2r +s−j−1 if i < 2s. Both of these i−2j binomial coeﬃcients are zero if i < 2j. If i ≥ 2j then we have 2j ≤ i < 2s, so j < s, hence s − j − 1 ≥ 0. The term 2r then makes no diﬀerence in if r is large since this 2r contributes only a single 1 to the dyadic expansion of 2r + s − j − 1, far to the left of all the nonzero entries in the dyadic expansions of s − j − 1 and i − 2j. This gives the Adem relations for the classes ι of dimension n = 2r − 1 + s with r large. This implies the relations hold for all classes of these dimensions, by naturality. 2r +s−j−1 i−2j Since we can suspend repeatedly to make any class have dimension of this form, the Adem relations must hold for all cohomology classes. ⊔⊓ Steenrod Squares and Powers Section 4.L 509 Steenrod Powers Our remaining task is to verify the axioms and Adem relations for the Steenrod powers for an odd prime p. Unfortunately this is quite a bit more complicated than the p = 2 case, largely because one has to be very careful in computing the many coeﬃcients in Zp that arise. Even for the innocent-looking axiom P 0 = 11 it will take three pages to calculate the normalization constants needed to make the axiom hold. One could wish that the whole process was a lot cleaner. Lemma 4L.14. θi = 0 unless i = 2k(p − 1) or 2k(p − 1) + 1. Proof: The group of automorphisms of Zp is the multiplicative group Z∗ p of nonzero elements of Zp. Since p is prime, Zp is a ﬁeld and Z∗ p is cyclic of order p − 1. Let r p. Deﬁne a map ϕ : S ∞× X ∧p→S ∞ × X ∧p permuting the factors Xj be a generator of Z∗ of X ∧p by �
�(s, Xj) = (sr, Xr j) where subscripts are taken mod p and sr means raise each coordinate of s, regarded as a unit vector in C∞, to the r th power and renormalize the resulting vector to have unit length. Then if γ is a generator of the Zp action on S ∞ × X ∧p, we have ϕ(γ(s, Xj)) = ϕ(e2π i/ps, Xj−1) = (e2r π i/psr, Xr j−r ) = γr (ϕ(s, Xj )). This says that ϕ takes orbits to orbits, so ϕ induces maps ϕ : X and ϕ : ϕ : L∞→L∞. Taking X = Kn, these maps ﬁt into the diagram at the right. The square obviously X. Restricting to the ﬁrst coordinate, there is also an induced map X→ X→ Λ Λ Γ Γ commutes. The triangle commutes up to homotopy and a sign of (−1)n since it suﬃces to verify this on the pn skeleton (S n)∧p, and here the map ϕ is an odd permutation of the S n factors since it is a cyclic permutation of order p−1, which is even, and a transposition of two S n factors has degree 1 if n is even and degree −1 if n is odd. Suppose ﬁrst that n is even. Then commutativity of the diagram means that i ω(p−1)n−i ⊗ θi(ι) is invariant under ϕ∗ ⊗ 11, hence ϕ∗(ω(p−1)n−i) = ω(p−1)n−i if θi(ι) is nonzero. The map ϕ induces multiplication by r in π1(L∞), hence also P in H1(L∞) and H 1(L∞; Zp), sending ω1 to r ω1. Since ω2 was chosen to be the Bockstein of ω1, it is also multiplied by r. We chose r to have order p − 1 in Z∗ p, so �
�∗(ωℓ) = ωℓ only when the total number of ω1 and ω2 factors in ωℓ is a multiple of p − 1. For ωℓ = ωk 2 this means ℓ = (p − 1)n − i = 2k(p − 1), while for ωℓ = ω1ωk−1 it means ℓ is 1 less than this, 2k(p − 1) − 1. Solving these equations for i gives i = (n − 2k)(p − 1) or i = (n − 2k)(p − 1) + 1. Since n is even this says 2 that i is congruent to 0 or 1 mod 2(p − 1), which is what the lemma asserts. When n is odd the condition ϕ∗(ωℓ) = ωℓ becomes ϕ∗(ωℓ) = −ωℓ. In the cyclic group Z∗ p the element −1 is the only element of order 2, and this element is (p − 1)/2 times a generator, so the total number of ω1 and ω2 factors in ωℓ must be (2k + 1)(p − 1)/2 for some integer k. This implies that ℓ = (p − 1)n − i = 510 Chapter 4 Homotopy Theory 2(2k + 1)(p − 1)/2 or 1 less than this, hence i = (n − 2k − 1)(p − 1) or 1 greater than ⊔⊓ this. As n is odd, this again says that i is congruent to 0 or 1 mod 2(p − 1). Since θ0 : H n(X)→H n(X) is a cohomology operation that preserves dimension, it must be deﬁned by a coeﬃcient homomorphism Zp→Zp, multiplication by some an ∈ Zp. We claim that these an ’s satisfy am+n = (−1)p(p−1)mn/2aman and an = (−1)p(p−1)n(n−1)/4an 1 To see this, recall the formula λ(α ` β) = (−1
)p(p−1)mn/2λ(α)λ(β) for \|α\| = m and \|β\| = n. From the deﬁnition of the θi ’s it then follows that θ0(α ` β) = (−1)p(p−1)mn/2θ0(α)θ0(β), which gives the ﬁrst part of the claim. The second part follows from this by induction on n. Lemma 4L.15. a1 = ±m! for m = (p − 1)/2, so p = 2m + 1. Proof: It suﬃces to compute θ0(α) where α is any nonzero 1 dimensional class, so the simplest thing is to choose α to be a generator of H 1(S 1), say a generator coming from a generator of H 1(S 1; Z). This determines α up to a sign. Since H i(S 1) = 0 for i > 1, we have θi(α) = 0 for i > 0, so the deﬁning formula for θ0(α) has the form ∇∗(λ(α)) = ωp−1 ⊗ θ0(α) = a1ωp−1 ⊗ α in H p(L∞ × S 1). To compute a1 there is no harm in replacing L∞ by a ﬁnite-dimensional lens space, say Lp, the p skeleton of L∞. S 1→L∞ to a bundle pS 1→Lp with the same ﬁbers Thus we may restrict the bundle (S 1)∧p = S p. We regard S 1 as the one-point compactiﬁcation of R with basepoint the added point at inﬁnity, and then (S 1)∧p becomes the one-point compactiﬁcation of Rp with Zp acting by permuting the coordinates of Rp cyclically, preserving the origin pS 1→Lp with ﬁbers S p, and the point at inﬁnity. This action deﬁnes the bundle pS 1 is obtained by collapsing pS 1 as the one-point compactiﬁcation of p
S 1, since the base space Lp is compact. the complement of the section at inﬁnity in The complement of the section at inﬁnity is a bundle E→Lp with ﬁbers Rp. In general, the one-point compactiﬁcation of a ﬁber bundle E over a compact base space with ﬁbers Rn is called the Thom space T (E) of the bundle, and a class in H n(T (E)) that restricts to a generator of H n of the one-point compactiﬁcation of each ﬁber Rn is called a Thom class. In our situation, λ(α) is such a Thom class. containing a zero section and a section at inﬁnity, and the section at inﬁnity. We can also describe Λ Λ Λ Λ Γ Γ Our ﬁrst task is to construct subbundles E0, E1, ···, Em of E, where E0 has ﬁber R and the other Ej ’s have ﬁber R2, so p = 2m + 1. The bundle E comes from the linear transformation T : Rp→Rp permuting the coordinates cyclically. We claim there is a decomposition Rp = V0 ⊕ V1 ⊕ ··· ⊕ Vm with V0 1 dimensional and the other Vj ’s 2 dimensional, such that T (Vj) = Vj for all j, with T \|\| V0 the identity and T \|\| Vj a rotation by the angle 2π j/p for j > 0. Thus T deﬁnes an action of Zp on Vj and we can deﬁne Ej just as E was deﬁned, as the quotient (S p × Vj)/Zp with respect to the diagonal action. Steenrod Squares and Powers Section 4.L 511 An easy way to get the decomposition Rp = V0 ⊕ V1 ⊕ ··· ⊕ Vm is to regard Rp as a module over the principal ideal domain R[t] by setting tv = T (v) for v ∈ Rp. Then Rp is isomorphic as a module to the module R[t]/(tp
− 1) since T permutes coordinates cyclically; this amounts to identifying the standard basis vectors v1, ···, vp in Rp with 1, t, ···, tp−1. The polynomial tp − 1 factors over C into the linear factors t − e2π ij/p for j = 0, ···, p − 1. Combining complex conjugate 1≤j≤m (t2 − 2(cos ϕj)t + 1), factors, this gives a factorization over R, tp − 1 = (t − 1) where ϕj = 2π j/p. These are distinct monic irreducible factors, so the module 1≤j≤mR[t]/(t2 − 2(cos ϕj)t + 1) by the basic R[t]/(tp − 1) splits as R[t]/(t − 1) structure theory of modules over a principal ideal domain. This translates into a decomposition Rp = V0 ⊕ V1 ⊕ ··· ⊕ Vm with T (Vj ) ⊂ Vj. Here V0 corresponds to R[t]/(t − 1) ≈ R with t acting as the identity, and Vj for j > 0 corresponds to R[t]/(t2 − 2(cos ϕj)t + 1). The latter module is isomorphic to R2 with t acting as rotation by the angle ϕj since the characteristic polynomial of this rotation is readily computed to be t2 −2(cos ϕj)t +1, hence this rotation satisﬁes t2 −2(cos ϕj)t +1 = 0 so there is a module homomorphism R[t]/(t2 − 2(cos ϕj)t + 1)→R2 which is obviously an isomorphism. L Q From the decomposition Rp = V0 ⊕ V1 ⊕ ··· ⊕ Vm and the action of T on each factor we can see that the only vectors ﬁxed by T are those in the line V0. The vectors (x, ···, x) are ﬁxed by T, so V0 must be this diagonal line. Next we compute Thom classes for the bundles Ej. This is easy
for E0 which is the product Lp × R, so the projection E0→R one-point compactiﬁes to a map T (E0)→S 1 and we can pull back the chosen generator α ∈ H 1(S 1) to a Thom class for E0. The other Ej ’s have 2 dimensional ﬁbers, which we now view as C rather than R2. Just as Ej is the quotient of S p × C via the identiﬁcations (v, z) ∼ (e2π i/pv, e2π ij/pz), we can deﬁne a bundle Ej→CPm with ﬁber C by the identiﬁcations (v, z) ∼ (λv, λjz) for λ ∈ S 1 ⊂ C. We then have the left half of the commutative diagram shown at the right, where the quotient map q restricts to a homeomorphism on each ﬁber. The maps f and f are induced by the map S p × C→S p × C sending e (v, z) to (v j, z) where v j means raise each coordinate of v to the j th power and e f is well-deﬁned since equivalent then rescale to get a vector of unit length. The map pairs (v, z) ∼ (λv, λjz) in Ej are carried to pairs (v j, z) and (λjv j, λjz) that are equivalent in E1. e e q and Since both f restrict to homeomorphisms in each ﬁber, they extend to maps of Thom spaces that pull a Thom class for E1 back to Thom classes for Ej and Ej. To construct a Thom class for E1, observe that the Thom space T (E1) is homeomorphic to CPm+1, namely, view the sphere S p = S 2m+1 as the unit sphere in Cm+1, and then the inclusion S p × C֓Cm+1 × C = Cm+2 induces a map g : E1→CPm+1 since the equivalence relation deﬁning E1 is (v, z) ∼ (λv, λz) for λ ∈
S 1. It is e 512 Chapter 4 Homotopy Theory evident that g is a homeomorphism onto the complement of the point [0, ···, 0, 1] in CPm+1, so sending the point at inﬁnity in T (E1) to [0, ···, 0, 1] gives an extension of g to a homeomorphism T (E1) ≈ CPm+1. Under this homeomorphism the one-point compactiﬁcations of the ﬁbers of E1 correspond to the 2 spheres S 2 v consisting of [0, ···, 0, 1] and the points [v, z] ∈ CPm+1 with ﬁxed v ∈ S p and varying z ∈ C. v is a CP1 in CPm+1 equivalent to the standard CP1 under a homeomorphism Each S 2 of CPm+1 coming from a linear isomorphism of Cm+2, so a generator γ of H 2(CPm+1) is a Thom class, restricting to a generator of H 2(S 2 v ) for each v. We choose γ to be the Zp reduction of a generator of H 2(CPm+1; Z), so γ is determined up to a sign. A slightly diﬀerent view of Thom classes will be useful. For the bundle E→Lp, for example, we have isomorphisms H ∗(T (E)) ≈ H ∗(T (E), ∞) where ∞ is the compactiﬁcation point e ≈ H ∗(T (E), T (E) − Lp) where Lp is embedded in T (E) as the zero section, so T (E) − Lp deformation retracts onto ∞ ≈ H ∗(E, E − Lp) by excision Thus we can view a Thom class as lying in H ∗(E, E − Lp), and similarly for the bundles Ej. We have projections πj : E→Ej via the projections V0 ⊕ V1 ⊕ ··· ⊕ Vm→Vj in ﬁbers. If τj ∈ H ∗(Ej, Ej − Lp) denotes the Thom class constructed above, then we have the pull
back π ∗ j (τj ) in H ∗(E, E − Lp) is a Thom class for E, as one sees by applying the calculation at the end H ∗(T (E)), the of Example 3.11 in each ﬁber. Under the isomorphism H ∗(E, E − Lp) ≈ j (τj ) corresponds to ±λ(α) since both classes restrict to ±α⊗p in each class ﬁber S p ⊂ T (E) and λ(α) is uniquely determined by its restriction to ﬁbers. j (Lp), and the cup product j (τj ) ∈ H ∗ E, E − π − Now we can ﬁnish the proof of the lemma. The class ∇∗(λ(α)) is obtained by restricting λ(α) ∈ H p(T (E)) to the diagonal T (E0), then pulling back to Lp × S 1 via the quotient map Lp × S 1→T (E0) which collapses the section at inﬁnity to a point. ` ··· ` em where Restricting ej ∈ H 2(E0) is the image of τj under H 2(Ej, Ej − Lp)→H 2(Ej) ≈ H 2(Lp) ≈ H 2(E0), these last two isomorphisms coming from including Lp in Ej and E0 as the zero section, to which they deformation retract. To compute ej, we use the diagram j (τj) to H p(E0, E0 − Lp) ≈ H p(T (E0)) gives τ0 j π ∗ ` e1 Q The Thom class for E1 lies in the upper right group. Following this class across the top of the diagram and then down to the lower left corner gives the element ej. To Steenrod Squares and Powers Section 4.L 513 compute ej we take the alternate route through the lower right corner of the diagram. The image of the Thom class for E1 in the lower right H 2(CPm) is the generator γ since T (E1) = CPm+1. The map f ∗ is multiplication by j since f has
degree j on CP1 ⊂ CPm. And q∗(γ) = ±ω2 since q restricts to a homeomorphism on the 2 cell of Lp in the CW structure deﬁned in Example 2.43. Thus ej = ±jω2, and so ` ωp−1. Since τ0 was the pullback of α τ0 via the projection T (E0)→S 1, when we pull τ0 back to Lp × S 1 via ∇ we get 1 ⊗ α, ⊔⊓ so τ0 ` ··· ` em pulls back to ±m!ωp−1 ⊗ α. Hence a1 = ±m!. ` ··· ` em = ±m!τ0 2 = ±m!τ0 ` ωm ` e1 ` e1 The lemma implies in particular that an is not zero in Zp, so an has a multi- plicative inverse a−1 n in Zp. We then deﬁne P i(α) = (−1)ia−1 n θ2i(p−1)(α) for α ∈ H n(X) The factor a−1 n guarantees that P 0 is the identity. The factor (−1)i is inserted in order to make P i(α) = αp if \|α\| = 2i, as we show next. We know that θ2i(p−1)(α) = αp, so what must be shown is that (−1)ia−1 To do this we need a number theory fact:!2 ≡ (−1)(p+1)/2 mod p. To derive this, note ﬁrst that the product of all the elements ±1, ±2, ···, ±(p − 1)/2!2(−1)(p−1)/2. On the other hand, this group is cyclic of even of Z∗ order, so the product of all its elements is the unique element of order 2, which is (p − 1)/2 p is 2i = 1, or equivalently, a2i = (−1)i. (p − 1)/2 −1, since all the other nontrivial elements cancel their inverses in this product. Thus (p − 1)/2!2 ≡ (−1)(p+1)/2 mod p.!2(
−1)(p−1)/2 ≡ −1 and hence (p − 1)/2 Using the formulas an = (−1)p(p−1)n(n−1)/4an 1 and a1 = ± (p − 1)/2! we then have a2i = (−1)p(p−1)2i(2i−1)/4 (p − 1)/2!2i = (−1)p[(p−1)/2]i(2i−1)(−1)i(p+1)/2 = (−1)i(p−1)/2(−1)i(p+1)/2 = (−1)ip = (−1)i since p is odd. since p and 2i − 1 are odd Theorem 4L.16. The operations P i satisfy the properties (1)–(6) and the Adem relations. Proof: Naturality and the fact that P i(α) = 0 if 2i > \|α\| are inherited from the θi ’s. Property (6) and the other half of (5) have just been shown above. For the Cartan formula we have, for α ∈ Hm and β ∈ H n, λ(α ` β) = (−1)p(p−1)mn/2λ(α)λ(β) and hence i X ω(p−1)(m+n)−i ⊗ θi(α ` β) = (−1)p(p−1)mn/2 Recall that ω2r = ωr 1 = 0. Therefore terms with i even on the left side of the equation can only come from terms with j and k even on the 2 and ω2r +1 = ω1ωr 2, with ω2 j X ω(p−1)m−j ⊗ θj(α) k ω(p−1)n−k ⊗ θk(β) X 514 Chapter 4 Homotopy Theory right side. This leads to the second equality in the following sequence: P i(α ` β) = (−1)ia−1 m+nθ2i(p−1)(α ` β) m+n(−1)p(p−1)mn/2 = (−1)ia−1 θ2
(i−j)(p−1)(α)θ2j(p−1)(β) j (−1)i−j a−1 m θ2(i−j)(p−1)(α)(−1)j a−1 n θ2j(p−1)(β) X P i−j(α)P j(β) = = X j j X Property (4), the invariance of P i under suspension, follows from the Cartan formula just as in the case p = 2, using the fact that P 0 is the only P i that can be nonzero on 1 dimensional classes, by (5). The additivity property follows just as before. It remains to prove the Adem relations for Steenrod powers. We will need a Bockstein calculation: Lemma 4L.17. βθ2k = −θ2k+1. Proof: Let us ﬁrst reduce the problem to showing that β∇∗(λ(ι)) = 0. If we compute β∇∗(λ(ι)) using the product formula for β, we get β i ω(p−1)n−i ⊗ θi(ι) = i X X βω(p−1)n−i ⊗ θi(ι) + (−1)iω(p−1)n−i ⊗ βθi(ι) k ω(p−1)n−2k ⊗ βθ2k(ι) and Since βω2j−1 = ω2j and βω2j = 0, the terms with i = 2k and i = 2k + 1 give k ω(p−1)n−2k ⊗ θ2k+1(ι) − k ω(p−1)n−2k−1 ⊗ βθ2k+1(ι), respectively. Thus the coeﬃcient of ω(p−1)n−2k in β∇∗(λ(ι)) is βθ2k(ι)+θ2k+1(ι), so P if we assume that β∇∗(λ(ι)) = 0, this coeﬃcient must vanish since we are in the tensor product H �
�(L∞) ⊗ H ∗(Kn). So we get βθ2k(ι) = −θ2k+1(ι) and hence βθ2k(α) = −θ2k+1(α) for all α. Note that βθ2k+1 = 0 from the coeﬃcient of ω(p−1)n−2k−1. This also follows from the formula βθ2k = −θ2k+1 since β2 = 0. P P In order to show that β∇∗(λ(ι)) = 0 we ﬁrst compute βλ(ι). We may assume Kn has a single n cell and a single (n + 1) cell, attached by a map of degree p. Let ϕ and ψ be the cellular cochains assigning the value 1 to the n cell and the (n+1) cell, respectively, so δϕ = pψ. In K∧p n we then have (∗) δ(ϕ⊗p) = i (−1)inϕ⊗i ⊗ δϕ ⊗ ϕ⊗(p−i−1) = p (−1)inϕ⊗i ⊗ ψ ⊗ ϕ⊗(p−i−1) X Kn since the latter space has only one (np + 1) cell not in K∧p where the tensor notation means cellular cross product, so for example ϕ⊗p is the cellular cochain dual to the np cell en × ··· × en of K∧p n. The formula (∗) holds also in n, with cellular boundary zero. Namely, this cell is the product of the 1 cell of L∞ and the np cell of Λ K∧p n with one end of this product attached to the np cell by the identity map and the other end by the cyclic permutation T, which has degree +1 since p is odd, so these i X two terms in the boundary of this cell cancel, and there are no other terms since the rest of the attachment of this cell is at the basepoint. Bockstein homomorphisms can be computed using cellular cochain complexes, i(−1)in
ϕ⊗i ⊗ ψ ⊗ ϕ⊗(p−i−1) represents βλ(ι). Via the so the formula (∗) says that P Steenrod Squares and Powers Section 4.L 515 ∆ Γ Λ quotient map Kn→ Kn, the class λ(ι) pulls back to a class γ(ι) with βγ(ι) also repi(−1)inϕ⊗i ⊗ ψ ⊗ ϕ⊗(p−i−1). To see what happens when we pull βγ(ι) Γ Kn, consider the commutative diP resented by back to β∇∗(λ(ι)) via the inclusion L∞× Kn ֓ agram at the right. In the lefthand square the maps π ∗ are induced by the covering space projections π : S ∞ × K∧p Kn and π : S ∞ × Kn→L∞× Kn arising from the free Zp actions. The vertical maps are inΓ duced by the diagonal inclusion S ∞× K ֓ S ∞ × K∧p n. The maps τ are the transfer X→X is a p sheeted homomorphisms deﬁned in §3.G. Recall the deﬁnition: If π : covering space, a chain map C∗(X)→C∗( X) is deﬁned by sending a singular simplex e k→X to the sum of its p lifts to σ : X, and τ is the induced map on cohomology. e The key property of τ is that τπ ∗ : H ∗(X)→H ∗(X) is multiplication by p, for any k→X back to X choice of coeﬃcient group, since when we project the p lifts of σ : X is given the lifted CW structure, then τ we get pσ. When X is a CW complex and n → ∆ e can also be deﬁned in cellular cohomology by the same procedure. e Let us compute the value of the upper τ in the diagram on 1 ⊗ ψ ⊗ ϕ⊗(p−1) where ‘
1 ’ is the cellular cocycle assigning the value 1 to each 0 cell of S ∞. By the deﬁnition of τ we have τ(1 ⊗ ψ ⊗ ϕ⊗(p−1)) = n permutes the factors cyclically. It does not matter whether T moves coordinates one unit left- i T i(ψ ⊗ ϕ⊗(p−1)) where T : K∧p n →K∧p P wards or one unit rightwards since we are summing over all the powers of T, so let us say T moves coordinates rightward. Then T (ψ ⊗ ϕ⊗(p−1)) = ϕ ⊗ ψ ⊗ ϕ⊗(p−2), with the last ϕ moved into the ﬁrst position. This move is achieved by transposing this ϕ with each of the preceding p − 2 ϕ ’s and with ψ. Transposing two ϕ ’s introduces a sign (−1)n2, and transposing ϕ with ψ introduces a sign (−1)n(n+1) = +1, by the commutativity property of cross product. Thus the total sign introduced by T is (−1)n2(p−2), which equals (−1)n since p is odd. Each successive iterate of T also introduces a sign of (−1)n, so T i introduces a sign (−1)in for 0 ≤ i ≤ p − 1. Thus τ(1 ⊗ ψ ⊗ ϕ⊗(p−1)) = T i(ψ ⊗ ϕ⊗(p−1)) = i X i X (−1)inϕ⊗i ⊗ ψ ⊗ ϕ⊗(p−i−1) As observed earlier, this last cocycle represents the class βγ(ι). Since βγ(ι) is in the image of the upper τ in the diagram, the image of βγ(ι) in H ∗(L∞× Kn), which is ∇∗(βλ(ι)), is in the image of the lower τ since the righthand square commutes. The map π ∗ in the lower row is obviously onto since S
∞ is contractible, so ∇∗(βλ(ι)) is in the image of the composition τπ ∗ across the bottom of the diagram. But this composition is multiplication by p, which is zero for Zp coeﬃcients, so β∇∗(λ(ι)) = ∇∗(βλ(ι)) = 0. ⊔⊓ The derivation of the Adem relations now follows the pattern for the case p = 2. We had the formula ∇∗ 2 λ∗ 2 (ι) = i,j ω(p−1)pn−i ⊗ θi ω(p−1)n−j ⊗ θj(ι). Since we are P 516 Chapter 4 Homotopy Theory letting p = 2m + 1, this can be rewritten as. The only nonzero θi ’s are θ2i(p−1) = (−1)ianP i and θ2i(p−1)+1 = −βθ2i(p−1) so we have ω2mn−j ⊗ θj(ι) i,j ω2mpn−i ⊗ θi P ω2mpn−i ⊗ θi i,j X = ω2mn−j ⊗ θj(ι) (−1)i+j a2mnanω2m(pn−2i) ⊗ P i i,j (−1)i+j a2mnanω2m(pn−2i) ⊗ P i (−1)i+j a2mnanω2m(pn−2i)−1 ⊗ βP i (−1)i+j a2mnanω2m(pn−2i)−1 ⊗ βP i i,j i,j X X X − − + i,j X ω2m(n−2j) ⊗ P j(ι) ω2m(n−2j)−1 ⊗ βP j(ι) ω2m(n−2j) ⊗ P j(ι) ω2m(n−2j)−1 ⊗ βP j(ι) Since m and n will
be ﬁxed throughout the discussion, we may factor out the nonzero constant a2mnan. Then applying the Cartan formula to expand the P i terms, using also the formulas P k(ω2r ) = ω2r +2k(p−1)+1 de- rived earlier in the section, we obtain ω2r +2k(p−1) and P k(ω2r +1) = r k r k i,j,k X i,j,k X i,j,k X i,j,k X i,j,k − − + − X (−1)i+j m(n−2j) k ω2m(pn−2i) ⊗ ω2m(n−2j+2k) ⊗ P i−kP j(ι) (−1)i+j (−1)i+j m(n−2j)−1 k m(n−2j) k ω2m(pn−2i) ⊗ ω2m(n−2j+2k)−1 ⊗ P i−kβP j(ι) ω2m(pn−2i)−1 ⊗ ω2m(n−2j+2k) ⊗ βP i−kP j(ι) (−1)i+j m(n−2j)−1 k (−1)i+j m(n−2j)−1 k ω2m(pn−2i)−1 ⊗ ω2m(n−2j+2k) ⊗ P i−kβP j(ι) ω2m(pn−2i)−1 ⊗ ω2m(n−2j+2k)−1 ⊗ βP i−kβP j(ι) Letting ℓ = mn+j −k, so that n−2j +2k = pn−2ℓ, the ﬁrst of these ﬁve summations becomes i,j,ℓ (−1)i+j m(n−2j) mn+j−ℓ ω2m(pn−2i) ⊗ ω2m(pn−2�
�) ⊗ P i+ℓ−mn−jP j(ι) and similarly for the other four summations. X 2 λ∗ Now we bring in the symmetry property ϕr s = (−1)r s+mnp ϕsr, where, as before, ∇∗ 2 (ι) = r,s ωr ⊗ ωs ⊗ ϕr s. Of the ﬁve summations, only the ﬁrst has both ω terms with even subscripts, namely r = 2m(pn − 2i) and s = 2m(pn − 2ℓ), so the coeﬃcient of ωr ⊗ ωs in this summation must be symmetric with respect to switching i and ℓ, up to a sign which will be + if we choose n to be even, as we will do. This P gives the relation (1) (−1)i+j j X m(n−2j) mn+j−ℓ P i+ℓ−mn−jP j(ι) = j X (−1)ℓ+j m(n−2j) mn+j−i P i+ℓ−mn−jP j(ι) Similarly, the second, third, and fourth summations involve ω ’s with subscripts of opposite parity, yielding the relation (2) j X j X (−1)ℓ+j (−1)i+j m(n−2j)−1 mn+j−ℓ P i+ℓ−mn−jβP j(ι) = βP i+ℓ−mn−jP j(ι) − j X m(n−2j) mn+j−i (−1)ℓ+j m(n−2j)−1 mn+j−i P i+ℓ−mn−jβP j(ι) The relations (1) and (2) will yield the two Adem relations, so we will not need to consider the relation arising from the ﬁfth summation. Steenrod Squares and Powers Section 4.L 517 To get the ﬁrst Adem relation from (1) we choose n and ℓ
so that the left side of (1) has only one term, namely we take n = 2(1 + p + ··· + pr −1) + 2s and ℓ = mn + s for given integers r and s. Then m(n−2j) mn+j−ℓ = pr −1−(p−1)(j−s) j−s and if r is large, this binomial coeﬃcient is 1 if j = s and 0 otherwise since if the rightmost nonzero digit in the p adic expansion of the ‘denominator’ j − s is x, then the corresponding digit of the ‘numerator’ (p − 1)[(1 + p + ··· + pr −1) − (j − s)] is = 0. Then (1) obtained by reducing (p − 1)(1 − x) mod p, giving x − 1, and x−1 x becomes (−1)i+s P iP s(ι) = or P iP s(ι) = = = (−1)ℓ+j m(n−2j) mn+j−i P i+s−jP j(ι) (−1)i+j (−1)i+j (−1)i+j m(n−2j) mn+j−i P i+s−jP j(ι) P i+s−jP j(ι) pr +(p−1)(s−j)−1 i−pj m(n−2j) i−pj +s−jP j(ι) since ℓ ≡ s mod 2 since x y = x x−y If r is large and i < ps, the term pr in the binomial coeﬃcient can be omitted since we may assume i ≥ pj, hence j < s, so −1 + (p − 1)(s − j) ≥ 0 and the pr has no eﬀect on the binomial coeﬃcient if r is large. This shows the ﬁrst Adem relation holds for the class ι, and the general case follows as in the case p = 2. To get the second Adem relation we choose n = 2pr + 2s and ℓ = mn + s. Reasoning as before, the left
side of (2) then reduces to (−1)i+s P iβP s(ι) and (2) becomes P iβP s(ι) = j X (−1)i+j − (p−1)(pr +s−j) i−pj βP i+s−jP j(ι) (p−1)(pr +s−j)−1 i−pj−1 (−1)i+j P i+s−jβP j(ι) j X This time the term pr can be omitted if r is large and i ≤ ps. ⊔⊓ Exercises 1. Determine all cohomology operations H 1(X; Z)→H n(X; Z), H 2(X; Z)→H n(X; Z), and H 1(X; Zp)→H n(X; Zp) for p prime. 2. Use cohomology operations to show that the spaces (S 1 × CP∞)/(S 1 × {x0}) and S 3 × CP∞ are not homotopy equivalent. 3. Since there is a ﬁber bundle S 2→CP5→HP2 by Exercise 35 in §4.2, one might ask whether there is an analogous bundle S 4→HP5→OP2. Use Steenrod powers for the prime 3 to show that such a bundle cannot exist. [The Gysin sequence can be used to determine the map on cohomology induced by the bundle projection HP5→OP2.] 4. Show there is no ﬁber bundle S 7→S 23→OP2. [Compute the cohomology ring of the mapping cone of the projection S 23→OP2 via Poincar´e duality or the Thom isomorphism.] 518 Chapter 4 Homotopy Theory 5. Show that the subalgebra of A 2 generated by Sqi for i ≤ 2 has dimension 8 as a vector space over Z2, with multiplicative structure encoded in the following diagram, where diagonal lines indicate left-multiplication by Sq1 and horizontal lines indicate left-multiplication by Sq2. Topology of Cell Complexes Here we collect a number of basic topological facts about CW complexes for con- venient reference. A few related facts about manifolds are also proved. Let us ﬁrst recall from Chapter 0 that
a CW complex is a space X constructed in the following way: (1) Start with a discrete set X 0, the 0 cells of X. (2) Inductively, form the n skeleton X n from X n−1 by attaching n cells en ϕα : S n−1→X n−1. This means that X n is the quotient space of X n−1 the identiﬁcations x ∼ ϕα(x) for x ∈ ∂Dn image of Dn α under the quotient map. α via maps α Dn α under α is the homeomorphic ` α. The cell en α − ∂Dn (3) X = n X n with the weak topology: A set A ⊂ X is open (or closed) iﬀ A ∩ X n is open (or closed) in X n for each n. S Note that condition (3) is superﬂuous when X is ﬁnite-dimensional, so that X = X n for some n. For if A is open in X = X n, the deﬁnition of the quotient topology on X n implies that A ∩ X n−1 is open in X n−1, and then by the same reasoning A ∩ X n−2 is open in X n−2, and similarly for all the skeleta X n−i. Each cell en α ֓ X n−1 α has its characteristic map α, which is by deﬁnition the composiα→X n ֓ X. This is continuous since it is a composition of tion Dn continuous maps, the inclusion X n ֓ X being continuous by (3). The restriction of α to the interior of Dn α is a homeomorphism onto en α. α Dn ` Φ Φ open (or closed) iﬀ −1 α (A) is open (or closed) in Dn In one direction this follows from continuity of the An alternative way to describe the topology on X is to say that a set A ⊂ X is α. α ’s, and in the other direction, α, and suppose by induction on n that suppose A ∩ X n−1 is open in X n−1. Then since α for all α, A ∩ X n is open Φ in X n by
the deﬁnition of the quotient topology on X n. Hence by (3), A is open in X. α for each −1 α (A) is open in Dn Φ −1 α (A) is open in Dn α for each characteristic map Φ Φ Φ A consequence of this characterization of the topology on X is that X is a quotient Φ space of n,α Dn α. ` 520 Appendix Topology of Cell Complexes A subcomplex of a CW complex X is a subspace A ⊂ X which is a union of cells of X, such that the closure of each cell in A is contained in A. Thus for each cell in A, the image of its attaching map is contained in A, so A is itself a CW complex. Its CW complex topology is the same as the topology induced from X, as one sees by noting inductively that the two topologies agree on An = A ∩ X n. see by induction over skeleta that a subcomplex is a closed subspace. Conversely, a It is easy to subcomplex could be deﬁned as a closed subspace which is a union of cells. A ﬁnite CW complex, that is, one with only ﬁnitely many cells, is compact since attaching a single cell preserves compactness. A sort of converse to this is: Proposition A.1. A compact subspace of a CW complex is contained in a ﬁnite subcomplex. Proof: First we show that a compact set C in a CW complex X can meet only ﬁnitely many cells of X. Suppose on the contrary that there is an inﬁnite sequence of points xi ∈ C all lying in distinct cells. Then the set S = {x1, x2, ···} is closed in X. Namely, assuming S ∩ X n−1 is closed in X n−1 by induction on n, then for each cell en α of X, ϕ−1 α (S) is closed in ∂Dn α, so −1 α. Therefore S ∩X n is closed in X n for each n, hence S is closed α (S) is closed in Dn in X. The same argument shows that any subset of S is closed, so S has the discrete Φ topology. But it is compact, being a closed
subset of the compact set C. Therefore S −1 α (S) consists of at most one more point in Dn α, and Φ must be ﬁnite, a contradiction. Since C is contained in a ﬁnite union of cells, it suﬃces to show that a ﬁnite union of cells is contained in a ﬁnite subcomplex of X. A ﬁnite union of ﬁnite subcomplexes is again a ﬁnite subcomplex, so this reduces to showing that a single cell en α is contained in a ﬁnite subcomplex. The image of the attaching map ϕα for en α is compact, hence by induction on dimension this image is contained in a ﬁnite subcomplex A ⊂ X n−1. So en α is contained in the ﬁnite subcomplex A ∪ en α. ⊔⊓ Now we can explain the mysterious letters ‘CW’, which refer to the following two properties satisﬁed by CW complexes: (1) Closure-ﬁniteness: The closure of each cell meets only ﬁnitely many other cells. This follows from the preceding proposition since the closure of a cell is compact, being the image of a characteristic map. (2) Weak topology: A set is closed iﬀ it meets the closure of each cell in a closed set. For if a set meets the closure of each cell in a closed set, it pulls back to a closed set under each characteristic map, hence is closed by an earlier remark. In J. H. C. Whitehead’s original deﬁnition of CW complexes these two properties played a more central role. The following proposition contains essentially this deﬁnition. Topology of Cell Complexes Appendix 521 Proposition A.2. Given a Hausdorﬀ space X and a family of maps then these maps are the characteristic maps of a CW complex structure on X iﬀ : α : Dn α→X, (i) Each α restricts to a homeomorphism from int Dn and these cells are all disjoint and their union is X. α onto its image, a cell en α ⊂ X, Φ Φ (ii) For each cell en α, α(∂Dn
α) is contained in the union of a ﬁnite number of cells of dimension less than n. Φ (iii) A subset of X is closed iﬀ it meets the closure of each cell of X in a closed set. Condition (iii) can be restated as saying that a set C ⊂ X is closed iﬀ −1 α (C) is closed in Dn α for all α, since a map from a compact space onto a Hausdorﬀ space is a quotient map. In particular, if there are only ﬁnitely many cells then (iii) is automatic α→X is a map from a compact space onto a since in this case the projection α Dn Φ Hausdorﬀ space, hence is a quotient map. ` For an example where all the conditions except the ﬁniteness hypothesis in (ii) are satisﬁed, take X to be D2 with its interior as a 2 cell and each point of ∂D2 as a 0 cell. The identity map of D2 serves as the α for the 2 cell. Condition (iii) is satisﬁed since it is a nontrivial condition only for the 2 cell. Φ Proof: We have already taken care of the ‘only if’ implication. For the converse, suppose inductively that X n−1, the union of all cells of dimension less than n, is a α ’s as characteristic maps. The induction can start CW complex with the appropriate α→X n be given by the inclusion on X n−1 and the with X −1 = ∅. Let f : X n−1 α Dn Φ α for all the n cells of X. This is a continuous surjection, and if we can show maps it is a quotient map, then X n will be obtained from X n−1 by attaching the n cells en α. Thus if C ⊂ X n is such that f −1(C) is closed, we need to show that C ∩ e m β is closed for all cells em β of X, the bar denoting closure. ` Φ β is closed since e m There are three cases. If m < n then f −1(C) closed implies C ∩ X n−1 closed, β ⊂ X n−1. If m = n then em
hence C ∩ e m α, so f −1(C) closed implies f −1(C) ∩ Dn α is closed, hence compact, hence its image C ∩ e n α under f is compact and therefore closed. Finally there is the case m > n. Then C ⊂ X n implies C ∩ e m β ). The latter space is contained in a ﬁnite union of e ℓ γ ’s with ℓ < m. By induction on m, each C ∩ e ℓ γ is closed. Hence the intersection Φ of C with the union of the ﬁnite collection of e ℓ γ ’s is closed. Intersecting this closed β, we conclude that C ∩ e m set with e m β is closed. β is one of the cells en β(∂Dm β ⊂ It remains only to check that X has the weak topology with respect to the X n ’s, that is, a set in X is closed iﬀ it intersects each X n in a closed set. The preceding argument with C = X n shows that X n is closed, so a closed set intersects each X n in a closed set. Conversely, if a set C intersects X n in a closed set, then C intersects each e n ⊔⊓ α in a closed set, so C is closed in X by (iii). 522 Appendix Topology of Cell Complexes ε (A) = A ∩ X 0. Then we deﬁne N n+1 Next we describe a convenient way of constructing open neighborhoods Nε(A) of subsets A of a CW complex X, where ε is a function assigning a number εα > 0 to each α of X. The construction is inductive over the skeleta X n, so suppose we have cell en ε (A), a neighborhood of A ∩ X n in X n, starting the process already constructed N n with N 0 (A) by specifying its preimage under −1 is the the characteristic map α α α (A) − ∂Dn+1 in Dn+1 − ∂Dn+1, −1 union of two parts: an open εα neighborhood of Φ and a product (1 − εα, 1]× with respect to
‘spherical’ coordinates (r, θ) in Dn+1, where r ∈ [0, 1] is the radial coordinate and θ lies in ∂Dn+1 = S n. Then we deﬁne Nε(A) = ε (A). This is an open set in X since it pulls back to an open set under each characteristic map. α : Dn+1→X of each cell en+1 ε (A), namely, N n+1 ε n N n N n (A) −1 α Φ Φ Φ ε S Proposition A.3. CW complexes are normal, and in particular, Hausdorﬀ. Φ distance apart, since otherwise by compactness we would have a sequence in ε (A) and N n α : Dn+1→X, observe that Proof: Points are closed in a CW complex X since they pull back to closed sets under α. For disjoint closed sets A and B in X, we show that Nε(A) all characteristic maps and Nε(B) are disjoint for small enough εα ’s. In the inductive process for building these open sets, assume N n ε (B) have been chosen to be disjoint. For a −1 α (B) are a positive and characteristic map −1 α (B) Φ Φ α (B) in ∂Dn+1 of distance zero from −1 N n, but −1 α (B) ∩ ∂Dn+1 in ∂Dn+1 −1 N n is a neighborhood of ε (B) α Φ −1 −1 N n α (A) are a positive distance. Similarly, ε (B) ε (A) α Φ −1 −1 α (B) are a positive distance apart. So a small enough εα α (A) and Φ Φ Φ in Dn+1. N n+1 N n+1 ⊔⊓ disjoint from (B) (A) ε ε Φ converging to a point of this is impossible since ε (A) Φ ε (A) disjoint from Proposition A.4. Each point in a CW complex has arbitrarily small contractible open neighborhoods, so CW complexes are locally
contractible. apart. Also, −1 α Φ will make −1 α N n N n and −1 α −1 α −1 α Φ Φ Φ Φ Φ (x) by sliding outward along radial segments in cells en Proof: Given a point x in a CW complex X and a neighborhood U of x in X, we can choose the εα ’s small enough so that Nε(x) ⊂ U by requiring that the closure of N n ε (x) be contained in U for each n. It remains to see that Nε(x) is contractible. If x ∈ Xm − Xm−1 and n > m we can construct a deformation retraction of N n ε (x) onto N n−1 β, the images under the ε β of radial segments in Dn. A deformation retraction of Nε(x) characteristic maps onto N m ε (x) onto N n−1 (x) being stationary outside this t interval. Finally, N m ε (x) is an open ball about x, and so ⊔⊓ deformation retracts onto x. ε (x) is then obtained by performing the deformation retraction of N n (x) during the t interval [1/2n, 1/2n−1], points of N n ε (x)−N n−1 Φ ε ε In particular, CW complexes are locally path-connected. So a CW complex is path- connected iﬀ it is connected. Topology of Cell Complexes Appendix 523 Proposition A.5. For a subcomplex A of a CW complex X, the open neighborhood Nε(A) deformation retracts onto A if εα < 1 for all α. Proof: In each cell of X − A, Nε(A) is a product neighborhood of the boundary of this cell, so a deformation retraction of Nε(A) onto A can be constructed just as in ⊔⊓ the previous proof. Note that for subcomplexes A and B of X, we have Nε(A) ∩ Nε(B) = Nε(A ∩ B). This implies for example that the van Kampen theorem and Mayer-Vietoris sequences hold for decompositions X = A ∪ B into subcomplexes A and
B as well as into open sets A and B. α of all cells en A map f : X→Y with domain a CW complex is continuous iﬀ its restrictions to the closures e n α are continuous, and it is useful to know that the same is true for homotopies ft : X→Y. With this objective in mind, let us introduce a little terminology. A topological space X is said to be generated by a collection of α Xα and a set A ⊂ X is closed iﬀ A ∩ Xα is closed in Xα for subspaces Xα if X = each α. Equivalently, we could say ‘open’ instead of ‘closed’ here, but ‘closed’ is more S convenient for our present purposes. As noted earlier, though not in these words, a CW complex X is generated by the closures e n α of its cells en α. Since every ﬁnite subcomplex of X is a ﬁnite union of closures e n α, X is also generated by its ﬁnite subcomplexes. It follows that X is also generated by its compact subspaces, or more brieﬂy, X is compactly generated. Proposition A.15 later in the Appendix asserts that if X is a compactly generated Hausdorﬀ space and Z is locally compact, then X × Z, with the product topology, is compactly generated. In particular, X × I is compactly generated if X is a CW complex. Since every compact set in X × I is contained in the product of a compact subspace of X with I, hence in the product of a ﬁnite subcomplex of X with I, such product subspaces also generate X × I. Since such a product subspace is a ﬁnite union of products e n α × I, it is also true that X × I is generated by its subspaces e n α × I. This implies that a homotopy F : X × I→Y is continuous iﬀ its restrictions to the subspaces e n α × I are continuous, which is the statement we were seeking. Products of CW Complexes There are some unexpected point-set-topological subtleties that arise with prod- ucts of CW complexes. As we shall show, the product of two CW complexes does have a
natural CW structure, but its topology is in general ﬁner, with more open sets, than the product topology. However, the distinctions between the two topologies are rather small, and indeed nonexistent in most cases of interest, so there is no real problem for algebraic topology. Given a space X and a collection of subspaces Xα whose union is X, these subspaces generate a possibly ﬁner topology on X by deﬁning a set A ⊂ X to be open 524 Appendix Topology of Cell Complexes iﬀ A ∩ Xα is open in Xα for all α. The axioms for a topology are easily veriﬁed for this deﬁnition. In case {Xα} is the collection of compact subsets of X, we write Xc for this new compactly generated topology. It is easy to see that X and Xc have the same compact subsets, and the two induced topologies on these compact subsets coincide. If X is compact, or even locally compact, then X = Xc, that is, X is compactly generated. Theorem A.6. For CW complexes X and Y with characteristic maps β, the product maps β are the characteristic maps for a CW complex structure on (X × Y )c. If either X or Y is compact or more generally locally compact, then (X × Y )c = X × Y. Also, (X × Y )c = X × Y if both X and Y have countably many cells. α and α × Φ Φ Ψ Ψ Proof: For the ﬁrst statement it suﬃces to check that the three conditions in Proposition A.2 are satisﬁed when we take the space ‘ X ’ there to be (X × Y )c. The ﬁrst two conditions are obvious. For the third, which says that (X × Y )c is generated by the products e m β, observe that every compact set in X × Y is contained in the product of its projections onto X and Y, and these projections are compact and hence α × e n contained in ﬁnite subcomplexes of X and Y, so the original compact set is contained in a ﬁnite union of products e m β generate (X × Y )c. The second assertion of
the theorem is a special case of Proposition A.15, hav- β. Hence the products ing nothing to do with CW complexes, which says that a product X × Y is compactly generated if X is compactly generated Hausdorﬀ and Y is locally compact. For the last statement of the theorem, suppose X and Y each have at most countably many cells. For an open set W ⊂ (X × Y )c and a point (a, b) ∈ W we need to ﬁnd a product U × V ⊂ W with U an open neighborhood of a in X and V an open neighborhood of b in Y. Choose ﬁnite subcomplexes X1 ⊂ X2 ⊂ ··· of X with X = i Xi, and similarly for Y. We may assume a ∈ X1 and b ∈ Y1. Since the two topologies agree on X1 × Y1, there is a compact product neighborhood K1 × L1 ⊂ W of (a, b) in X1 × Y1. Assuming inductively that Ki × Li ⊂ W has been constructed in Xi × Yi, we would like to construct Ki+1 × Li+1 ⊂ W as a compact neighborhood of Ki × Li in Xi+1 × Yi+1. To do this, we ﬁrst choose for each x ∈ Ki compact neighborhoods Kx of x in Xi+1 and Lx of Li in Yi+1 such that Kx × Lx ⊂ W, using the compactness of Li. By compactness of Ki, a ﬁnite number of the Kx ’s cover Ki. Let Ki+1 be the union of these Kx ’s and let Li+1 be the intersection of the corresponding Lx ’s. This deﬁnes the desired Ki+1 × Li+1. Let Ui be the interior of Ki in Xi, so Ui ⊂ Ui+1 for each i. The union U = i Ui is then open in X since it intersects each Xi in a union of open sets and the Xi ’s generate X. In the same way the Li ’s yield an open set V ⊔⊓ in Y. Thus we have a product of open sets U × V ⊂ W containing (a, b). S
S We will describe now an example from [Dowker 1952] where the product topology on X × Y diﬀers from the CW topology. Both X and Y will be graphs consisting of Topology of Cell Complexes Appendix 525 inﬁnitely many edges emanating from a single vertex, with uncountably many edges for X and countably many for Y. W Let X = s Is where Is is a copy of the interval [0, 1] and the index s ranges over all inﬁnite sequences s = (s1, s2, ···) of positive integers. The wedge sum is formed at the 0 endpoint of Is. Similarly we let Y = j Ij but with j varying just over positive integers. Let psj be the point (1/sj, 1/sj) ∈ Is × Ij ⊂ X × Y and let P be the union of all these points psj. Thus P consists of a single point in each 2 cell of X × Y, so P is closed in the CW topology on X × Y. We will show it is not closed in the product topology by showing that (x0, y0) lies in its closure, where x0 is the common endpoint of the intervals Is and y0 is the common endpoint of the intervals Ij. W s [0, as) and V = A basic open set containing (x0, y0) in the product topology has the form U × V where U = It suﬃces to show that P has nonempty intersection with U × V. Choose a sequence t = (t1, t2, ···) with tj > j and tj > 1/bj for all j, and choose an integer k > 1/at. Then tk > k > 1/at hence 1/tk < at. We also have 1/tk < bk. So (1/tk, 1/tk) is a point of P that lies in [0, at)× [0, bk) and hence in U × V. j [0, bj). W W Euclidean Neighborhood Retracts At certain places in this book it is desirable to know that a given compact space is a retract of a ﬁnite simplicial complex, or equivalently (as we shall see) a retract of a neighborhood in some Euclidean space. For example,
this condition occurs in the Lefschetz ﬁxed point theorem, and it was used in the proof of Alexander duality. So let us study this situation in more detail. Theorem A.7. A compact subspace K of Rn is a retract of some neighborhood iﬀ K is locally contractible in the weak sense that for each x ∈ K and each neighborhood U of x in K there exists a neighborhood V ⊂ U of x such that the inclusion V ֓ U is nullhomotopic. Note that if K is a retract of some neighborhood, then it is a retract of every smaller neighborhood, just by restriction of the retraction. So it does not matter if we require the neighborhoods to be open. Similarly it does not matter if the neighbor- hoods U and V in the statement of the theorem are required to be open. Proof: Let us do the harder half ﬁrst, constructing a retraction of a neighborhood of K onto K under the local contractibility assumption. The ﬁrst step is to put a CW structure on the open set X = Rn − K, with the size of the cells approaching zero near K. Consider the subdivision of Rn into unit cubes of dimension n with vertices at the points with integer coordinates. Call this collection of cubes C0. For an integer k > 0, we can subdivide the cubes of C0 by taking n dimensional cubes of edgelength 1/2k with vertices having coordinates of the form i/2k for i ∈ Z. Denote this collection of cubes by Ck. Let A0 ⊂ C0 be the set of cubes disjoint from K, and 526 Appendix Topology of Cell Complexes inductively, let Ak ⊂ Ck be the set of cubes disjoint from K and not contained in cubes of Aj for j < k. The open set X is then the union of all the cubes in the combined collection A = k Ak. Note that the collection A is locally ﬁnite: Each point of X has a neighborhood meeting only ﬁnitely many cubes in A, since the point has a positive S distance from the closed set K. If two cubes of A intersect, their intersection is an i dimensional face of one of them for some i < n. Likewise, when two faces of cubes of A intersect, their intersection is a face of one of them. This
implies that the open faces of cubes of A that are minimal with respect to inclusion among such faces form the cells of a CW structure on X, since the boundary of such a face is a union of such faces. The vertices of this CW structure are thus the vertices of all the cubes of A, and the n cells are the interiors of the cubes of A. Next we deﬁne inductively a subcomplex Z of this CW structure on X and a map r : Z→K. The 0 cells of Z are exactly the 0 cells of X, and we let r send each 0 cell to the closest point of K, or if this is not unique, any one of the closest points of K. Assume inductively that Z k and r : Z k→K have been deﬁned. For a cell ek+1 of X with boundary in Z k, if the restriction of r to this boundary extends over ek+1 then we include ek+1 in Z k+1 and we let r on ek+1 be such an extension that is not too large, say an extension for which the diameter of its image r (ek+1) is less than twice the inﬁmum of the diameters for all possible extensions. This deﬁnes Z k+1 and r : Z k+1→K. At the end of the induction we set Z = Z n. It remains to verify that by letting r equal the identity on K we obtain a continuous retraction Z ∪ K→K, and that Z ∪ K contains a neighborhood of K. Given a point x ∈ K, let U be a ball in the metric space K centered at x. Since K is locally contractible, we can choose a ﬁnite sequence of balls in K centered at x, of the form U = Un ⊃ Vn ⊃ Un−1 ⊃ Vn−1 ⊃ ··· ⊃ U0 ⊃ V0, each ball having radius equal to some small fraction of the radius of the preceding one, and with Vi contractible in Ui. Let B ⊂ Rn be a ball centered at x with radius less than half the radius of V0, and let Y be the subcomplex of X formed by the cells whose closures are contained in B. Thus Y ∪ K contains a neighborhood of x in Rn. By the choice
of B and the deﬁnition of r on 0 cells we have r (Y 0) ⊂ V0. Since V0 is contractible in U0, r is deﬁned on the 1 cells of Y. Also, r (Y 1) ⊂ V1 by the deﬁnition of r on 1 cells and the fact that U0 is much smaller than V1. Similarly, by induction we have r deﬁned on Y i with r (Y i) ⊂ Vi for all i. In particular, r maps Y to U. Since U could be arbitrarily small, this shows that extending r by the identity map on K gives a continuous map r : Z ∪ K→K. And since Y ⊂ Z, we see that Z ∪ K contains a neighborhood of K by the earlier observation that Y ∪ K contains a neighborhood of x. Thus r : Z ∪ K→K retracts a neighborhood of K onto K. Now for the converse. Since open sets in Rn are locally contractible, it suﬃces to show that a retract of a locally contractible space is locally contractible. Let r : X→A Topology of Cell Complexes Appendix 527 be a retraction and let U ⊂ A be a neighborhood of a given point x ∈ A. If X is locally contractible, then inside the open set r −1(U) there is a neighborhood V of x that is contractible in r −1(U), say by a homotopy ft : V→r −1(U). Then V ∩ A is ⊔⊓ contractible in U via the restriction of the composition r ft. A space X is called a Euclidean neighborhood retract or ENR if for some n there exists an embedding i : X ֓ Rn such that i(X) is a retract of some neighborhood in Rn. The preceding theorem implies that the existence of the retraction is independent of the choice of embedding, at least when X is compact. Corollary A.8. A compact space is an ENR iﬀ it can be embedded as a retract of a ﬁnite simplicial complex. Hence the homology groups and the fundamental group of a compact ENR are ﬁnitely generated. Proof: A ﬁnite simplicial complex
K with n vertices is a subcomplex of a simplex n−1, and hence embeds in Rn. The preceding theorem then implies that K is a retract of some neighborhood in Rn, so any retract of K is also a retract of such a ∆ neighborhood, via the composition of the two retractions. Conversely, let K be a compact space that is a retract of some open neighborhood U in Rn. Since K is compact n, say by repeated it is bounded, lying in some large simplex barycentric subdivision, so that all simplices of the subdivision have diameter less n ⊂ Rn. Subdivide than the distance from K to the complement of U. Then the union of all the sim- ∆ ∆ plices in this subdivision that intersect K is a ﬁnite simplicial complex that retracts onto K via the restriction of the retraction U→K. ⊔⊓ Corollary A.9. Every compact manifold, with or without boundary, is an ENR. Proof: Manifolds are locally contractible, so it suﬃces to show that a compact manifold M can be embedded in Rk for some k. If M is not closed, it embeds in the closed manifold obtained from two copies of M by identifying their boundaries. So it suﬃces to consider the case that M is closed. By compactness there exist ﬁnitely many closed balls Bn i ⊂ M whose interiors cover M, where n is the dimension of M. Let fi : M→S n be the quotient map collapsing the complement of the interior to a point. These fi ’s are the components of a map f : M→(S n)m which is of Bn i injective since if x and y are distinct points of M with x in the interior of Bn i, say, then fi(x) ≠ fi(y). Composing f with an embedding (S n)m ֓ Rk, for example the product of the standard embeddings S n ֓ Rn+1, we obtain a continuous injection M ֓ Rk, and this is a homeomorphism onto its image since M is compact. ⊔⊓ Corollary A.10. Every ﬁnite CW complex is an ENR. Proof: Since CW complexes are locally contractible, it
suﬃces to show that a ﬁnite CW complex can be embedded in some Rn. This is proved by induction on the number 528 Appendix Topology of Cell Complexes of cells. Suppose the CW complex X is obtained from a subcomplex A by attaching a cell ek via a map f : S k−1→A, and suppose that we have an embedding A ֓ Rm. Then we can embed X in Rk × Rm × R as the union of Dk × {0}× {0}, {0}× A× {1}, and all line segments joining points (x, 0, 0) and (0, f (x), 1) for x ∈ S k−1. ⊔⊓ Spaces Dominated by CW Complexes We have been considering spaces which are retracts of ﬁnite simplicial complexes, and now we show that such spaces have the homotopy type of CW complexes. In fact, we can just as easily prove something a little more general than this. A space Y is r-----→ Y with r i ≃ 11. This makes the notion of a retract into something that depends only on the homotopy said to be dominated by a space X if there are maps Y i-----→ X types of the spaces involved. Proposition A.11. A space dominated by a CW complex is homotopy equivalent to a CW complex. Proof: Recall from §3.F that the mapping telescope T (f1, f2, ···) of a sequence of obtained maps X1 by identifying (x, i + 1) ∈ Xi × [i, i + 1] with (f (x), i + 1) ∈ Xi+1 × [i + 1, i + 2]. We shall need the following elementary facts: f2------------→ X3 ------------→ ··· is the quotient space of Xi × [i, i + 1] f1------------→ X2 ` i (1) T (f1, f2, ···) ≃ T (g1, g2, ···) if fi ≃ gi for each i. (2) T (f1, f2, ···) ≃ T (f2, f3, ···). (3) T (f1, f2, ···) ≃ T (f2f1,
f4f3, ···). i ` ` by attaching Xi × {i} The second of these is obvious. To prove the other two we will use Proposition 0.18, whose proof applies not just to CW pairs but to any pair (X1, A) for which there is a deformation retraction of X1 × I onto X1 × {0} ∪ A× I. To prove (1) we regard T (f1, f2, ···) as being obtained from Xi × [i, i + 1]. i Then we can obtain T (g1, g2, ···) by varying the attaching map by homotopy. To prove (3) we view T (f1, f2, ···) as obtained from the disjoint union of the mapping. By sliding the attachment of cylinders M(f2i) by attaching X2i−1 × [2i − 1, 2i] to X2i ⊂ M(f2i) down the latter mapping cylinder to X2i+1 we convert M(f2i−1) ∪ M(f2i) into M(f2if2i−1) ∪ M(f2i). This last space deformation retracts onto M(f2if2i−1). Doing this for all i gives the homotopy equivalence in (3). Now to prove the proposition, suppose that the space Y is dominated by the CW r-----→ Y with r i ≃ 11. By (2) and (3) we have T (ir, ir, ···) ≃ T (r, i, r, i, ···) ≃ T (i, r, i, r, ···) ≃ T (r i, r i, ···). Since r i ≃ 11, T (r i, r i, ···) is homotopy equivalent to the telescope of the identity maps Y →Y →Y → ···, which is Y × [0, ∞) ≃ Y. On the other hand, the map ir is homotopic to a cellular map f : X→X, so T (ir, ir, ···) ≃ T (f, f, ···), which is a CW complex. ⊔⊓ X2i−1 × [2i − 1, 2i
] complex X via maps Y i-----→ X i ` The Compact-Open Topology Appendix 529 One might ask whether a space dominated by a ﬁnite CW complex is homotopy equivalent to a ﬁnite CW complex. In the simply-connected case this follows from Proposition 4C.1 since such a space has ﬁnitely generated homology groups. But there are counterexamples in the general case; see [Wall 1965]. In view of Corollary A.9 the preceding proposition implies: Corollary A.12. A compact manifold is homotopy equivalent to a CW complex. ⊔⊓ One could ask more reﬁned questions. For example, do all compact manifolds have CW complex structures, or even stronger, do they have simplicial complex struc- tures? Answers here are considerably harder to come by. Restricting attention to closed manifolds for simplicity, it is known that simplicial complex structures ex- ist for all manifolds of dimension 1, 2, and 3, but in each higher dimension starting with 4 there exist manifolds that have no simplicial complex structure. In dimensions greater than 4, CW structures always exist, but it seems to be still unknown whether all manifolds of dimension 4 are CW complexes. For more on these questions, see [Kirby & Siebenmann 1977], [Freedman & Quinn 1990], and [Manolescu 2016]. Exercises 1. Show that a covering space of a CW complex is also a CW complex, with cells projecting homeomorphically onto cells. 2. Let X be a CW complex and x0 any point of X. Construct a new CW complex structure on X having x0 as a 0 cell, and having each of the original cells a union of the new cells. The latter condition is expressed by saying the new CW structure is a subdivision of the old one. 3. Show that a CW complex is path-connected iﬀ its 1 skeleton is path-connected. 4. Show that a CW complex is locally compact iﬀ each point has a neighborhood that meets only ﬁnitely many cells. 5. For a space X, show that the identity map Xc→X induces an isomorphism on π1, where Xc denotes X with the compactly generated topology. The Compact-Open Topology By deﬁnition, the compact-open topology on the space
X Y of maps f : Y →X has a subbasis consisting of the sets M(K, U) of mappings taking a compact set K ⊂ Y to an open set U ⊂ X. Thus a basis for X Y consists of sets of maps taking a ﬁnite number of compact sets Ki ⊂ Y to open sets Ui ⊂ X. If Y is compact, which is the only case we consider in this book, convergence to f ∈ X Y means, loosely speaking, that ﬁner and ﬁner compact covers {Ki} of Y are taken to smaller and smaller open covers {Ui} of f (Y ). One of the main cases of interest in homotopy theory is when Y = I, so X I is the space of paths in X. In this case one can check that a system of 530 Appendix The Compact-Open Topology basic neighborhoods of a path f : I→X consists of the open sets i M(Ki, Ui) where the Ki ’s are a partition of I into nonoverlapping closed intervals and Ui is an open neighborhood of f (Ki). T The compact-open topology is the same as the topology of uniform convergence in many cases: Proposition A.13. If X is a metric space and Y is compact, then the compact-open topology on X Y is the same as the metric topology deﬁned by the metric d(f, g) = supy∈Y d(f (y), g(y)). Proof: First we show that every open ε ball Bε(f ) about f ∈ X Y contains a neighborhood of f in the compact-open topology. Since f (Y ) is compact, it is covered by ﬁnitely many balls Bε/3 Ki is compact, Y = To show that say y ∈ Ki, we have d < ε/2, so d d Since y was arbitrary, this shows g ∈ Bε(f )., so i M(Ki, Ui). i M(Ki, Ui). For any y ∈ Y, < ε/2 since g(Ki) ⊂ Ui. Likewise we have < ε.. Let Ki ⊂ Y be the closure of f −1 f (yi) f (yi
) i Ki, and f (Ki) ⊂ Bε/2 i M(Ki, Ui) ⊂ Bε(f ), suppose that g ∈ f (y), f (yi) f (y), f (yi) g(y), f (yi) g(y), f (yi) Bε/3(f (yi)) = Ui, hence f ∈ f (y), g(y) ≤ d + d T T T S Conversely, we show that for each open set M(K, U) and each f ∈ M(K, U) there is a ball Bε(f ) ⊂ M(K, U). Since f (K) is compact, it has a distance ε > 0 from the complement of U. Then d(f, g) < ε/2 implies g(K) ⊂ U since g(K) is contained in an ε/2 neighborhood of f (K). So Bε/2(f ) ⊂ M(K, U). ⊔⊓ The next proposition contains some useful properties of the compact-open topol- ogy from the viewpoint of algebraic topology. Proposition A.14. (a) The evaluation map e : X Y × Y →X, e(f, y) = f (y), is continuous If Y is locally compact. (b) If f : Y × Z→X is continuous then so is the map (c) The converse to (b) holds when Y is locally compact. f (z)(y) = f (y, z). f : Z→X Y, b b Diﬀerent deﬁnitions of local compactness are common, but the deﬁnition we are using is that Y is locally compact if for each point y ∈ Y and each neighborhood U of y there is a compact neighborhood V of y contained in U. X, Y i = hX, In particular, parts (b) and (c) of the proposition provide the point-set topology Y i in §4.3, since they imply that a map justifying the adjoint relation h X→Y is continuous iﬀ the associated map X→ Y is continuous, and similarly for X→Y as homotopies of such maps. Namely, think of a bas
epoint-preserving map Σ a map f : I × X→Y taking ∂I × X ∪ {x0}× I to the basepoint of Y, so the associated X→Y gives map a map F : I × X × I→Y taking ∂I × X × I ∪ I × {x0}× I to the basepoint, with F a map b X × I→ Y ⊂ Y I deﬁning a basepoint-preserving homotopy f : X→Y I has image in the subspace Y ⊂ Y I. A homotopy ft : Ω Ω Ω Σ Σ Σ ft. b Ω b The Compact-Open Topology Appendix 531 Proof: (a) For (f, y) ∈ X Y × Y let U ⊂ X be an open neighborhood of f (y). Since Y is locally compact, continuity of f implies there is a compact neighborhood K ⊂ Y of y such that f (K) ⊂ U. Then M(K, U)× K is a neighborhood of (f, y) in X Y × Y taken to U by e, so e is continuous at (f, y). (b) Suppose f : Y × Z→X is continuous. To show continuity of that for a subbasic set M(K, U) ⊂ X Y, the set is open in Z. Let K, z) ⊂ U }. Since f −1(U) is an open neighborhood of the compact set K × {z}, there exist open sets V ⊂ Y and W ⊂ Z whose product V × W satisﬁes K × {z} ⊂ V × W ⊂ f −1(U). So W is a neighborhood of z in (c) Note that f : Y × Z→X is the composition Y × Z→Y × X Y →X of 11× evaluation map, so part (a) gives the result. f it suﬃces to show f and the M(K, U) M(K, U) M(K, U) f −1 f −1 f −1 ⊔⊓ b b b b. b We will give three separate applications of Proposition A.14. Here is the �
��rst: Proposition A.15. If X is a compactly generated Hausdorﬀ space and Y is locally compact, then the product topology on X × Y is compactly generated. Proof: First a preliminary observation: A function f : X × Y →Z is continuous iﬀ its restrictions f : C × Y →Z are continuous for all compact C ⊂ X. For, using (b) and (c) f : X→Z Y being of the preceding Proposition A.14, the ﬁrst statement is equivalent to f : C→Z Y being continuous for b f : X→Z Y is continuous and the second statement is equivalent to all compact C ⊂ X. Then since X is compactly generated, continuity of b equivalent to continuity of f : C→Z Y for all compact C ⊂ X. To prove the proposition we just need to show the identity map X × Y →(X × Y )c is continuous. By the previous paragraph, this is equivalent to continuity of the inclusion maps C × Y →(X × Y )c for all compact C ⊂ X. Since Y is locally compact, it is compactly generated, and C is compact Hausdorﬀ hence locally compact, so the same reasoning shows that continuity of C × Y →(X × Y )c is equivalent to continuity of C × C ′→(X × Y )c for all compact C ′ ⊂ Y. But on the compact set C × C ′, the two topologies on X × Y agree, so we are done. (This proof is from [Dugundji 1966].) ⊔⊓ b b Returning to the context of Proposition A.14, part (b) of that proposition implies that there is a well-deﬁned function X Y × Z→(X Y )Z sending f to and part (c) implies that it is surjective if Y is locally compact. f. This is injective, b Proposition A.16. The map X Y × Z→(X Y )Z, f ֏ locally compact Hausdorﬀ and Z is Hausdorﬀ. f, is a homeomorphism if Y is b Proof: First we show that a subbasis for X Y × Z is formed by the sets M(A× B, U) as A
and B range over compact sets in Y and Z respectively and U ranges over open sets in X. Given a compact K ⊂ Y × Z and a map f ∈ M(K, U), let KY and KZ be the projections of K onto Y and Z. Then KY × KZ is compact Hausdorﬀ and 532 Appendix The Compact-Open Topology hence normal. A normal space has the property that for each closed set C and each open set O containing C there is another open set O′ containing C whose closure is contained in O. To see this, apply the normality property to the two closed sets C and the complement C ′ of O, taking O′ to be the resulting open set containing C and disjoint from an open set containing C ′, so the closure of O′ is contained in O. Applying this observation to the normal space KY × KZ with C a point k ∈ K and O = (KY × KZ ) ∩ f −1(U), the result is an open neighborhood of k in KY × KZ whose closure is contained in f −1(U). We can take this open neighborhood to be a product Vk × Wk ⊂ KY × KZ, so its closure is a compact neighborhood Ak × Bk ⊂ f −1(U) of k in KY × KZ. The sets Vk × Wk for varying k ∈ K form an open cover of the compact set K so a ﬁnite number of the products Ak × Bk cover K. After discarding the others we k M(Ak × Bk, U) ⊂ M(K, U), which shows that the sets M(A× B, U) then have f ∈ form a subbasis for X Y × Z as claimed. T Under the bijection X Y × Z→(X Y )Z the sets M(A× B, U) correspond to the sets M(B, M(A, U)), so it will suﬃce to show the latter sets form a subbasis for (X Y )Z. We will show more generally that for any space Q a subbasis for QZ is formed by the sets M(K, V ) as V ranges over a subbasis for Q and K ranges over compact sets in Z, assuming that Z is Hausdorﬀ
. Then we let Q = X Y with subbasis the sets M(A, U). Given f ∈ M(K, U) with K compact in Z and U open in Q, write U as a union of basic sets Uα with each Uα an intersection of ﬁnitely many sets Vα,j of the given subbasis for Q. The cover of K by the open sets f −1(Uα) has a ﬁnite subcover, say by the open sets f −1(Ui). Since K is compact Hausdorﬀ, hence normal, we can write K as a union of compact subsets Ki with Ki ⊂ f −1(Ui), namely, each k ∈ K has a compact neighborhood Kk contained in some f −1(Ui) with k ∈ f −1(Ui), so compactness of K implies that ﬁnitely many of these sets Kk cover K and we let Ki be the union of those contained in f −1(Ui). Now f lies in M(Ki, Ui) = M(Ki, j Vij) = i M(Ki, Ui) ⊂ M(K, U). i,j M(Ki, Vij) is a ﬁnite intersection, this shows that the sets M(K, V ) form a ⊔⊓ j M(Ki, Vij) for each i. Hence f lies in Since T subbasis for QZ. i,j M(Ki, Vij) = T T T T Finally, we use Proposition A.14 to prove a very useful fact relating product spaces and quotient spaces: Proposition A.17. If f : X→Y is a quotient map then so is f × 11 : X × Z→Y × Z whenever Z is locally compact. This can be applied when Z = I to show that a homotopy deﬁned on a quotient space is continuous. Proof: Consider the diagram at the right, where W is Y × Z with the quotient topology from X × Z, with g the quotient map and h the identity. Every open set in Y × Z is open in W since f × 11 is continuous, so it will suﬃce to show that h is continuous. The Homotopy Extension Property
Appendix 533 Since g is continuous, so is the associated map g : X→W Z, by Proposition A.14. h : Y →W Z is continuous since f is a quotient map. Applying Propo⊔⊓ b sition A.14 again, we conclude that h is continuous. This implies that b The Homotopy Extension Property Near the end of Chapter 0 we stated, and partially proved, an equivalence between the homotopy extension property and a certain retraction property: Proposition A.18. A pair (X, A) has the homotopy extension property if and only if X × {0} ∪ A× I is a retract of X × I. Proof: We already gave the easy argument showing that the homotopy extension property implies the retraction property. The converse was also easy when A is closed in X, and what we need now is an argument for the converse that does not use this extra assumption. The argument is from [Strøm 1968]. To simplify the notation, let Y = X × {0} ∪ A× I with the subspace topology from X × I. We will identify X with the subspace X × {0} of Y. Assuming there exists a retraction r : X × I→Y, we will show that a subset O ⊂ Y is open in Y if its intersections with X and A× I are open in these two subspaces. This implies that a function on Y is continuous if its restrictions to X and A× I are continuous. Composing such a function with the retraction then provides the extension to X × I required for the homotopy extension property. To show that O is open in Y it suﬃces to ﬁnd, for each point x ∈ O, a product of open sets V × W ⊂ X × I containing x such that (V × W ) ∩ Y ⊂ O. If x ∈ A× (0, 1] there is no problem doing this, so we may assume x ∈ X. In this case there is also no problem if x in not in the closure A of A, so we will assume x ∈ A from now on. For an integer n ≥ 1 let Un be the largest open set in X such that (Un ∩A)× [0, 1/n) ⊂ O. The existence of a largest such
set Un follows from the fact that a union of open sets with this property is again an open set with this property. Let U = n Un. Note that A∩O ⊂ U since O intersects A× I in an open set in A× I by assumption. It will suﬃce to show that x ∈ U since if x ∈ Un then we can choose V × W = (Un ∩ O)× [0, 1/n) ∩ Y ⊂ O, where Un ∩ O is open in X since Un is open because S (Un ∩ O)× [0, 1/n) in X. In order to show that x ∈ U we ﬁrst consider the point (x, t) for ﬁxed t > 0. Writing r (x, t) = (r1(x, t), r2(x, t)) ∈ X × I we have r2(x, t) = t since x ∈ A and r (a, t) = (a, t) for a ∈ A. Thus r (x, t) ∈ A× {t} so r1(x, t) ∈ A. We claim next that if r1(x, t) ∈ Un then x ∈ Un. For if r1(x, t) ∈ Un then continuity of r1 implies that r1(V × (t − ε, t + ε)) ⊂ Un for some open neighborhood V of x in X and some ε > 0. ⊂ Un, or in other words, V ∩ A ⊂ Un. By the deﬁnition In particular r1 of Un this implies that V ⊂ Un and hence x ∈ Un since x ∈ V. Thus we have shown that r1(x, t) ∈ Un implies x ∈ Un. (V ∩ A)× {t} 534 Appendix Simplicial CW Structures Suppose now that x is not in U. From what we have just shown, this implies that r1(x, t) ∈ A − U. It follows that r1(x, t) ∈ A − O since A ∩ O ⊂ U, as noted earlier. The relation r1(x, t) ∈ A − O holds for arbitrary t > 0
, so by letting t approach 0 we conclude that r1(x, 0) ∈ A − O since r1 is a continuous map to X and X ∩ O is open in X. Since r1(x, 0) = x we deduce that x is not in O. However, this contradicts the fact that x was chosen to be a point in O. From this contradiction we conclude that x must be in U, and the proof is ﬁnished. ⊔⊓ For an example of a set O ⊂ X × {0} ∪ A× I which is not open even though it intersects both X × {0} and A× I in open sets, let X = [0, 1] with A = (0, 1] and (x, t) ∈ X × I \|\|\|\| t < x or t = 0 O =. Note that in this case there exists no retraction X × I→X × {0}∪A× I since the image of a compact set must be compact. This example also illustrates how the topology on X × {0} ∪ A× I as a subspace of X × I can be diﬀerent from the topology as the mapping cylinder of the inclusion A ֓ X, which is the quotient topology from X ∐ A× I. Simplicial CW Structures n−1 of A D complex can be deﬁned as a CW complex X in which each cell en with a distinguished characteristic map σα : to each face α is provided n→X such that the restriction of σα It is n−1 have a speciﬁed ordering of their vertices, n induces an ordering of the vertices of each face, n−1. Intuitively, one thinks of the vertices of each n cell of X as ordered by attaching the labels 0, 1, ···, n near n is the distinguished σβ for some (n − 1) cell en−1 understood that the simplices ∆ which allows each face to be identiﬁed canonically with and the ordering of the vertices of n and ∆ ∆ ∆ ∆ ∆ β. the vertices, just inside the cell. The vertices themselves do not have to be distinct ∆ points of X. n is allowed to be the composition of σβ : If we
no longer pay attention to orderings of vertices of simplices, we obtain a weaker structure which could be called an unordered D complex. Here each cell en α n→X, but the restriction of σα to a face has a distinguished characteristic map σα : n−1→X with a symmetry of of permuting its vertices. Alternatively, we could say that each cell en (n+1)! distinguished characteristic maps α has a family of n, such that the restrictions of these characteristic maps to faces give the distinguished ∆ characteristic maps for (n − 1) cells. The barycentric subdivision of any unordered ∆ n→X diﬀering only by symmetries of n−1 ∆ ∆ ∆ ∆ complex is an ordered complex since the vertices of the barycentric subdivision are the barycenters of the simplices of the original complex, hence have a canonical ∆ ordering according to the dimensions of these simplices. The simplest example of an ∆ unordered division is complex that cannot be made into an ordered 2 with its three edges identiﬁed by a one-third rotation of ∆ complex without sub2 permuting ∆ the three vertices cyclically. ∆ ∆ Simplicial CW Structures Appendix 535 In the literature unordered complex structures are sometimes called general- ized triangulations. They can be useful in situations where orderings of vertices are not needed. One disadvantage of unordered complexes is that they do not behave as well with respect to products. The product of two ordered simplices has a canonical subdivision into ordered simplices using the shuﬄing operation described in §3.B, and ∆ this allows the product of two ordered complexes to be given a canonical ordered ∆ complex structure. Without orderings this no longer works. A CW complex is called regular if its characteristic maps can be chosen to be ∆ embeddings. The closures of the cells are then homeomorphic to closed balls, and so it makes sense to speak of closed cells in a regular CW complex. The closed cells can be regarded as cones on their boundary spheres, and these cone structures can be used to subdivide a regular CW complex into a regular complex, by induction over skeleta. In particular, regular CW complexes are homeomorphic to complexes. The barycentric subdivision of an unordered plicial complex is a regular unordered complex is a regular complex. A sim-
∆ complex in which each simplex is uniquely ∆ ∆ determined by its vertices. In the literature a regular unordered complex is some- times called a simplicial multicomplex, or just a multicomplex, to convey the idea that there can be many simplices with the same set of vertices. The barycentric subdivi- sion of a regular unordered complex is a simplicial complex. Hence barycentrically subdividing an unordered complex twice produces a simplicial complex. ∆ ∆ ∆ ∆ ∆ A major disadvantage of ∆ complexes is that they do not allow quotient con- structions. The quotient X/A of a complex X by a subcomplex A is not usually a complex. More generally, attaching a complex X to a complex Y via a simpli- cial map from a subcomplex A ⊂ X to Y is not usually a ∆ ∆ map f : A→Y is one that sends each cell en β of Y so that the square at the right commutes, with q a linear surjection send- complex. Here a simplicial ∆ α of A onto a cell ek ∆ ∆ ∆ ing vertices to vertices, preserving order. To ﬁx this problem we need complex to allow cells to be attached to broaden the deﬁnition of a by arbitrary simplicial maps. Thus we deﬁne a singular D complex, or s to be a CW complex with distinguished characteristic maps σα : strictions to faces are compositions σβq : taking vertices to vertices, preserving order. Simplicial maps between s complex, n→X whose rek→X for q a linear surjection complexes n−1→ ∆ ∆ ∆ are deﬁned just as for ∆ complexes. With s ∆ complexes one can perform attaching constructions in the same way as for CW complexes, using simplicial maps instead ∆ of cellular maps to specify the attachments. ∆ In particular one can form quotients, ∆ mapping cylinders, and mapping cones. One can also take products by the same subdivision procedure as for complexes. We can view any s complex X as being constructed inductively, skeleton by n skeleton, where the skeleton X n is obtained from X n−1 by attaching simplices via simplicial maps ∂ ∆ n→X n−1 that preserve the ordering of vert
ices in each face ∆ ∆ ∆ 536 Appendix Simplicial CW Structures n. Conversely, any CW complex built in this way is an s of the usual CW structure on S n consisting of one 0 cell and one n cell is an s structure since the attaching map of the n cell, the constant map, is a simplicial complex. For example, complex ∆ ∆ n to a point. One can regard this s complex structure as assigning map from ∂ barycentric coordinates to all points of S n other than the 0 cell. In fact, an arbitrary complex structure can be regarded as just a way of putting barycentric coordinates s ∆ ∆ ∆ in all the open cells, subject to a compatibility condition on how the coordinates ∆ change when one passes from a cell to the cells in its boundary. Combinatorial Descriptions The data which speciﬁes a complex is combinatorial in nature and can be for- ∆ ∆ X be a mulated quite naturally in the language of categories. To see how this is done, let complex and let Xn be its set of n simplices. The way in which simplices of X ﬁt together is determined by a ‘face function’ which assigns to each element of n an element of Xn−1. Thinking of the Xn and each (n − 1) dimensional face of n combinatorially as its set of vertices, which we view as the ordered set n simplex ∆ n = { 0, 1, ···, n }, the face-function for X assigns to each order-preserving injecn a map Xn→Xn−1. By composing these maps we get, for each ordertion ∆ n a map g∗ : Xn→Xk specifying how the k simplices of preserving injection g : X are arranged in the boundary of each n simplex. The association g ֏ g∗ satisﬁes (gh)∗ = h∗g∗, and we can set 11∗ = 11, so X determines a contravariant functor from n, n ≥ 0, and whose morphisms are the category whose objects are the ordered sets ∆ n−1→ k→ ∆ ∆ ∆ ∆ the order-preserving injections, to the category of sets, namely the functor sending to Xn and the injection g to
g∗. Such a functor is exactly equivalent to a Explicitly, we can reconstruct the n complex. ∆ ∆ complex X from the functor by setting ∆ X = ∆ n(Xn × n)/(g∗(x), y) ∼ (x, g∗(y)) ` for (x, y) ∈ Xn × of k to the g(i)th vertex of ∆ k, where g∗ is the linear inclusion k→ n sending the ith vertex n, and we perform the indicated identiﬁcations letting g range over all order-preserving injections ∆ ∆ ∆ k→ k→ linear maps preserving surjections If we wish to generalize this to s arbitrary order-preserving maps complexes, we will have to consider surjective n as well as injections. This corresponds to considering ordern in addition to injections. Every map of sets decomposes canonically as a surjection followed by an injection, so we may as well consider n. These form the morphisms in a category ∗, with objects the n ’s. We are thus led to consider contravariant functors from ∗ to the category of sets. Such a functor is called a simplicial set. This terminology ∆ has the virtue that one can immediately deﬁne, for example, a simplicial group to be ∆ a contravariant functor from ∗ to the category of groups, and similarly for simplicial rings, simplicial modules, and so on. One can even deﬁne simplicial spaces as k→ ∆ ∆ ∆ ∆ ∆ ∆ Simplicial CW Structures Appendix 537 contravariant functors from ∗ to the category of topological spaces and continuous maps. ∆ ∆ ∆ gular complex of X, whose n simplices are all the continuous maps For any space X there is an associated rather large simplicial set S(X), the sinn→X. For a n the induced map g∗ from n simplices of S(X) to k simplices n. We introduced S(X) in §2.1 in complex, morphism g : of S(X) is obtained by composition with g∗ : connection with the deﬁnition of singular homology and described it as a k→ k→ ∆ ∆ ∆ but
in fact it has the additional structure of a simplicial set. ∆ ∆ In a similar but more restricted way, an s (X) whose k simplices are all the simplicial maps complex X gives rise to a simplicial k→X. These are uniquely set n→X of simplicial surjections q (preservexpressible as compositions σαq : ing orderings of vertices) with characteristic maps of simplices of X. The maps g∗ n. These are obtained just as for S(X), by composition with the maps g∗ : examples (X) in fact account for all simplicial sets: k→ k→ ∆ ∆ ∆ ∆ ∆ ∆ Proposition A.19. Every simplicial set is isomorphic to one of the form some s complex X which is unique up to isomorphism. ∆ ∆ (X) for ∆ Here an isomorphism of simplicial sets means an isomorphism in the category ∆ travariant functors from of simplicial sets, where the morphisms are natural transformations between con∗ to the category of sets. This translates into just what one would expect, maps sending n simplices to n simplices that commute with the maps g∗. Note that the proposition implies in particular that a nonempty simplicial (X). This set contains simplices of all dimensions since this is evidently true for ∆ is also easy to deduce directly from the deﬁnition of a simplicial set. Thus simplicial sets are in a certain sense large inﬁnite objects, but the proposition says that their essential geometrical core, an s complex, can be much smaller. ∆ Proof: Let Y be a simplicial set, with Yn its set of n simplices. A simplex τ in Yn is called degenerate if it is in the image of g∗ : Yk→Yn for some noninjective k. Since g can be factored as a surjection followed by an injection, there is g : (X) the degenerate simplices n→X that are not injective on the interior of no loss in requiring g to be surjective. For example, in are those that are the simplicial maps n→ ∆ ∆ n. Thus the main diﬀerence between X and (X) is the degenerate simplices. ∆ ∆ Every
degenerate simplex of Y has the form g∗(τ) for some nondegenerate simk. We claim that such a g and τ are unique. For n→ 1 (τ1) = g∗ 2 (τ2) with τ1 and τ2 nondegenerate and g1 : surjective. Choose order-preserving injections h1 : ∆ plex τ and surjection g : n→ suppose we have g∗ k1 k1→ and g2 : n and ∆ ∆ k2→ h2 : 2 (τ2) implies that ∆ ∆ 2 (τ2) = h∗ 2 g∗ 1 (τ1) = h∗ 2 g∗ h∗ 1 (τ1) = τ1, so the nondegeneracy ∆ ∆ of τ1 and τ2 implies that g1h2 and g2h1 are injective. This in turn implies that k1 = k2 and g1h2 = 11 = g2h1, hence τ1 = τ2. If g1 ≠ g2 then g1(i) ≠ g2(i) for n→ n with g1h1 = 11 and g2h2 = 11. Then g∗ 2 (τ2) = τ2 and h∗ 1 (τ1) = g∗ 1 g∗ 1 g∗ ∆ ∆ ∆ ∆ k2 ∆ 538 Appendix Simplicial CW Structures some i, and if we choose h1 so that h1g1(i) = i, then g2h1g1(i) = g2(i) ≠ g1(i), contradicting g2h1 = 11 and ﬁnishing the proof of the claim. Just as we reconstructed a associate to the simplicial set Y an s complex from its categorical description, we can complex \|Y \|, its geometric realization, by setting ∆ \|Y \| = n(Yn × ∆ n)/(g∗(y), z) ∼ (y, g∗(z)) for (y, z) ∈ Yn × n. Since every g factors canonically as a surjec∆ tion followed by an injection, it suﬃces to perform the indicated identiﬁcations just when g is a surjection or an
injection. Letting g range over surjections amounts to ` k and g : k→ ∆ ∆ ∆ collapsing each simplex onto a unique nondegenerate simplex by a unique projection, by the claim in the preceding paragraph, so after performing the identiﬁcations just for surjections we obtain a collection of disjoint simplices, with one n simplex for each nondegenerate n simplex of Y. Then doing the identiﬁcations as g varies over injections attaches these nondegenerate simplices together to form an s complex, which is \|Y \|. The quotient map from the collection of disjoint simplices to \|Y \| gives the collection of distinguished characteristic maps for the cells of \|Y \|. ∆ If we start with an s complex X and form \| as X. In the other direction, if we start with a simplicial set Y and form there is an evident bijection between the n simplices of these two simplicial sets, and ∆ this commutes with the maps g∗ so the two simplicial sets are equivalent. ⊔⊓ ∆ ∆ (X)\|, then this is clearly the same (\|Y \|) then As we observed in the preceding proof, the geometric realization \|Y \| of a sim- plicial set Y can be built in two stages, by ﬁrst collapsing all degenerate simplices by making the identiﬁcations (g∗(y), z) ∼ (y, g∗(z)) as g ranges over surjections, and then glueing together these nondegenerate simplices by letting g range over injec- tions. We could equally well perform these two types of identiﬁcations in the opposite order. If we ﬁrst do the identiﬁcations for injections, this amounts to regarding Y as a category-theoretic to sets, to the subcategory of geometric realization \|Y ∆ complex Y by restricting Y, regarded as a functor from ∗ ∗ consisting of injective maps, and then taking the complex. After doing this, if ∆ ∆ \| to produce a geometric we perform the identiﬁcations for surjections g we obtain a natural quotient map \|→\|Y \|. This is a homotopy equivalence, but we will not prove
this fact here. The \|Y \| is sometimes called the thick geometric realization of Y. complex \|Y ∆ ∆ ∆ ∆ Since simplicial sets are very combinatorial objects, many standard constructions ∆ can be performed on them. A good example is products. For simplicial sets X and Y there is an easily-deﬁned product simplicial set X × Y, having (X × Y )n = Xn × Yn and g∗(x, y) =. The nice surprise about this deﬁnition is that it is compatible with geometric realization: the realization \|X × Y \| turns out to be homeo- g∗(x), g∗(y) morphic to \|X\|× \|Y \|, the product of the CW complexes \|X\| and \|Y \| (with the com- pactly generated CW topology). The homeomorphism is just the product of the maps ∆ Simplicial CW Structures Appendix 539 \|X × Y \|→\|X\| and \|X × Y \|→\|Y \| induced by the projections of X × Y onto its two fac1). tors. As a very simple example, consider the case that X and Y are both Letting [v0, v1] and [w0, w1] be the two copies of product X × Y has two nondegenerate 2 simplices: 1, the ∆ ∆ ( ∆ ([v0, v1, v1], [w0, w0, w1]) = [(v0, w0), (v1, w0), (v1, w1)] ([v0, v0, v1], [w0, w1, w1]) = [(v0, w0), (v0, w1), (v1, w1)] 1 × These subdivide the square 1 into two 2 simplices. There are ﬁve nondegenerate 1 simplices in X × Y, as shown ∆ in the ﬁgure. One of these, the diagonal of the square, is the pair ([v0, v1], [w0, w1]) formed by the two nondegenerate 1 simplices [v0, v1] and [w0, w1], while the other four are pairs like ([v0, v0], [w0, w1])
where one factor is a degenerate 1 simplex and the other is a nondegenerate 1 simplex. Obviously there are no nondegenerate n simplices in X × Y for n > 2. ∆ It is not hard to see how this example generalizes to the product q. Here one obtains the subdivision of the product into (p + q) simplices described in §3.B p × in terms of the shuﬄing operation. Once one understands the case of a product of ∆ ∆ simplices, the general case easily follows. One could also deﬁne unordered s complexes in a similar way to unordered complexes, and then work out the ‘simplicial set’ description of these objects. How- ever, this sort of structure is more cumbersome to work with and has not been used ∆ much. ∆ Books J. F. Adams, Algebraic Topology: a Student’s Guide, Cambridge Univ. Press, 1972. J. F. Adams, Stable Homotopy and Generalised Homology, Univ. of Chicago Press, 1974. J. F. Adams, Inﬁnite Loop Spaces, Ann. of Math. Studies 90, 1978. A. Adem and R. J. Milgram, Cohomology of Finite Groups, Springer-Verlag, 1994. M. Aguilar, S. Gitler, and C. Prieto, Algebraic Topology from a Homotopical Viewpoint, Springer-Verlag, 2002. M. Aigner and G. M. Ziegler, Proofs from THE BOOK, Springer-Verlag, 1999. P. Alexandroff and H. Hopf, Topologie, Chelsea, 1972 (reprint of original 1935 edition). M. A. Armstrong, Basic Topology, Springer-Verlag, 1983. M. F. Atiyah, K–Theory, W. A. Benjamin, 1967. H. J. Baues, Homotopy Type and Homology, Oxford Univ. Press, 1996. D. J. Benson, Representations and Cohomology, Volume II: Cohomology of Groups and Modules, Cambridge Univ. Press, 1992. D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge Univ. Press, 1993. R. Bott and L. W. Tu, Diﬀerential Forms in
Algebraic Topology, Springer-Verlag GTM 82, 1982. G. E. Bredon, Topology and Geometry, Springer-Verlag GTM 139, 1993. K. S. Brown, Cohomology of Groups, Springer-Verlag GTM 87, 1982. R. F. Brown, The Lefschetz Fixed Point Theorem, Scott Foresman, 1971. M. M. Cohen, A Course in Simple-Homotopy Theory, Springer-Verlag GTM 10, 1973. T. tom Dieck, Algebraic Topology, E.M.S., 2008. J. Dieudonn´e, A History of Algebraic and Diﬀerential Topology 1900-1960, Birkh¨auser, 1989. A. Dold, Lectures on Algebraic Topology, Springer-Verlag, 1980. J. Dugundji, Topology, Allyn & Bacon, 1966. H.-D. Ebbinghaus et al., Numbers, Springer-Verlag GTM 123, 1991. S. Eilenberg and N. Steenrod, Foundations of Algebraic Topology, Princeton Univ. Press, 1952. Y. F´elix, S. Halperin, and J.-C. Thomas, Rational Homotopy Theory, Springer-Verlag GTM 205, 2001. R. A. Fenn, Techniques of Geometric Topology, Cambridge Univ. Press, 1983. Bibliography 541 A. T. Fomenko and D. B. Fuks, A Course in Homotopic Topology, Izd. Nauka, 1989. (In Russian; an English translation of an earlier version was published by Akad´emiai Kiad´o, Budapest, 1986.) M. H. Freedman and F. Quinn, Topology of 4-Manifolds, Princeton Univ. Press, 1990. W. Fulton, Algebraic Topology: A First Course, Springer-Verlag GTM 153, 1995. B. Gray, Homotopy Theory, Academic Press, 1975. M. J. Greenberg and J. R. Harper, Algebraic Topology: A First Course, Addison-Wesley, 1981. P. Grifﬁths and J. Morgan, Rational Homotopy Theory and Diﬀerential Forms, Birkh¨auser, 1981. P. J. Hilton, An
Introduction to Homotopy Theory, Cambridge Univ. Press, 1953. P. J. Hilton, Homotopy Theory and Duality, Gordon and Breach, 1966. P. J. Hilton and S. Wylie, Homology Theory, Cambridge Univ. Press, 1967. P. J. Hilton and U. Stammbach, A Course in Homological Algebra, Springer-Verlag GTM 4, 1970. P. J. Hilton, G. Mislin, and J. Roitberg, Localization of Nilpotent Groups and Spaces, North- Holland, 1975. D. Husemoller, Fibre Bundles, McGraw-Hill, 1966 (later editions by Springer-Verlag). I. M. James, ed., Handbook of Algebraic Topology, North-Holland, 1995. I. M. James, ed., History of Topology, North-Holland, 1999. K. J¨anich, Topology, Springer-Verlag, 1984. R. M. Kane, The Homology of Hopf Spaces, North-Holland, 1988. R. C. Kirby and L. C. Siebenmann, Foundational Essays on Topological Manifolds, Smoothings, and Triangulations, Ann. of Math. Studies 88, 1977. S. O. Kochman, Bordism, Stable Homotopy and Adams Spectral Sequences, Fields Institute Monographs 7, A.M.S., 1996. D. K¨onig, Theory of Finite and Inﬁnite Graphs, Birkh¨auser, 1990. A. T. Lundell and S. Weingram, The Topology of CW Complexes, Van Nostrand Reinhold, 1969. S. MacLane, Homology, Springer-Verlag, 1963. S. MacLane, Categories for the Working Mathematician, Springer-Verlag GTM 5, 1971. I. Madsen and R. J. Milgram, The Classifying Spaces for Surgery and Cobordism of Manifolds, Ann. of Math. Studies 92, 1979. W. S. Massey, Algebraic Topology: An Introduction, Harcourt, Brace & World, 1967 (reprinted by Springer-Verlag). W. S. Massey, A Basic Course in Algebraic Topology, Springer-Verlag GTM 127, 1993. J. Matou
ˇsek, Using the Borsuk-Ulam Theorem, Springer-Verlag, 2003. 542 Bibliography C. R. F. Maunder, Algebraic Topology, Cambridge Univ. Press, 1980 (reprinted by Dover Pub- lications). J. P. May, Simplicial Objects in Algebraic Topology, Van Nostrand, 1967 (reprinted by Univ. Chicago Press). J. P. May, A Concise Course in Algebraic Topology, Univ. Chicago Press, 1999. J. W. Milnor, Topology from the Diﬀerentiable Viewpoint, Univ. Press of Virginia, 1965. J. W. Milnor and J. D. Stasheff, Characteristic Classes, Ann. of Math. Studies 76, 1974. R. E. Mosher and M. C. Tangora, Cohomology Operations and Applications in Homotopy Theory, Harper and Row, 1968. V. V. Prasolov, Elements of Homology Theory, A.M.S., 2007. D. C. Ravenel, Complex Cobordism and Stable Homotopy Groups of Spheres, Academic Press, 1986. D. C. Ravenel, Nilpotence and Periodicity in Stable Homotopy Theory, Ann. of Math. Studies 128, 1992. E. Rees and J. D. S. Jones, eds., Homotopy Theory: Proceeding of the Durham Symposium 1985, Cambridge Univ. Press, 1987. D. Rolfsen, Knots and Links, Publish or Perish, 1976. H. Seifert and W. Threlfall, Lehrbuch der Topologie, Teubner, 1934. P. Selick, Introduction to Homotopy Theory, Fields Institute Monographs 9, A.M.S., 1997. J.-P. Serre, A Course in Arithmetic, Springer-Verlag GTM 7, 1973. E. H. Spanier, Algebraic Topology, McGraw-Hill, 1966 (reprinted by Springer-Verlag). N. Steenrod, The Topology of Fibre Bundles, Princeton Univ. Press, 1951. N. E. Steenrod and D. B. A. Epstein, Cohomology Operations, Ann. of Math. Studies 50, 1962. R. E. Stong, Notes on Cobordism Theory,
Princeton Univ. Press, 1968. D. Sullivan, Geometric Topology, xeroxed notes from MIT, 1970. R. M. Switzer, Algebraic Topology, Springer-Verlag, 1975. H. Toda, Composition Methods in Homotopy Groups of Spheres, Ann. of Math. Studies 49, 1962. K. Varadarajan, The Finiteness Obstruction of C. T. C. Wall, Wiley, 1989. C. A. Weibel, An Introduction to Homological Algebra, Cambridge Univ. Press, 1994. G. W. Whitehead, Elements of Homotopy Theory, Springer-Verlag GTM 62, 1978. J. A. Wolf, Spaces of Constant Curvature, McGraw Hill, 1967. 6th ed. AMS Chelsea 2010. Papers J. F. Adams, On the non-existence of elements of Hopf invariant one, Ann. of Math. 72 (1960), 20–104. J. F. Adams, Vector ﬁelds on spheres, Ann. of Math. 75 (1962), 603–632. Bibliography 543 J. F. Adams, On the groups J(X) IV, Topology 5 (1966), 21–71. J. F. Adams and M. F. Atiyah, K–theory and the Hopf invariant, Quart. J. Math. 17 (1966), 31–38. J. F. Adams, A variant of E. H. Brown’s representability theorem, Topology 10 (1971), 185–198. J. F. Adams and C. W. Wilkerson, Finite H–spaces and algebras over the Steenrod algebra, Ann. of Math. 111 (1980), 95–143. K. K. S. Andersen and J. Grodal, The Steenrod problem of realizing polynomial cohomology rings, Journal of Topology 1 (2008), 747–760. M. G. Barratt and J. Milnor, An example of anomalous singular homology, Proc. A.M.S. 13 (1962), 293–297. M. Bestvina and N. Brady, Morse theory and ﬁniteness properties of groups, Invent. Math. 129 (1997), 445–470. A. Borel, Sur la cohomologie des espaces
ﬁbr´es principaux, Ann. of Math. 57 (1953), 115–207. R. Bott and H. Samelson, On the Pontryagin product in spaces of paths, Comment. Math. Helv. 27 (1953), 320–337. R. Bott and J. Milnor, On the parallelizability of spheres, Bull. A.M.S. 64 (1958), 87–89. R. Bott, The stable homotopy of the classical groups, Ann. of Math. 70 (1959), 313–337. E. H. Brown and A. H. Copeland, An homology analog of Postnikov systems, Michigan Math. J. 6 (1959), 313–330. E. H. Brown, Cohomology theories, Ann. of Math. 75 (1962), 467–484. M. Brown, A proof of the generalized Schoenﬂies theorem, Bull. A.M.S. 66 (1960), 74–76. J. W. Cannon and G. R. Conner, The combinatorial structure of the Hawaiian earring group, Top. and its Appl. 106 (2000), 225–271. G. Carlsson and R. J. Milgram, Stable homotopy and iterated loopspaces, pp. 505–583 in Handbook of Algebraic Topology, ed. I. M. James, Elsevier 1995. J. F. Davis and R. J. Milgram, A survey of the spherical space form problem, Math. Reports 2 (1985), 223–283, Harwood Acad. Pub. A. Dold and R. Thom, Quasifaserungen und unendliche symmetrische Produkte, Ann. of Math. 67 (1958), 239–281. C. H. Dowker, Topology of metric complexes, Am. J. Math. 74 (1952), 555–577. W. G. Dwyer, H. R. Miller, and C. W. Wilkerson, Homotopical uniqueness of classifying spaces, Topology 31 (1992), 29–45. W. G. Dwyer and C. W. Wilkerson, A new ﬁnite loop space at the prime two, Journal A.M.S. 6 (1993), 37–63. E. Dyer and A. T. Vasquez, Some small
aspherical spaces, J. Austral. Math. Soc. 16 (1973), 332–352. 544 Bibliography R. D. Edwards, A contractible, nowhere locally connected compactum, Abstracts A.M.S. 20 (1999), 494. S. Eilenberg, Singular homology theory, Ann. of Math. 45 (1944), 407–447. S. Eilenberg and J. A. Zilber, Semi-simplicial complexes and singular homology, Ann. of Math. 51 (1950), 499–513. S. Feder and S. Gitler, Mappings of quaternionic projective spaces, Bol. Soc. Mat. Mexicana 18 (1978), 33–37. J. Gubeladze, The isomorphism problem for commutative monoid rings, J. Pure Appl. Alg. 129 (1998), 35–65. J. C. Harris and N. J. Kuhn, Stable decompositions of classifying spaces of ﬁnite abelian p groups, Math. Proc. Camb. Phil. Soc. 103 (1988), 427–449. P. Hoffman and G. J. Porter, Cohomology realizations of Q [x], Ox. Q. 24 (1973), 251–255. R. Holzsager, Stable splitting of K(G,1), Proc. A.M.S. 31 (1972), 305–306. H. Hopf, ¨Uber die Abbildungen der dreidimensionalen Sph¨are auf die Kugelﬂ¨ache, Math. Ann. 104 (1931), 637–665. D. C. Isaksen, Stable stems, arXiv: 1407.8418. I. M. James, Reduced product spaces, Ann. of Math. 62 (1955), 170–197. M. A. Kervaire, Non-parallelizability of the n sphere for n > 7, Proc. N.A.S. 44 (1958), 280– 283. I. Madsen, C. B. Thomas, and C. T. C. Wall, The topological spherical space form problem, II: existence of free actions, Topology 15 (1976), 375–382. C. Manolescu, Pin(2)-equivariant Seiberg-Witten Floer
homology and the Triangulation Con- jecture, Jour. A.M.S. 29 (2016), 147–176. J. P. May, A general approach to Steenrod operations, Springer Lecture Notes 168 (1970), 153– 231. J. P. May, Weak equivalences and quasiﬁbrations, Springer Lecture Notes 1425 (1990), 91–101. C. Miller, The topology of rotation groups, Ann. of Math. 57 (1953), 95–110. J. W. Milnor, Construction of universal bundles I, II, Ann. of Math. 63 (1956), 272–284, 430– 436. J. W. Milnor, Groups which act on S n without ﬁxed points, Am. J. Math 79 (1957), 623–630. J. W. Milnor, Some consequences of a theorem of Bott, Ann. of Math. 68 (1958), 444–449. J. W. Milnor, On spaces having the homotopy type of a CW complex, Trans. A.M.S. 90 (1959), 272–280. J. W. Milnor, On axiomatic homology theory, Pac. J. Math. 12 (1962), 337–341. J. W. Milnor and J. C. Moore, On the structure of Hopf algebras, Ann. of Math. 81 (1965), 211–264. Bibliography 545 J. W. Milnor, On the Steenrod homology theory, pp. 79–96 in Novikov Conjectures, Index Theorems, and Rigidity, ed. S. Ferry, A. Ranicki, and J. Rosenberg, Cambridge Univ. Press, 1995. J. Neisendorfer, Primary homotopy theory, Mem. A.M.S. 232 (1980). D. Notbohm, Spaces with polynomial mod–p cohomology, Math. Proc. Camb. Phil. Soc., (1999), 277–292. D. Quillen, Rational homotopy theory, Ann. of Math. 90 (1969), 205–295. D. L. Rector, Loop structures on the homotopy type of S 3, Springer Lecture Notes 249 (1971), 99–105. C. A. Robinson, Moore-Postnikov systems for non-simple �
��brations, Ill. J. Math. 16 (1972), 234–242. P. Scott and T. Wall, Topological methods in group theory, London Math. Soc. Lecture Notes 36 (1979), 137–203. J.-P. Serre, Homologie singuli`ere des espaces ﬁbr´es, Ann. of Math. 54 (1951), 425–505. J.-P. Serre, Groupes d’homotopie et classes de groupes ab´eliens, Ann. of Math. 58 (1953), 258–294. S. Shelah, Can the fundamental group of a space be the rationals?, Proc. A.M.S. 103 (1988), 627–632. A. J. Sieradski, Algebraic topology for two dimensional complexes, pp. 51–96 in Two-Dimen- sional Homotopy and Combinatorial Group Theory, ed. C. Hog-Angeloni, W. Metzler, and A. J. Sieradski., Cambridge Univ. Press, 1993. J. Stallings, A ﬁnitely presented group whose 3 dimensional integral homology is not ﬁnitely generated, Am. J. Math. 85 (1963), 541–543. A. Strøm, Note on coﬁbrations II, Math. Scand. 22 (1968), 130–142. D. Sullivan, Genetics of homotopy theory and the Adams conjecture, Ann. of Math. 100 (1974), 1–79. H. Toda, Note on the cohomology ring of certain spaces, Proc. A.M.S. 14 (1963), 89–95. E. van Kampen, On the connection between the fundamental groups of some related spaces, Am. J. Math. 55 (1933), 261–267. C. T. C. Wall, Finiteness conditions for CW complexes, Ann. of Math. 81 (1965), 56–69. G. W. Whitehead, Generalized homology theories, Trans. A.M.S. 102 (1962), 227–283. J. H. C. Whitehead, Combinatorial homotopy II, Bull. A.M.S. 55 (1949), 453–496. abelian space 342, 417 boundary homomorphism 105, 108,
technical restrictions on what subsets can or cannot be events, according to the mathematical subject of measure theory. But we will not concern ourselves with such technicalities here.) Finally, and most importantly, a probability model requires a probability measure, usually written P. This probability measure must assign, to each event A, a probability P A. We require the following properties: 1. P A is always a nonnegative real number, between 0 and 1 inclusive. 2. P 0, i.e., if A is the empty set, then P A 0. 3. P S 1, i.e., if A is the entire sample space S, then P A 1. 4. P is (countably) additive, meaning that if A1 A2 is a ﬁnite or countable sequence of disjoint events, then P A1 A2 P A1 P A2 (1.2.1) The ﬁrst of these properties says that we shall measure all probabilities on a scale from 0 to 1, where 0 means impossible and 1 (or 100%) means certain. The second property says the probability that nothing happens is 0; in other words, it is impossible that no outcome will occur. The third property says the probability that something happens is 1; in other words, it is certain that some outcome must occur. The fourth property is the most subtle. It says that we can calculate probabilities of complicated events by adding up the probabilities of smaller events, provided those smaller events are disjoint and together contain the entire complicated event. Note that events are disjoint if they contain no outcomes in common. For example, rain and snow, clear are disjoint, whereas rain and rain, clear are not disjoint. (We are assuming for simplicity that it cannot both rain and snow tomorrow.) Thus, we should, but do not expect to have P rain have P rain (the latter being the same as P rain clear P snow clear P rain clear ). P rain snow clear P rain rain clear We now formalize the deﬁnition of a probability model. Deﬁnition 1.2.1 A probability model consists of a nonempty set called the sample space S; a collection of events that are subsets of S; and a probability measure P assigning a probability between 0 and 1 to each event, with P 1 and with P additive as in (1.2.1). 0 and P S 6 Section 1
.2: Probability Models EXAMPLE 1.2.1 Consider again the weather example, with S rain, snow, clear. Suppose that the probability of rain is 40%, the probability of snow is 15%, and the probability of a clear day is 45%. We can express this as P rain 0 15, and P clear 0 40, P snow 0 45. 0, i.e., it is impossible that nothing will happen For this example, of course P tomorrow. Also P rain, snow, clear 1, because we are assuming that exactly one of rain, snow, or clear must occur tomorrow. (To be more realistic, we might say that we are predicting the weather at exactly 11:00 A.M. tomorrow.) Now, what is the probability that it will rain or snow tomorrow? Well, by the additivity property, we see that P rain snow P rain P snow 0 40 0 15 0 55 We thus conclude that, as expected, there is a 55% chance of rain or snow tomorrow. EXAMPLE 1.2.2 Suppose your candidate has a 60% chance of winning an election in progress. Then S 1. win, lose, with P win 0 4. Note that P win 0 6 and P lose P lose EXAMPLE 1.2.3 Suppose we ip a fair coin, which can come up either heads (H ) or tails (T ) with equal probability. Then S 1. 0 5. Of course, P H H T, with P H P T P T EXAMPLE 1.2.4 Suppose we ip three fair coins in a row and keep track of the sequence of heads and tails that result. Then, Furthermore, each of these eight outcomes is equally likely. Thus, etc. Also, the probability that the ﬁrst coin is heads and the second coin is tails, but the third coin can be anything, is equal to the sum of the probabilities of the events H T H and H T T, i.e.,. 1 8 1 8 EXAMPLE 1.2.5 Suppose we ip three fair coins in a row but care only about the number of heads 0 1 2 3. However, the probabilities of these four outcomes that result. Then S are not all equally likely; we will see later that in fact P 0 1 8, while P 1 3 8. P 2 P 3 We note that it is possible to deﬁne probability models on
more complicated (e.g., uncountably inﬁnite) sample spaces as well. EXAMPLE 1.2.6 Suppose that S S by saying that [0 1] is the unit interval. We can deﬁne a probability measure P on P [a b] b a whenever 0 a b 1 (1.2.2) Chapter 1: Probability Models 7 In words, for any1 subinterval [a b] of [0 1], the probability of the interval is simply the length of that interval. This example is called the uniform distribution on [0 1]. The uniform distribution is just the ﬁrst of many distributions on uncountable state spaces. Many further examples will be given in Chapter 2. 1.2.1 Venn Diagrams and Subsets Venn diagrams provide a very useful graphical method for depicting the sample space S and subsets of it. For example, in Figure 1.2.1 we have a Venn diagram showing the subset A S and the complement Ac s : s A of A The rectangle denotes the entire sample space S The circle (and its interior) de notes the subset A the region outside the circle, but inside S denotes Ac S 1 A A S Ac Figure 1.2.1: Venn diagram of the subsets A and Ac of the sample space S. Two subsets A S and B the next page. The intersection S are depicted as two circles, as in Figure 1.2.2 on A B s : s A and s B of the subsets A and B is the set of elements common to both sets and is depicted by the region where the two circles overlap. The set A B A Bc s : s A and s B is called the complement of B in A and is depicted as the region inside the A circle, but not inside the B circle. This is the set of elements in A but not in B Similarly, we have the complement of A in B namely, Ac B Observe that the sets A B A Bc, and Ac B are mutually disjoint. 1For the uniform distribution on [0 1], it turns out that not all subsets of [0 1] can properly be regarded as events for this model. However, this is merely a technical property, and any subset that we can explicitly write down will always be an event. See more advanced probability books, e.g., page 3 of A First Look at Rig
orous Probability Theory, Second Edition, by J. S. Rosenthal (World Scientiﬁc Publishing, Singapore, 2006). 8 The union Section 1.2: Probability Models A B s : s A or s B of the sets A and B is the set of elements that are in either A or B In Figure 1.2.2, it is depicted by the region covered by both circles. Notice that A A B A Bc Ac B B There is one further region in Figure 1.2.2. This is the complement of A namely, the set of elements that are in neither A nor B So we immediately have B Similarly, we can show that A B c Ac Bc A B c Ac Bc namely, the subset of elements that are not in both A and B is given by the set of ele ments not in A or not in B S B Ac  B A  B A A  Bc Ac  Bc Figure 1.2.2: Venn diagram depicting the subsets A, B, A B, A Bc, Ac and A B B, Ac Bc, Finally, we note that if A and B are disjoint subsets, then it makes sense to depict these as drawn in Figure 1.2.3, i.e., as two nonoverlapping circles because they have no elements in common. S A B 1 A Figure 1.2.3: Venn diagram of the disjoint subsets A and B Chapter 1: Probability Models 9 Summary of Section 1.2 A probability model consists of a sample space S and a probability measure P assigning probabilities to each event. Different sorts of sets can arise as sample spaces. Venn diagrams provide a convenient method for representing sets and the rela tionships among them. EXERCISES s 8. 1 6. 1 2. 1 8 for 1 0 5, P 3 0 2, P 2 0 7, P 1 1 2, P 2 1, P 1 2 1 3, and P 3 2 3. What must 1 2 3, with P 1 1 2 3, with, with P s 1 2? 1 2 and P 1 2 1 2 3, and we try to deﬁne P by P 1 2 3 1.2.1 Suppose S (a) What is P 1 2? (b) What is P 1 2 3? (c) List all
events A such that P A 1.2.2 Suppose S (a) What is P 1 2? (b) What is P 1 2 3? (c) How many events A are there such that P A 1.2.3 Suppose S P 2 be? 1.2.4 Suppose S 0 7, P 1 3 0 5, P 2 3 P a valid probability measure? Why or why not? 1.2.5 Consider the uniform distribution on [0 1]. Let s is P s? Do you ﬁnd this result surprising? 1.2.6 Label the subregions in the Venn diagram in Figure 1.2.4 using the sets A B and C and their complements (just as we did in Figure 1.2.2). 1.2.7 On a Venn diagram, depict the set of elements that are in subsets A or B but not in both. Also write this as a subset involving unions and intersections of A B and their complements. 1.2.8 Suppose S P 1, P 2, and P 3. 1.2.9 Suppose S P 1 2 3 1.2.10 Suppose S P 2, and P 3. 1.2.11 Suppose S Compute P 1, P 2, and P 3. 1.2.12 Suppose S Compute P 1, P 2, P 3, and P 4. 1 3. Compute P 1, P 2, P 3, and P 4. [0 1] be any outcome. What 2 P 2. Compute P 1, 1 2 3 4, and P 1 1 2 3, and P 1 2 1 2 3 4, and P 1 1 12, and P 1 2 1 2 3, and P 1 1 2 3, and P 1 1 3, and P 2 3 2 3. Compute 1 6, and, and 3 P 3 0 3. Is P 3 P 2 P 2 1 8 10 Section 1.3: Properties of Probability Models Figure 1.2.4: Venn diagram of subsets A B and C. PROBLEMS 1.2.13 Consider again the uniform distribution on [0 1]. Is it true that P [0 1] P s? s [0 1] How does this relate to the additivity property of probability measures? 1.2.14 Suppose S is a ﬁnite or countable set. Is it possible that P s single s
1.2.15 Suppose S is an uncountable set. Is it possible that P s s S? Why or why not? S? Why or why not? 0 for every 0 for every single DISCUSSION TOPICS 1.2.16 Does the additivity property make sense intuitively? Why or why not? 1.2.17 Is it important that we always have P S change if this were not the case? 1? How would probability theory 1.3 Properties of Probability Models The additivity property of probability measures automatically implies certain basic properties. These are true for any probability model at all. If A is any event, we write Ac (read “A complement”) for the event that A does not occur. In the weather example, if A snow, clear. In the coin examples, if A is the event that the ﬁrst coin is heads, then Ac is the event that the ﬁrst coin is tails. rain, then Ac Now, A and Ac are always disjoint. Furthermore, their union is always the entire sample space: A Ac S. Hence, by the additivity property, we must have P A Chapter 1: Probability Models P Ac P S. But we always have P S 1. Thus, P A P Ac 1, or P Ac 1 P A 11 (1.3.1) In words, the probability that any event does not occur is equal to one minus the prob ability that it does occur. This is a very helpful fact that we shall use often. Now suppose that A1 A2 are events that form a partition of the sample space are disjoint and, furthermore, that their union is equal S. This means that A1 A2 to S, i.e., A1 S. We have the following basic theorem that allows us to decompose the calculation of the probability of B into the sum of the probabilities of the sets Ai B. Often these are easier to compute. A2 Theorem 1.3.1 (Law of total probability, unconditioned version) Let A1 A2 be events that form a partition of the sample space S. Let B be any event. Then P B P A1 B P A2 B PROOF The events A1 B the result follows immediately from the additivity property (1.2.1). A2 B are disjoint, and their union is B. Hence, A somewhat more useful version of

Subsets and Splits

No community queries yet

The top public SQL queries from the community will appear here once available.