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j! Xi be the projection onto the i-th factor. Then Q Q ŒY; j 2J Xj! j 2J ŒY; Xj ; ` Œf 7!.Œpi ı f / is a well-deﬁned bijection. Let ik W Xk! the k-th summand. Then j 2J Xj be the canonical inclusion of ` j 2J Xj ; Y! Q j 2J ŒXj ; Y ; Œ Œf 7!.Œf ı ik/ is a well-deﬁned bijection. In other words: sum and product in TOP also represent sum and product in h-TOP. (Problems arise when it comes to pullbacks and pushouts.) Let P be a point. A map P! Y can be identiﬁed with its image and a homotopy P I! Y can be identiﬁed with a path. The Hom-functor ŒP; is therefore essentially the same thing as the functor 0. 2.1.5 Linear homotopy. Given maps f; g W X! A, A Rn. Suppose that the line-segment from f.x/ to g.x/ is always contained in A. Then H.x; t/ D.1 t/f.x/ C tg.x/ is a homotopy from f to g (linear homotopy). It will turn out that many homotopies are constructed from linear homotopies. A set A Rn is star-shaped with respect to a0 2 A if for each a 2 A the line-segment from a0 to a is contained in A. If A is star-shaped, then H.a; t/ D.1 t/a C ta0 is a null homotopy of the identity. Hence star-shaped sets are contractible. A set C Rn is convex if and only if it is star-shaped with respect to each of its points. Note: If A D Rn and a0 D 0, then each Ht, t < 1, is a homeomorphism, and only in the very last moment is H1 constant! This is less mysterious, if we look at Þ the paths t 7! H.x; t/. The reader should now recall the notion of a quot
ient map (identiﬁcation), its universal property, and the fact that the product of a quotient map by the identity of a locally compact space is again a quotient map (see (2.4.6)). (2.1.6) Proposition. Let p W X! Y be a quotient map. Suppose Ht W Y! Z is a family of set maps such that Ht ı p is a homotopy. Then Ht is a homotopy. 2As a general principle we use a lower index for covariant functors and an upper index for contravariant functors. If we apply a (covariant) functor to a morphism f we often call the result the induced morphism and denote it simply by f if the functor is clear from the context. 30 Chapter 2. The Fundamental Group Proof. The product p id W X I! Y I is an identiﬁcation, since I is compact. The composition H ı.p id/ is continuous and therefore H is continuous. (2.1.7) Proposition. Let Ht W X! X be a homotopy of the identity H0 D id.X/ such that the subspace ; 6D A X is always mapped into itself, Ht.A/ A. Suppose H1 is constant on A. Then the projection p W X! X=A (A identiﬁed to a point) is an h-equivalence. Proof. Since H1.A/ is a point, there exists a map q W X=A! X such that qp D H1. By assumption, this composition is homotopic to the identity. The map p ı Ht W X! X=A factorizes over p and yields Kt W X=A! X=A such that Kt p D pHt. By (2.1.6), Kt is a homotopy, K0 D id and K1 D pq. Problems 1. Suppose there exists a homeomorphism R! X Y. Then X or Y is a point. 2. Let f W X! Y be surjective. If X is (path) connected, then Y is (path) connected. 3. Let C be a countable subset of Rn, n 2. Show that Rn X C is path connected. 4. The unitary group U.n
/ and the general linear group GLn.C/ are path connected. The orthogonal group O.n/ and the general linear group GLn.R/ have two path components; one of them consists of matrices with positive determinant. 5. Let U Rn be open. The path components of U are open and coincide with the components. The set of path components is ﬁnite or countably inﬁnite. An open subset of R is a disjoint union of open intervals. 6. List theorems of point-set topology which show that the product homotopy and the inverse homotopy are continuous. Do the same for the linear homotopy in 2.1.5. 7. A space X is contractible if and only if the identity id.X/ is null homotopic. 8. gf is null homotopic, if f or g is null homotopic. 9. Let A be contractible. Then any two maps X! A are homotopic. 10. The inclusions O.n/ GLn.R/ and U.n/ GLn.C/ are homotopy equivalences. Let P.n/ denote the space of positive deﬁnite real.n; n/-matrices. Then O.n/ P.n/! GLn.R/,.X; P / 7! XP is a homeomorphism; P.n/ is star-like with respect to the unit matrix. 11. There exist contractible and non-contractible spaces consisting of two points. 2.2 Further Homotopy Notions The homotopy notion can be adapted to a variety of other contexts and categories: Consider homotopies which preserve some additional structure of a category. We describe some examples from which the general idea emerges. This section only contains terminology. The construction of group structures on homotopy sets uses the category of pointed spaces, as we will see shortly. We call a pair.X; x0/ consisting of a space 2.2. Further Homotopy Notions 31 X and a base point x0 2 X a pointed space. A pointed map f W.X; x0/!.Y; y0/ is a continuous map f W X! Y which sends the base point to the base point. A homotopy H W X I! Y is pointed if Ht is pointed for each t 2
I. We denote by ŒX; Y 0 the set of pointed homotopy classes (ﬁxed base points assumed) or by Œ.X; x0/;.Y; y0/. We obtain related notions: pointed homotopy equivalence, pointed contractible, pointed null homotopy. We denote the category of pointed spaces and pointed maps by TOP0, and by h-TOP0 the associated homotopy category. Often a base point will just be denoted by. Also a set with a single element will be denoted by its element. The choice of a base point is an additional structure. There is a functor ˛ from TOP to TOP0 which sends a space X to X C D X C fg, i.e., to X with an additional base point added (topological sum), with the obvious extension to pointed maps. This functor is compatible with homotopies. We also have the forgetful functor ˇ from TOP0 to TOP. They are adjoint TOP0.˛.X/; Y / Š TOP.X; ˇY /, and similarly for the homotopy categories. j Xj ; this yields the pointed product. Let The category TOP0 has sums and products. Suppose.Xj ; xj / is a family of pointed spaces. The family.xj / of base points is taken as base point in the product Q j 2J Xj where all base points are identiﬁed to a single new base point. We have canonical pointed maps ik W Xk! j Xj which arise from the canonical inclusions Xk! ` j Xj of the pointed spaces Xj j 2J Xj be the quotient of j Xj. The wedge, also called the bouquet, together with the ik is the pointed sum in TOP0. ` W W W The sum and the product in TOP0 also represent the sum and the product in h-TOP0 (use (2.1.6)). Let.A; a/ and.B; b/ be pointed spaces. Their smash product is A ^ B D A B=A b [ a B D A B=A _ B: (This is not a categorical product. It is rather analogous to the tensor product.) The smash product is a functor in two variables and also compatible with homotopies: Given f W A! C
; g W B! D we have the induced map ;.a; b/ 7!.f.a/; g.b//; and homotopies ft ; gt induce a homotopy ft ^gt. Unfortunately, there are point-set topological problems with the associativity of the smash product (see Problem 14). A pair.X; A/ of topological spaces consists of a space X and a subspace A. A morphism f W.X; A/!.Y; B/ between pairs is a map f W X! Y such that f.A/ B. In this way we obtain the category of pairs TOP.2/. A homotopy H in this category is assumed to have each Ht a morphism of pairs. We write Œ.X; A/;.Y; B/ for the associated homotopy sets and h-TOP.2/ for the homotopy category. If.X; A/ is a pair, we usually consider the quotient space X=A as a pointed space (A identiﬁed to a point) with base point fAg. If A D ;, then X=A D X C is X with a separate base point. 32 Chapter 2. The Fundamental Group (2.2.1) Note. A continuous map f W.X; A/!.Y; / into a pointed space induces a pointed map xf W X=A! Y. The assignment f 7! xf induces a bijection Œ.X; A/;.Y; / Š ŒX=A; Y 0. A veriﬁcation uses (2.1.6). We use the notation.X; A/.Y; B/ D.X Y; X B [ A Y /; although this is not a categorical product. With this notation.I m; @I m/.I n; @I n/ D.I mCn; @I mCn/. In a similar manner we treat other conﬁgurations, e.g., triples.X; A; B/ of spaces A B X and the category TOP.3/ of triples. Let K and B be ﬁxed spaces. The category TOPK of spaces under K has as objects the maps i W K! X. A morphism from i W K! X to j W K! Y
is a map f W X! Y such that f i D j. The category TOPB of spaces over B has as objects the maps p W X! B. A morphism from p W X! B to q W Y! B is a map f W X! Y such that qf D p. If B is a point, then TOPB can be identiﬁed with TOP, since each space has a unique map to a point. If K D fg is a point, then TOPK is the same as TOP0. If p W X! B is given, then p1.b/ is called the ﬁbre of p over b; in this context, B is the base space and X the total space of p. A map in TOPB will also be called ﬁbrewise or ﬁbre preserving. Categories like TOPK or TOPB have an associated notion of homotopy. A homotopy Ht is in TOPK if each Ht is a morphism in this category. A similar deﬁnition is used for TOPB. A homotopy in TOPB will also be called ﬁbrewise or ﬁbre preserving. Again, being homotopic is an equivalence relation in these categories. We denote by ŒX; Y K the set of homotopy classes in TOPK, and by ŒX; Y B the set of homotopy classes in TOPB. The homotopy categories are h-TOPK and h-TOPB. Note that a homotopy equivalence in TOPB, i.e., a ﬁbrewise homotopy equivalence, from p W X! B to q W Y! B induces for each b 2 B a homotopy equivalence p1.b/! q1.b/ between the ﬁbres over b, so this is a continuous family of ordinary homotopy equivalences, parametrized by B ([96], [97], [38], [128]). It is called a retraction of i. A morphism r from i W K! X to id W K! K in TOPK is a map r W X! K If it exists, then i is an such that ri D id.X/. If i W K X we then call K a retract of X. The retraction r of embedding. i W K X is a hom
A. Show that p W X! X=A is a pointed h-equivalence. Is.X; A/ h-equivalent to.X; fag/? 6. Show that (2.1.7) yields a homotopy equivalence of pairs.X; A/!.X=A; /. 7. The inclusion.I; @I /!.I; I X f1=2g/ is not an h-equivalence in TOP.2/ although the component maps I! I and @I! I X f1=2g are h-equivalences. 8. Let E R2 consist of k points. Show, heuristically, that the complement R2 X E is h-equivalent to the k-fold sum 9. Remove a point from the torus S 1 S 1 and show that the result is h-equivalent to S 1 _S 1. Is there an analogous result when you remove a point from S m S n? 10. Construct an inclusion A X which is a retract and a homotopy equivalence but not a deformation retract. 11. Construct a map p W E! B such that all ﬁbres p1.b/ are contractible but which does not have a section. Construct an h-equivalence p W E! B which has a section but which is not shrinkable. 12. What is the sum of two objects in TOPK? What is the product of two objects in TOPB? 13. A pullback of a shrinkable map is shrinkable. A pushout of a deformation retract is a deformation retract. 14. Let Y; Z be compact or X; Z be locally compact. Then the canonical bijection (the identity).Y ^ Z/ is a homeomorphism. (In the category of compactly generated spaces (with its associated product and smash product!) the map is always a homeomorphism. See also [155, Satz 18].) ^ B be the canonical map which is on each sum15. Let mand Ak ^ B induced by the inclusion Ak! j Aj. Show that this map is a homeomorphism if the index set is ﬁnite. Show that in this case both spaces are quotients of.qj Aj / B Š qj.Aj B/. 16. Let A be a compact subset of X and p W X! X=A be the
quotient map. Then for each space Y the product p id.Y / is a quotient map. If X is a Hausdorff space, then p is proper and p id closed. 17. The canonical map X I! X I =@I! X ^ I =@I is a quotient map which induces a homeomorphism †X D X I =.X @I [ fg I / Š X ^ I =@I. 18. There is a canonical bijective continuous map.X Y /=.X B [AY /! X=A^Y =B j.Aj ^ B/! j Aj W W W 34 Chapter 2. The Fundamental Group (the identity on representatives). It is a homeomorphism if X Y! X=A ^ Y =B is a quotient map, e.g., if X and Y =B are locally compact (or in the category of compactly generated spaces). 2.3 Standard Spaces Standard spaces are Euclidean spaces, disks, cells, spheres, cubes and simplices. We collect notation and elementary results about such spaces. The material will be used almost everywhere in this book. We begin with a list of spaces. The Euclidean norm is kxk. Rn Dn D fx 2 Rn j kxk 1g S n1 D fx 2 Dn j kxk D1g D@D n E n D Dn X S n1 I n D fx 2 Rn j 0 xi 1g @I n D fx 2 I n j xi D 0; 1 for some ig P n D Œn D fx 2 RnC1 j xi 0; @n D f.xi / 2 n j some xi D 0g Euclidean space n-dimensional disk.n 1/-dimensional sphere n-dimensional cell n-dimensional cube boundary of I n i xi D 1g n-dimensional simplex boundary of n The spaces Dn, I n, En and n are convex and hence contractible. We think of R0 D f0g. The spaces D0, I 0, and 0 are singletons, and S 1 D @D0, @0 are empty. In the case of n we use the indexing t D.t0; : : : ; tn/ 2 n; the subset @i n D ft 2 n j ti D
0g is the i-th face of n; hence @n D S n iD0 @i n. It is useful to observe that certain standard spaces are homeomorphic. A general result of this type is: (2.3.1) Proposition. Let K Rn be a compact convex subset with non-empty interior Kı. Then there exists a homeomorphism of pairs.Dn; S n1/!.K; @K/ which sends 0 2 Dn to a pre-assigned x 2 Kı. Proof. Let K Rn be closed and compact and 0 2 Kı. Verify that a ray from 0 intersects the boundary @K of K in Rn in exactly one point. The map f W @K! S n1, x 7! x=kxk is a homeomorphism. The continuous map'W S n1 Œ0; 1! K,.x; t/ 7! tf 1.x/ factors over q W S n1 Œ0; 1! Dn,.x; t/ 7! tx and yields a bijective map k W Dn! K, hence a homeomorphism (use (1.4.3)). This proposition can be used to deduce a homeomorphism.Dn; S n1/ Š.I n; @I n/. The simplex n is a compact convex subset with interior points in the hyperplane fx 2 RnC1 j i xi D 1g. From this fact we deduce a homeomorphism.Dn; @Dn/ Š.n; @n/. P The sphere S n, as a homeomorphism type, will appear in many different shapes. 2.3. Standard Spaces 35 (2.3.2) Example. Let N D enC1 D.0; : : : ; 0; 1/ 2 RnC1. We deﬁne the stereographic projection 'N W S n X fenC1g!R n; the point 'N.x/ is the intersection of the line through enC1 and x with the hyperplane Rn 0 D Rn. One computes 'N.x1; : : : ; xnC1/ D.1 xnC1/1.x1; : : : ; xn/. The inverse map is N W x 7!..1 C kx
k/2/1.2x; kxk2 1/. We also have the stereographic projection 'S W S n X fenC1g!R n and the transition map is 'S ı '1 N.y/ D kyk2y. From the stereographic projection we obtain S n as a speciﬁc model of the onepoint compactiﬁcation Rn [ f1g by extending N.1/ D enC1. We also write S V D V [ f1g for the one-point compactiﬁcation of a ﬁnite-dimensional real Þ vector space V. 2.3.3 Spheres. Let y 2 S n. From (2.3.2) we see that S n X y is homeomorphic to Rn and hence contractible. Thus, if X! S n is not surjective, it is null homotopic. The inclusion i W S n! RnC1 X f0g is an h-equivalence with homotopy inverse N W RnC1 X f0g!S n, x 7! kxk1x. A homotopy (rel S n) from i ı N to the identity is the linear homotopy.x; t/ 7! tx C.1 t/iN.x/. Moreover N ı i D id. We see that i is a deformation retract. Under suitable circumstances each map in a small neighbourhood of f is already homotopic to f. For a general theorem to this effect see (15.8.3). Here we only give a simple, but typical, example. Let f; g W X! S n be maps such that kf.x/ g.x/k < 2. Then they are homotopic by a linear homotopy when viewed as maps into RnC1 X f0g. We compose with N and see that f'g. If f W S m! S n is a continuous map, then there exists (by the theorem of Stone–Weierstrass, say) a C 1-map g W S m! S n such that kf.x/ g.x/k < 2. This indicates another use of homotopies: Improve maps up to homotopy. If one uses some analysis, namely (the easy part of) the theorem of Sard about
the density of regular values, one sees that for m < n a C 1-map S m! S n is not surjective and hence null homotopic. (Later we prove this fact by other methods.) There exist surjective continuous maps S 1! S 2 (Peano curves); this ungeometric behaviour of continuous maps is the source for many of the technical difﬁculties in topology.Þ (2.3.4) Proposition. The map p W S n1 I! Dn,.x; t/ 7!.1 t/x is a quotient map. Given F W Dn! X, the composition Fp W S n1 I! X is a null homotopy of f D F jS n1. Each null homotopy of a map f W S n1! X arises from a unique F. Proof. Since a null homotopy H of f W S n1! X sends S n1 1 to a point, it factors through the quotient map q W S n1 I! S n1 I =S n1 1. Thus null homotopies H correspond via xH 7! xH q to maps xH W S n1 I =S n1 1! X. The map p induces a homeomorphism xp W S n1 I =S n1 1! Dn (use (1.4.3)). Hence there exists a unique F such that F xp D xH. 36 Chapter 2. The Fundamental Group Let us use the notation S.n/ D I n=@I n; S.n/ D Rn [ f1g; since these spaces are homeomorphic to S n. The canonical map I nCm=@I nCm! I m@I m ^ I n=@I n which is the identity on representatives is a pointed homeomorphism. If V and W are ﬁnite-dimensional real vector spaces, we have a canonical pointed homeomorphism S V ^ S W Š S V ˚W which is the identity away from the base point. The homeomorphism 0; 1Œ! R, s 7! 2s1 s.1s/ induces a homeomorphism W S.1/! S.1/ which transports t 7! 1 t into the antipodal map x 7! x on R. We obtain an induced homeomorphism n
W S.n/ D S.1/ ^ ^ S.1/! S.1/ ^ ^ S.1/ D S.n/ of the n-fold smash products. (2.3.5) Example. A retraction r W Dn!S n1 I [ Dn 0 is r.x; t/ D.2˛.x; t/1 x; ˛.x; t/2Ct/ with ˛.x; t/ D max.2kxk; 2t/. (See Figure 2.1, a central projection from the point.0; 2/.) Given a map f W I n! X and a homotopy h W @I n I! X with h0 D f j@I n combine to a map g W I n 0 [ @I n I! X. We compose with a retraction and obtain a homotopy H W I n! X which extends h and begins at H0 D f. This homotopy extension property is later studied more Þ generally under the name of coﬁbration..0; 2/........................................ I x r.x/ Dn Figure 2.1. A retraction. (2.3.6) Example. The assignment H W.x; t/ 7!.˛.x; t/1.1 C t/ x; 2 ˛.x; t// with the function ˛.x; t/ D max.2kxk; 2 t/ yields a homeomorphism of pairs.Dn; S n1/.I; 0/ Š Dn.I; 0/, see Figure 2.2. Similarly for.I n; @I n/ in place of.Dn; S n1/, since these two pairs are homeoÞ morphic. 2.4. Mapping Spaces and Homotopy 37 b0 a0 I c0.................... Dn a b c H! a0 b0 c0.................... a c b Figure 2.2. A relative homeomorphism. Problems S j f 1 C D f.x0; : : : ; xn/ 2 S n j xn 0g. Show that the quotient map S n! S n=Dn C 1. Construct a homeomorphism.Dm; S m1/.Dn; S n1
/ Š.DmCn; S mCn1/. 2. RnC1 X fxg!S n, z 7!.z x/=kz xk is an h-equivalence. 3. Let Dn is an h-equivalence. 4. Let f1; : : : ; fk W Cn! C be linearly independent linear forms (k n). Then the complement Cn X 5. S n! f.x; y/ 2 S n S n j x 6D yg, x 7!.x; x/ is an h-equivalence. 6. Let f; g W X! S n be maps such that always f.x/ 6D g.x/. Then f'g. 7. Let A En be star-shaped with respect to 0. Show that S n1 Rn XA is a deformation retract. 8. The projection p W TS n D f.x; v/ 2 S n RnC1 j x? vg!S n,.x; v/ 7! x is called the tangent bundle of S n. Show that p admits a ﬁbrewise homeomorphism with pr,.x; y/ 7! x (with D the diagonal)..0/ is homotopy equivalent to the product of k factors S 1. j 2.4 Mapping Spaces and Homotopy It is customary to endow sets of continuous maps with a topology. In this section we review from point-set topology the compact-open topology. It enables us to consider a homotopy H W X I! Y as a family of paths in Y, parametrized by X. This dual aspect of the homotopy notion will be used quite often. It can be formalized; but we use it more like a heuristic principle to dualize various constructions and notions in homotopy theory (Eckmann–Hilton duality). We denote by Y X or F.X; Y / the set of continuous maps X! Y. For K X and U Y we set W.K; U / D ff 2 Y X j f.K/ U g. The compact-open topology (CO-topology) on Y X is the topology which has as a subbasis the sets of the form W.K; U / for compact K X and open U Y
. In the sequel the set Y X always carries the CO-topology. A continuous map f W X! Y induces continuous maps f Z W X Z! Y Z, g 7! fg and Zf W ZY! ZX, g 7! gf. (2.4.1) Proposition. Let X be locally compact. Then the evaluation eX;f; x/ 7! f.x/ is continuous. 38 Chapter 2. The Fundamental Group Proof. Let U be an open neighbourhood of f.x/. Since f is continuous and X locally compact, there exists a compact neighbourhood K of x such that f.K/ U. The neighbourhood W.K; U / K of.f; x/ is therefore mapped under e into U. This shows the continuity of e at.f; x/. (2.4.2) Proposition. Let f W X Y! Z be continuous. Then the adjoint map f ^ W X! ZY, f ^.x/.y/ D f.x; y/ is continuous. Proof. Let K Y be compact and U Z open. It sufﬁces to show that W.K; U / has an open pre-image under f ^. Let f ^.x/ 2 W.K; U / and hence f.fxg K/ U. Since K is compact, there exists by (1.4.1) a neighbourhood V of x in X such that V K f 1.U / and hence f ^.V / W.K; U /. From (2.4.2) we obtain a set map ˛ W ZXY!.ZY /X, f 7! f ^. Let eY;Z be continuous. A continuous map'W X! ZY induces a continuous map '_ D eY;Z ı.' idY / W X Y! ZY Y! Z. Hence we obtain a set map ˇ W.ZY /X! ZXY,'7! '_. (2.4.3) Proposition. Let eY;Z be continuous. Then ˛ and ˇ are inverse bijections. Thus'W X Y! Z is continuous if '_ W X Y! Z is continuous, and f W X Y! Z is continuous if f ^ W X! ZY is continuous. (2.4.4) Corollary. If h W
X Y I! Z is a homotopy, then h^ W X I! ZY is a homotopy (see (2.4.2)). Hence ŒX Y; Z! ŒX; ZY, Œf 7! Œf ^ is well-deﬁned. If, moreover, eY;Z is continuous, e.g., Y locally compact, then this map is bijective (see (2.4.3)). 2.4.5 Dual notion of homotopy. We have the continuous evaluation et W Y I! Y, w 7! w.t/. A homotopy from f0 W X! Y to f1 W X! Y is a continuous map h W X! Y I such that e" ı h D f" for " D 0; 1. The equivalence with our original deﬁnition follows from (2.4.3): Since I is locally compact, continuous Þ maps X I! Y correspond bijectively to continuous maps X! Y I. (2.4.6) Theorem. Let Z be locally compact. Suppose p W X! Y is a quotient map. Then p id.Z/ W X Z! Y Z is a quotient map. Proof. We verify for p id the universal property of a quotient map: If h W Y Z! C is a set map and h ı.p id/ is continuous, then h is continuous. The adjoint of h ı.p id/ is h^ ı p. By (2.4.2), it is continuous. Since p is a quotient map, h^ is continuous. Since Z is locally compact, h is continuous, by (2.4.3). (2.4.7) Theorem (Exponential law). Let X and Y be locally compact. Then the adjunction map ˛ W ZXY!.ZY /X is a homeomorphism. 2.4. Mapping Spaces and Homotopy 39 Proof. By (2.4.3), ˛ is continuous if ˛1 D eX;ZY ı.˛ id/ is continuous. And this map is continuous if ˛2 D eY;X ı.˛1 id/ is continuous. One veriﬁes that ˛2 D eXY;Z
. The evaluations which appear are continuous by (2.4.1). The inverse ˛1 is continuous if eXY;Z ı.˛1 id/ is continuous, and this map equals eY;Z ı.eX;ZY id/. Let.X; x/ and.Y; y/ be pointed spaces. We denote by F 0.X; Y / the space of pointed maps with CO-topology as a subspace of F.X; Y /. In F 0.X; Y / we use the constant map as a base point. The adjoint f ^ W X! F.Y; Z/ of f W X Y! Z is a pointed map into F 0.X; Y / if and only if f sends X y [ x Y to the base point of Z. Let =.X y [ x Y / be the quotient map. If g W X ^Y! Z is given, we denote the adjoint of gıp W X Y! X ^Y! Z by ˛0.g/ and consider it as an element of F 0.X; F 0.Y; Z//. In this manner we obtain a set map ˛0 W F 0.X ^ Y; Z/! F 0.X; F 0.Y; Z//. The evaluation F 0.X; Y / X! Y,.f; x/ 7! f.x/ factors over the quotient X;Y W F 0.X; Y / ^ X! Y. From (2.4.1) space F 0.X; Y / ^ X and induces e0 D e0 we conclude: (2.4.8) Proposition. Let X be locally compact. Then e0 X;Y is continuous. Let e0 X;Y be continuous. From a pointed map'W X! F 0.Y; Z/ we obX;Y ı.' ^ id/ W X ^ Y! Z, and hence a set map tain '_ D ˇ0.'/ D e0 ˇ0 W F 0.X; F 0.Y; Z//! F 0.X ^ Y; Z/. (2.4.9) Proposition. Let e0 X;Y be continuous. Then ˛0 and ˇ0 are inverse bijections. (2.4.10) Coroll
ary. Let h W.X ^ Y / I! Z be a pointed homotopy. Then ˛0.ht / W X! F 0.Y; Z/ is a pointed homotopy and therefore ŒX ^ Y; Z0! ŒX; F 0.Y; Z/0; Œf 7! Œ˛0.f / is well deﬁned. If, moreover, e0 X;Y is continuous, then this map is bijective. By a proof formally similar to the proof of (2.4.7), we obtain the pointed version of the exponential law. (2.4.11) Theorem (Exponential law). Let X and Y be locally compact. Then the pointed adjunction map ˛0 W F 0.X ^ Y; Z/! F 0.X; F 0.Y; Z// is a homeomor- phism. (2.4.12) Lemma. Let ka W Z! A denote the constant map with value a. Then W X Z A!.X A/Z,.'; a/ 7!.'; ka/ is continuous. 40 Chapter 2. The Fundamental Group Proof. Let.f; a/ 2 W.K; U /. This means: For x 2 K we have.f.x/; a/ 2 U. There exist open neighbourhoods V1 of f.K/ in X and V2 of a in A such that V1 V2 U. The inclusion.W.K; V1/ V2/ W.K; U / shows the continuity of at.f; a/. (2.4.13) Proposition. A homotopy Ht W X! Y induces homotopies H Z t and ZHt. Proof. In the ﬁrst case we obtain, with a map from (2.4.12), a continuous map H Z ı W X Z I!.X I /Z! Y Z: In the second case we use the composition e ı.˛ id/ ı.ZH id/ W ZY I! ZXI I!.ZX /I I! ZX which is continuous. (2.4.14) Corollary. Let f be a homotopy equivalence. Then the induced maps F.Z; X/! F.Z; Y / and F.
Y; Z/! F.X; Z/ are h-equivalences. If f is a pointed h-equivalence, the induced maps F 0.Z; X/! F 0.Z; Y / and F 0.Y; Z/! F 0.X; Z/ are pointed h-equivalences. Problems ` j Xj ; Y Q 1. Verify that f Z and Zf are continuous. 2. An inclusion i W Z Y induces an embedding i X W ZX! Y X. Q 3. The canonical map F 4. The canonical map F.X; continuous. If X is locally compact, it is a homeomorphism. 5. Let p W X! Y be a surjective continuous map. Suppose the pre-image of a compact set is compact. Then Zp W ZY! ZX is an embedding. W 6. We have a canonical bijective map F 0 W j F.Xj ; Y / is always a homeomorphism. j F.X; Yj /, f 7!.prj f / is always bijective and! j Xj is the sum in TOP0. If J is ﬁnite, it is a homeomorphism. j 2J F 0.Xj ; Y /, since! j Yj /! j 2J Xj ; Y Q Q 7. Let X; Y; U, and V be spaces. The Cartesian product of maps gives a map W U X V Y!.U V /XY ;.f; g/ 7! f g: Let X and Y be Hausdorff spaces. Then the map is continuous. 8. By deﬁnition of a product, a map X! Y Z is essentially the same thing as a pair of maps X! Y, X! Z. In this sense, we obtain a tautological bijection W.Y Z/X! Y X ZX. Let X be a Hausdorff space. Then the tautological map is a homeomorphism. 9. Let X and Y be locally compact. Then composition of maps ZY Y X! ZX,.g; f / 7! g ı f is continuous. 10. Let.Y; / be a pointed space,.X; A/ a pair of spaces and p W X! X=A the quotient map. The space X=
A is pointed with base point fAg. Let F..X; A/;.Y; // be the subspace of F.X; Y / of the maps which send A to the base point. Composition with p induces a bijective continuous map W F 0.X=A; Y /! F..X; A/;.Y; //; and a bijection of homotopy sets 2.5. The Fundamental Groupoid 41 ŒX=A; Y 0! Œ.X; A/;.Y; /. If p has compact pre-images of compact sets, then is a homeomorphism. 11. Consider diagrams where the right-hand one is obtained by multiplying the left-hand one with X : A C B D, A X B X C X D X. If the left-hand diagram is a pushout in TOP and X locally compact, then the right-hand diagram is a pushout in TOP. In TOP0 the smash product with a locally compact space yields again a pushout. 12. The CO-topology on the set of linear maps Rn! R is the standard topology. 13. Let X be a compact space and Y a metric space. Then the CO-topology on Y X is induced by the supremum-metric. 2.5 The Fundamental Groupoid A path in the plane can be quite ungeometric: nowhere differentiable, inﬁnite length, surjective onto I I (a so-called Peano curve). We introduce an equivalence relation on paths, and the equivalence classes still capture qualitative geometric properties of the path. In particular a reparametrization of a path (different “velocity”) does not change basic topological properties. We consider homotopy classes relative to @I of paths. A homotopy of paths is always assumed to be relative to @I. A homotopy of paths between paths u and v with the same end points x0 D u.0/ D v.0/, x1 D u.1/ D v.1/ is a map H W I I! X such that H.s; 0/ D u.s/; H.s; 1/ D v.s/; H.0; t/ D u.0/ D v.0/; H.1; t/ D u.1/ D v.1/: Thus for
each t 2 I we have a path Ht W s 7! H.s; t/ and all these paths have the same end points. We write H W u'v for this homotopy. x0 "t x0............................... H1 D v Ht H0 D u............................... x1 x1 H constant along dotted lines 42 Chapter 2. The Fundamental Group Being homotopic in this sense is an equivalence relation on the set of all paths from x to y. The product operation is compatible with this relation as the next proposition shows. (2.5.1) Proposition. The product of paths has the following properties: (1) Let ˛ W I! I be continuous and ˛.0/ D 0; ˛.1/ D 1. Then u'u˛. (2) u1.u2 u2/ '.u1 u2/ u3 (if the products are deﬁned ). (3) u1'u0 (4) u u is always deﬁned and homotopic to the constant path. (5) ku.0/ u'u'u ku.1/. 2 implies u1 u2'u0 1 and u2'u0 1 u0 2. Proof. (1) H W.s; t/ 7! u.s.1 t/ C t˛.s// is a homotopy from u to u˛. ˛.t/ D 2t for t 1 (2) The relation.u1.u2 u3//˛ D.u1 u2/ u3 holds for ˛ deﬁned as 4 t 1 4 for 1 2 for 1 2 t 1. i then G W u1 u2'u0 2 for G deﬁned as G.s; t / D 2, ˛.t/ D t 1 u0 (3) Given Fi W ui'u0 4, ˛.t/ D t C 1 2 C 1 F1.2s; t/ for 0 t 1 2 and G.s; t/ D F2.2s 1; t / for 1 2 t 1. (4) The map F W I I! X deﬁned as F.s; t/ D u.2s.1 t// for
0 s 1 2 and F.s; t/ D u.2.1 s/.1 t/ for 1 2 t 1 is a homotopy from u u to the constant path. (At time t we only use the path from 0 to.1 t/ and compose it with its inverse.) (5) is proved again with the parameter invariance (1). From homotopy classes of paths in X we obtain again a category, denoted ….X/. The objects are the points of X. A morphism from x to y is a homotopy class relative to @I of paths from x to y. A constant path represents an identity. If u is a path from a to b and w a path from b to c, then we have the product path u v from a to c, and the composition of morphisms is deﬁned by Œv ı Œu D Œu v. In this category each morphism has an inverse, i.e. is an isomorphism, represented by the inverse path. A category with this property is called groupoid. Note that in a groupoid the endomorphism set of each object becomes a group under composition of morphisms. The category ….X/ is called the fundamental groupoid of X. The automorphism group of the object x in this category is the fundamental group of X with respect to the base point x. The usual rules of categorical notation force us to deﬁne the multiplication in this group by Œu ı Œv D Œv u. As long as we are just interested in this group (and not in the categorical aspect), we use the opposite multiplication Œu Œv D Œu v and denote this group by 1.X; x/. This is the traditional fundamental group of the pointed space.X; x/ (Poincaré 1895 [151, §12]). An element in 1.X; x/ is represented by a closed path w based at x (i.e., w.0/ D w.1/ D x), also called a loop based at x. (2.5.2) Remark. We can obtain the fundamental groupoid ….X/ as a quotient category of the path category W.X/. In that case we call paths u W Œ0; a W I! X and v W Œ0; b! X from x to
y homotopic, if there exist constant paths with image y 2.5. The Fundamental Groupoid 43 such that the compositions with u and v, respectively, have the same domain of Þ deﬁnition Œ0; c and the resulting composed paths are homotopic rel f0; cg. (2.5.3) Remark. The set 1.X; x/ has different interpretations. A loop based at x is a map w W.I; @I /!.X; x/. It induces a pointed map xw W I =@I! X. The exponential function p0 W I! S 1, t 7! exp.2 it/ induces a pointed homeomorphism q W I =@I! S 1 which sends the base point f@I g to the base point 1. There exists a unique u W S 1! X such that uq D xw. Altogether we obtain bijections 1.X; x/ D Œ.I; @I /;.X; x/ Š ŒI =@I; X0 Š ŒS 1; X0; induced by Œw $ Œ xw D Œuq $ Œu. Þ It is a general fact for groupoids … that the automorphism groups Aut.x/ D ….x; x/ and Aut.y/ D ….y; y/ of objects x; y in … are isomorphic, provided there exists a morphism from x to y. If ˛ 2 ….x; y/, then ….x; x/! ….y; y/; ˇ 7! ˛ˇ˛1 is an isomorphism. It depends on the choice of ˛; there is, in general, no canonical isomorphism between these groups. Thus fundamental groups associated to base points in the same path component are isomorphic, but not canonically. A space is simply connected or 1-connected if it is path connected and its fundamental group is trivial (consists of the neutral element alone). A continuous map f W X! Y induces a homomorphism 1.f / W 1.X; x/! 1.Y; f.x//; Œu 7! Œf u and, more generally, a functor ….f / W ….X/! ….Y /; x
7! f.x/; Œu 7! Œf u: In this way, 1 becomes a functor from TOP0 to the category of groups, and … a functor from TOP to the category of small categories (small category: its objects form a set). Homotopies correspond to natural transformations: (2.5.4) Proposition. Let H W X I! Y be a homotopy from f to g. Each x 2 X yields the path H x W t 7! H.x; t/ and the morphism ŒH x in ….Y / from f.x/ to g.x/. The ŒH x constitute a natural transformation ….H / from ….f / to ….g/. Proof. The claim says that for each path u W I! X the relation f u H u.1/'H u.0/ gu holds. We use I I! Y,.s; t/ 7! H.u.s/; t/. We obtain f u H u.1/ as composition with a b and H u.0/ gu as composition with c d, where a, b, c, and d are the sides of the square: a.t/ D.t; 0/, b.t/ D.1; t/, c.t/ D.0; t/, d.t/ D.t; 1/. But a b and c d are homotopic by a linear homotopy. 44 Chapter 2. The Fundamental Group We express the commutativity of (2.5.4) in a different way. It says that tH ı g D f W …X.x; y/! …Y.f x; f y/; where tH W ….Y /.gx; gy/! ….Y /.f x; f y/ is the bijection a 7! ŒH y1aŒH x. The rule ….K L/ D ….L/….K/ is obvious. Hence if f is an h-equivalence with h-inverse g W Y! X, then ….f /….g/ and ….g/….f / are naturally isomorphic to the identity functor, i.e., ….f / is an equivalence of categories. The natural transformation ….H / only depends on the homotopy class relative to X @
I of H. It is a general categorical fact that a natural equivalence of categories induces a bijection of morphism sets. We prove this in the notation of our special case at hand. (2.5.5) Proposition. Let f W X! Y be a homotopy equivalence. Then the functor ….f / W ….X/! ….Y / is an equivalence of categories. The induced maps between morphism sets f W …X.x; y/! …Y.f x; f y/ are bijections. In particular, 1.f / W 1.X; x/! 1.Y; f.x//; Œw 7! Œf w is an isomorphism for each x 2 X. A contractible space is simply connected. Proof. Let g W Y! X be h-inverse to f. Consider …X.x; y/ f! …Y.f x; f y/ g! …X.gf x; gf y/ f! …Y.fgf x; fgf y/: Choose H W gf'id.X/. Then gf D.gf / D tH ı.id/ D tH is a bijection, hence g is surjective. In a similar manner one proves that fg is a bijection, hence g is also injective. Since gf and g are bijective we see that f is bijective. The fundamental group forces us to work with pointed spaces. Usually the base points serve some technical purpose and one has to study what happens when the base point is changed. For pointed h-equivalences f it would be immediately clear that 1.f / is an isomorphism. For the more general case (2.5.5) one needs some argument like the one above. (2.5.6) Proposition. Let.X; x0/ and.Y; y0/ be pointed spaces and!.x; y0/ and!.x0; y/. Then 1.X; x0/ 1.Y; y0/! 1.X Y;.x0; y0//;.u; v/ 7! i X u i Y v is a well-deﬁned isomorphism with inverse z 7!.prX z; prY z/. Proof. Since homotopy is compatible with
products we know already that the second map is an isomorphism. In order to show that the ﬁrst map is a homomorphism we have to verify that i X v. Let now u and v be actual paths u commutes with i Y 2.6. The Theorem of Seifert and van Kampen 45 and write w D.u v/ı with the diagonal ı. With a notation introduced in the proof of (2.5.4) we have.u v/.a b/ D i X u i Y v and.u v/.. We now use that ı, a b, and c d are homotopic (linear homotopy). It should be clear that the two maps of the proposition are inverse to each other. 2.6 The Theorem of Seifert and van Kampen Let a space X be the union of subsets X0; X1. A general problem is to derive properties of X from those of X0, X1, and X01 D X0 \ X1. (Similar problem for more general unions.) Usually the covering has to satisfy certain reasonable conditions. In this section we consider the fundamental groupoid and the fundamental group under this aspect. The basic result is the theorem (2.6.2) of Seifert [166] and van Kampen [100]. We ﬁrst prove an analogous and slightly more general result for groupoids [34]. The result is more formal but the proof is (notationally) simpler because we need not take care of base points. Note that the hypothesis of the next theorem implies that X is the pushout in TOP of the inclusions X0 X01 X1. Thus (2.6.1) says that the functor … preserves pushouts. (2.6.1) Theorem (R. Brown). Let X0 and X1 be subspaces of X such that the interiors cover X. Let i W X01! X and j W X! X be the inclusions. Then ….X01/ ….i0/ ….X0/ ….i1/ ….j0/ ….X1/ ….j1/ ….X/ is a pushout in the category of groupoids. Proof. Let h W ….X/! ƒ be functors into a groupoid such that h1….i1/ D h0….i0/. We
have to show: There exists a unique functor W ….X/! ƒ such that h1 D ….j1/ and h0 D ….j0/. We begin with a couple of remarks. A path w W Œa; b! U represents a morphism Œw in ….U / from w.a/ to w.b/ if we compose it with an increasing homeomorphism ˛ W Œ0; 1! Œa; b. If a D t0 < t1 < < tm D b, then w represents the composition of the morphisms ŒwjŒti ; tiC1. Suppose that w W I! X is a path. Then there exists a decomposition 0 D. Choose.i/. Consider wjŒti ; tiC1 as t0 < t1 < < tmC1 D 1 such that w.Œti ; tiC1/ is contained in a set X ı W f0; : : : ; mg! f0; 1g such that w.Œti ; tiC1/ X ı path wi in X.i/. Then Œw D ….j.m//Œwm ı ı ….j.0//Œw0: 46 Chapter 2. The Fundamental Group If exists, then.i/ Œw D h.m/Œwm ı ı h.0/Œw0; i.e., is uniquely determined. In order to show the existence of, we have to deﬁne Œw by.i/. We have to verify that this is well-deﬁned. The commutativity h0….i0/ D h1….h1/ shows that a different choice of yields the same result. Since h0 and h1 are functors, we obtain the same result if we reﬁne the decom- position of the interval. It remains to be shown that.i/ only depends on the homotopy class of the path. Let H W I I! X be a homotopy of paths from x to y. There exists n 2 N such that H sends each sub-square Œi=n;.i C 1/=n Œj=n;.j C 1/=n into one of the sets X ı (see (2.6.
4)). We consider edge-paths in the subdivided square I I which differ by a sub-square, as indicated in the following ﬁgure (n D 5)..1; 1/.0; 0/ We apply H and obtain two paths in X. They yield the same result.i/, since they differ by a homotopy on some subinterval which stays inside one of the sets X ı. Changes of this type allow us to pass inductively from the H on the lower to H on the upper boundary path from.0; 0/ to.1; 1/. These paths differ from H0 and H1 by composition with a constant path. Finally, from the construction we conclude that is a functor. (2.6.2) Theorem (Seifert–van Kampen). Let X0 and X1 be subspaces of X such that the interiors cover X. Let i W X01 D X0 \ X1! X and j W X! X be the inclusions. Suppose that X0; X1; X01 are path connected with base point 2X 01. Then 1.X01; / i1 1.X1; / i0 1.X0; / j0 j1 1.X; / is a pushout in the category of groups. Proof. The theorem is a formal consequence of (2.6.1). In general, if Z is path connected and z 2 Z we have a retraction functor r W ….Z/! 1.Z; z/ onto the full subcategory with object z. For each z 2 Z we choose a morphism ux 2 ….Z/ 2.7. The Fundamental Group of the Circle 47 from x to z such that uz D id. Then r assigns uy˛u1 x to a morphism ˛ W x! y. We apply this to Z D X01; X0; X1; X and z D and choose a morphism ux 2 ….Z/ if x is contained in Z. We obtain a commutative diagram of functors ….X0/ r1 ….X01/ r01 1.X0; / 1.X01; / ….X1/ r0 1.X1; /. Given homomorphisms'W 1.X; /! G into a group G (D a groupoid with a single object
) which agree on 1.X01; /, we compose with r and apply (2.6.1) to obtain a functor ….X/! G. Its restriction to 1.X; / is the unique solution of the pushout problem in (2.6.2). (2.6.3) Remark. From the proof of (2.6.1) we see that each morphism in ….X/ is a composition of morphisms in ….X0/ and ….X1/. Similarly, in (2.6.2) the group 1.X; / is generated by the images of j0 and j1. This algebraic fact is not Þ immediately clear from the deﬁnition of a pushout. We have used above the next fundamental result. It is impossible to prove a geometric results about continuous maps without subdivision and approximation procedures. In most of these procedures (2.6.4) will be used. (2.6.4) Proposition (Lebesgue). Let X be a compact metric space. Let A be an open covering of X. Then there exists " > 0 such that for each x 2 X the "-neighbourhood U".x/ is contained in some member of A. (An " with this property is called a Lebesgue number of the covering.) 2.7 The Fundamental Group of the Circle The space R is simply connected; ….R/ has a single morphism between any two objects. We consider ….R/ as a topological groupoid: The object space is R, the morphism space is R R, the source is.a; b/ 7! a, the range.a; b/ 7! b, the identity a 7!.a; a/, and.b; c/ ı.a; b/ D.a; c/ the composition. The continuous map p W R! S 1 induces a functor ….p/ W ….R/! ….S 1/. It turns out that this functor is surjective on morphisms and provides us with an algebraic description of ….S 1/. So let us deﬁne a topological groupoid G. The object space is S 1, the morphism space is S 1 R, the source.a; t/ 7! a, the range.a; t/ 7! a exp.2 it/, the identity
a 7!.a; 0/, and the composition.b; t/ı.a; s/ D.a; s C t/. The assignments a 7! exp.2 ia/ and.a; b/ 7!.exp.2 ia/; b a/ yield a continuous functor ….R/! G. We will show that G (forgetting the topology) is ….S 1/. 48 Chapter 2. The Fundamental Group We have the open covering of S 1 C by X0 D S 1 X f1g and X1 D S 1 X f1g with inclusions ik W X01! Xk and jk W Xk! S 1. The sets Xk are contractible, hence simply connected. Therefore there exists a single morphism.a; b/k W a! b between two objects a; b of ….Xk/. We have bijective maps f0 W 0; 1Œ! X0 and f1 W 1=2; 1=2Œ! X1 given 7! exp.2 it/. We deﬁne functors k W ….Xk/! G by the identity on by t objects and by k.a; b/k D.a; f 1 k.b/ f 1 k.a//. Moreover we have a functor W G! ….S 1/ which is the identity on objects and which sends the morphism.a; t/ of G to the class of the path I! S 1, s 7! a exp.2 its/ from a to a exp.2 it/. (The idea behind the deﬁnition of G is the fact that each path in S 1 is homotopic to one of this normal form, see (2.7.9).) The following diagram is commutative. ….X0/ …i0 …i1 0 1 ….X01/ …j0 G ….S 1/ ….X1/ …j1 (2.7.1) Proposition. The functor is an isomorphism. Proof. We apply (2.6.1) to the pair.0; 1/ and obtain a functor W ….S 1/! G. The uniqueness property of a pushout solution shows D id. In order to show D id we note that the morphisms of G are generated by the images of 0 and
1. Given.a; t/ 2 G.a; b/, choose a decomposition t D t1 C C tm such that jtr j < 1=2 for each r. Set a0 D a and ar D a exp.2 i.t1 C C tr //. Then.a; t/ D.am1; tm/ ı ı.a1; t2/ ı.a0; t1/ in the groupoid G. Since jtr j < 1=2 there exists for each r a k.r/ 2 f0; 1g such that ar1 exp.2 itr s/ 2 Xk.r/ for s 2 I. Then.ar1; tr / D k.r/.ar1; ar /k.r/. Thus G.a; b/ is generated by morphisms in the images of the k. The unit circle S 1 in the complex plane is the prototype of a loop. Typical elements in the fundamental group are obtained by running n times around the circle. Up to homotopy, there are no other possibilities. With (2.7.1) we have determined the fundamental group 1.S 1; 1/, namely as the automorphism group in ….S 1/ of the object 1. The automorphisms of the object 1 in G are the.1; n/; n 2 Z and.1; n/ is the loop t 7! exp.2 i nt/. (2.7.2) Theorem. Let sn W I! S 1 be the loop t 7! exp.2 i nt/. The assignment ı W Z! 1.S 1; 1/, n 7! Œsn is an isomorphism. The circle S 1 is a group with respect to multiplication of complex numbers. We show that the composition law in 1.S 1; 1/ can also be deﬁned using this multiplication. 2.7. The Fundamental Group of the Circle 49 More generally, assume that X is a space with a continuous multiplication m W X X! X;.x; y/ 7! m.x; y/ D xy and neutral element up to homotopy e 2 X (the base point), i.e., the maps x 7! m.e; x/ and x 7! m.x; e/ are both pointed homotopic to the identity. We call such an object a
monoid in h-TOP. (We do not assume that m is associative or commutative.) We deﬁne a composition law on the pointed homotopy set ŒY; X0, called the m-product, by Œf ; Œg 7! Œf m Œg D Œf g; here f g W y 7! m.f.y/; g.y// is the ordinary pointwise multiplication. The constant map represents a two-sided unit for the m-product. In a similar manner we deﬁne by pointwise multiplication of loops the m-product on 1.X; e/. The set 1.X; e/ Š ŒS 1; X0 now carries two composition laws: the m-product and the -product of the fundamental group. (2.7.3) Proposition. Let.X; m/ be a monoid in h-TOP. Then the -product and the m-product on 1.X; e/ coincide and the product is commutative. Proof. Let k be the constant loop. Then for any two loops u and v the relations u v '.u k/.k v/ D.u k/.k v//'u v; u v'u v '.k u/.v k/ D.k v/.u k/'v u hold. In order to see the equalities, write down the deﬁnition of the maps. (2.7.4) Lemma. The map v W ŒS 1; S 10! ŒS 1; S 1 which forgets the base point is a bijection. Proof. Given f W S 1! S 1 we choose a path w W I! S 1 from 1 to f.1/1. Then.x; t/ 7! f.x/w.t/ is a homotopy from f to a pointed map, hence v is surjective. Let H W S 1 I! S 1 be a homotopy between pointed maps; then.x; t/ 7! H.x; t/ H.1; t/1 is a pointed homotopy between the same maps, i.e., v is injective. If f; g W X! S 1 are continuous maps, then f g W x 7! f.x/g.x/ is again
continuous. This product of functions is compatible with homotopies and induces the structure of an abelian group on ŒX; S 1. (2.7.5) Theorem. From (2.7.2), (2.7.4) and (2.5.3) we obtain an isomorphism d, d W ŒS 1; S 1 Š ŒS 1; S 10 Š 1.S 1; 1/ Š Z: We call the integer d.f / D d.Œf / the degree of f W S 1! S 1. A standard map of degree n is n W z 7! zn. A null homotopic map has degree zero. (2.7.6) Example. A polynomial function g W C! C; g.z/ D zn C a1zn1 C C an has a root (n 1). 50 Chapter 2. The Fundamental Group Proof. Suppose g.z/ 6D 0 for jzj D1. Then f W S 1! S 1, z 7! g.z/=jg.z/j is deﬁned. Suppose g is non-zero for jzj 1. Then h.z; t/ D f.tz/ is a null homotopy of f. For t > 0 we have k.z; t/ D zn C t.a1zn1 C a2tzn2 C C ant n1/ D t ng.z=t/: If g is non-zero for jzj 1, then H.z; t/ D k.z; t /=jk.z; t/j is a homotopy from f to n. Thus if g has no root, then n is null homotopic; this contradicts (2.7.5). The classical approach to 1.S 1/ uses topological properties of the exponential function p W R! S 1, t 7! exp.2 it/. A lifting of w W Œa; b! S 1 along p is a map W W Œa; b! R with pW D w; the value W.a/ is the initial condition of the lifting. Liftings always exist and depend continuously on the path and the initial condition (see (2.7.8)). (2.7.7) Proposition. Let f W S 1
! S 1 be given. Let F W I! R be a lifting of fp0 along p. Then F.1/ F.0/ is the degree of f. Proof. Let g D f.1/1f. Then ı.d.f // D Œgp0, by the deﬁnition of d in (2.7.5). There exists a 2 R such that f.1/ D exp.2 ia/ and F.0/ D a. Then ˆ D F a is a lifting of gp0 with initial condition 0. Hence F.1/ F.0/ D ˆ.1/ ˆ.0/ D ˆ.1/ D n 2 Z. The homotopy.x; t/ 7!.1 t/ˆ.x/ C txˆ.1/ is a homotopy of paths. Hence the loop gp0 is homotopic to sn. This shows ı.n/ D Œgp0. The next proposition will be proved in the chapter on covering spaces. It ex- presses the fact that p W R! S 1 is ﬁbration. (2.7.8) Proposition. Given a homotopy h W X I! S 1 and an initial condition a W X! R such that pa.x/ D h.x; 0/. Then there exists a unique homotopy H W X I! R such that H.x; 0/ D a.x/ and pH D h. (2.7.9) Example. Let w W Œ0; 1! S 1 be a path with w.0/ D z D exp.2 ia/. Let W W Œ0; 1! R be a lifting of w with W.0/ D a. Suppose W.1/ D b. Then W is, by a linear homotopy, homotopic to t 7! a C t.b a/ and hence w homotopic to Þ the path in normal form t 7! z exp.2 i.b a/t/. 2.7.10 The winding number. Let x 2 C D R2. The map rx W C n fxg!S 1; z 7!.z x/=jz xj is an h-equivalence and therefore ŒS 1; C n fxg!
ŒS 1; S 1, Œf 7! Œrxf a bijection. The degree of rxf is the winding number of f with respect to x. We denote it by W.f; x/. Maps f0; f1 W S 1! C n fxg are homotopic if and only if they have the same winding number. If f W S 1! C is given and w W I! C a path with 2.7. The Fundamental Group of the Circle 51 f.S 1/ \ w.I / D ;, then.x; t/ 7! rw.t/f is a homotopy. Therefore the winding numbers of f with respect to w.0/ and w.1/ are equal. The complement C n f.S 1/ decomposes into open path components, and the winding number with respect to x is constant as long as x stays within a component. Let u W I! C X fxg be a loop. Then there exists a unique continuous map f W S 1! C X fxg such that f ı p0 D u. The winding number of f is then also Þ called the winding number of u, and we denote it by W.u; x/. The notion of the degree can be extended to other situations. Let h W S! S 1 be a homeomorphism and f W S! S any map; the degree of hf h1 is independent of the choice of a homeomorphism h and also called the degree d.f / of f. Problems 1. Let p be a polynomial function on C which has no root on S 1. Then the number of roots z with jzj < 1 (counted with multiplicities) is equal to the winding number W.pjS 1; 0/. What is the winding number of the function 1=p with respect to 0? 2. (Properties of the degree.) d.f ı g/ D d.f /d.g/. A homeomorphism S 1! S 1 has degree ˙1. If f W S 1! S 1 has degree d.f / ¤ 1, then there exists x 2 S 1 such that f.x/ D x. The map z 7! xz has degree 1. Let u D exp.2 i=n/ be an n-th root of unity. Suppose h W S 1
! S 1 satisﬁes h.uz/ D h.z/. Then d.h/ 0 mod n. Let k; j 2 Z and assume that k is coprime to n. Let f W S 1! S 1 satisfy the functional equation f.ukz/ D uj f.z/. Then k d.f / j mod n. If, conversely, this congruence is satisﬁed with some integer d.f /, then there exists a map f of degree d.f / which satisﬁes the functional equation. In particular an odd map f, i.e., f.z/ D f.z/, has odd degree. Suppose f.z/ ¤ f.z/ for all z; then the degree of f is odd. Suppose f.z/ ¤ g.z/ for all z; then d.f / D d.g/. Suppose d.g/ 0 mod n for some n > 0; then there exists h W S 1! S 1 such that g D hn. 3. Let U W I! C be a lifting of u W I! C D C X 0 along the covering P W C! C, z 7! exp.2 iz/. Then W.u; 0/ D U.1/ U.0/. 4. Let W Œ0; 1! C be a continuously differentiable path with initial point 1. Then d z W Œ0; 1! C, t 7! 1 z is a continuously differentiable lifting of along P 2 i with initial point 0. dz 5. If u W I! C X fxg is a continuously differentiable loop, then W.u; x/ D 1 zx. 2 i 6. Let A 2 GL2.R/. Then the winding number of lA W S 1 7! R2 X 0, x 7! Ax with respect to the origin is the sign of the determinant det.A/. 7. Let v W ŒS 1; X0! ŒS 1; X be the map which forgets about the base point (pointed homotopies versus free homotopies). Conjugate elements in the group ŒS 1; X0 have the same image under v. Hence v induces a well-deﬁned map xv W ŒS 1; X0=./! Œ
S 1; X from the set of conjugacy classes. This map is injective, and surjective if X is path connected. Thus v is bijective if X is path connected and the fundamental group abelian. jŒ0;t R R u 52 Chapter 2. The Fundamental Group 2.8 Examples The formal nature of the theorem of Seifert and van Kampen is simple, but the corresponding algebra can be complicated. The setup usually leads to groups presented by generators and relations. It may be difﬁcult to understand a group presented in this manner. For an introduction to this type of group theory see [122]; also [171] and [39] are informative in this context. We report about some relevant algebra and describe a number of examples and different applications of the fundamental group. 2.8.1 Spheres. If a space X is covered by two open simply connected subsets with path connected intersection, then X is simply connected, since the pushout of two trivial groups is trivial. Coverings of this type exist for the spheres S n for n 2. Þ Hence these spheres are simply connected. 2.8.2 Removing a point. The inclusion Dn X 0 Dn induces for n 3 an isomorphism of fundamental groups; actually both groups are zero, since Dn is contractible and Dn X 0'S n1. Let M be a manifold of dimension n 3 and U M homeomorphic to Dn under a homeomorphism that sends x to 0. Then M is the pushout of M X fxg and U. Theorem (2.6.2) implies that M X fxg M induces an isomorphism of fundamental groups. Often we view the space S n as the one-point compactiﬁcation Rn [ f1g of the Euclidean space, see (2.3.2). Let K be a compact subset of Rn for n 3. Then the inclusion Rn X K S n X K induces an isomorphism of fundamental groups. Þ 0 D S mCnC1 \.RmC1 0/ and S n 2.8.3 Complements of spheres. Let S m S mCnC1 \.0 RnC1/. Then X D S mCnC1 X S n A homeomorphism S m E n! X is.x; y/ 7! Y D
S mCnC1 X.S m p p. plement Y is h-equivalent to S m S n. 1 D 1 is homeomorphic to S m E n.. The space 1 / is homeomorphic to S m S n 0; 1Œ via.x; y; t/ 7! ty/. Therefore the complement X is h-equivalent to S m and the com- 1 kyk2x; y 0 [ S n 1 tx; p The fundamental group of S 3 X.f0g S 1/ is isomorphic to Z. If we view S 3 D R3[f1g, then we are considering the complement of the axis Z D f.0; 0; z/ j z 2 Rg [ f1g. The generator of the fundamental group is a loop that runs once about the axis Z, represented by the standard sphere W D S 1 f0g. It is impossible to span a 2-disk with boundary W in the complement of Z, because W represents a non-zero element in the fundamental group of the complement, see Figure 2.3. This is expressed by saying that W and Z are linked in S 3. Apply the stereographic projection (2.3.2) toS 1 0 [ 0 S 1 S 3. The image yields W [ Z. The complement of W [ Z in R3 is therefore isomorphic to the fundamental group Z Z of the torus S 1 S 1. The reader should draw generators Þ of 1.R3 X W [ Z/. 2.8. Examples 53 W 1 Z Figure 2.3. A standard circle in S 3. 2.8.4 Presentation of groups by generators and relations. Let S be a set. A free group with basis S consists of a group F.S/ and a set map i W S! F.S/ which has the following universal property: For each set map ˛ W S! G into a group G there exists a unique homomorphism A W F.S/! G such that A ı i D ˛. It turns out that i is injective. Let us consider i as an inclusion and set S 1 D fs1 j s 2 Sg. A word in the alphabet X D S q S 1 is a sequence.x1; : : : ; xm/ of elements xi 2 X. The elements in F.S/ are the products x1 :
: : xm corresponding to the words; the neutral element belongs to the empty word; a word.x; x1/ also represents the neutral element. Let R be a set of words and xR the image in F.S/. Let N.R/ be the normal subgroup generated by xR. The factor group G D F.S/=N.R/ is the group presented by the generators S and the relations R. We denote this group by hSjR i. It has the following universal property: Let ˛ W S! H be a set map into a group H. Assume that for each.x1; : : : ; xm/ 2 R the relation ˛.x1/ : : : ˛.xm/ D 1 holds in H. Then there exists a unique homomorphism A W G! H such that A.x/ D ˛.x/ for each x 2 S. Each group can be presented in the form hS j R i – in many different ways. In practice one uses a less formal notation. Here are a few examples. (i) The cyclic group of order n has the presentation ha j an i. (ii) Let S D fx; yg. Consider the word.x; x; y1; y1; y1/ and R consisting of this word. Then we can write G D hSjR i also in the form hx; y j x2y3 i or hx; y j x2 D y3 i. The universal property says in this case that homomorphisms G! H correspond bijectively to set maps ˛ W fx; yg!H such that ˛.x/2 D ˛.y/3. (iii) ha; b j ab D ba i is a presentation for the free abelian group with basis a, b. Þ 2.8.5 Free product and pushout of groups. The sum (D coproduct) in the category of groups is also called a free product. Let.Gj j j 2 J / be a family of groups. The free product of this family consists of a group∗k2J Gk together with a family 54 Chapter 2. The Fundamental Group of homomorphisms j W Gj! ∗k2J Gk which have the universal property of a sum in the category of groups
. (The notation G1 G2 is used for the free product of two groups.) Each family has a sum. Let Gj D hSj jRj i and assume that the Sj are disjoint. Let S D qj Sj. The Rj are then words in the alphabet S q S 1. Let R D j Rj. We have homomorphisms j W hSj jRj i! hSjR i which are induced by Sj S. These homomorphisms are a sum in the category of groups. S Let G and H be groups and i1 W G! G H, j1 W H! G H be the canonical maps which belong to the sum. Let J W P! G, I W P! H be homomorphisms from a further group P. Let N be the normal subgroup of G H generated by the elements fi1J.x/ j1I.x1/ j x 2 P g. Let Q D.G H /=N and denote by i W G! Q, j W H! Q the composition of i1; j1 with the quotient map. Then.i; j / is a pushout of.J; I / in the category of groups. In the case that I and J are inclusions (but sometimes also in the general case) one writes Q D G P H. Let S be a set and Z D Zs a copy of the additive group Z for each s 2 S. Then Þ the groups F.S/ with basis S is also the free product ∗s2S Z. 2.8.6 Free products of fundamental groups. The free product 1.X0/ 1.X1/ arises geometrically if X01 is simply connected. Let X D S 1_S 1 with X0 D Y _S 1, X1 D S 1_Y, where Y D S 1Xf1g; D 1. Then.Y; 1/ is pointed contractible; hence the inclusion of the summands S 1! X0; X1 are pointed h-equivalences. One can apply (2.6.2) to the covering of X by X0; X1. Since X0 \ X1 is pointed contractible, we see that 1.X/ is the free product 1.X0/ 1.X1/. Hence the inclusions of the summands S 1! S
1 _ S 1 yield a presentation of 1.S 1 _ S 1/ as a free product 1.S 1/ 1.S 1/ Š Z Z. Þ By induction one shows that 1 is the free group of rank k. W k 1 S 1 2.8.7 Plane without two points. The space R2 X f˙1g has as a deformation retract the union X of the circles about ˙1 with radius 1=2 and the segment from 1=2 to 1=2, see Figure 2.4. (The reader should try to get an intuitive understanding of a 1 0!! C1 u v Figure 2.4. Generators u; v of 1.R2 X ˙1/. retraction. In order to give a formal proof, without writing down explicit formulas, it is advisable to wait for the method of coﬁbrations.) The fundamental group 1.R2 X f˙1g; 0/ is the free group Z Z and generators are represented by two 2.8. Examples 55 small circles about ˙1 (of radius 1=2, say) connected linearly to the base point. One can apply (2.6.2) to the covering of R2 X f˙1g by the punctured half-spaces Þ f.x; y/ j x < 1=3; x 6D 1g and f.x; y/ j 1=3 < x; x 6D 1g. In the example in 2.8.6 one cannot apply (2.6.2) directly to the covering of S 1_S 1 by the two summands, since the interiors do not cover the space. The general method in cases like this is to ﬁrst “thicken” the subspaces up to h-equivalence. In the next theorem we add a hypothesis which allows for a thickening. (2.8.8) Theorem. Let..Xj ; xj / j j 2 J / be a family of pointed spaces with the property: The base point xj has an open neighbourhood Uj Xj which is pointed contractible to the base point. The inclusions of the summands induce W homomorphisms ij W 1.Xj ; xj /! 1. This family is a free product of the groups 1.Xj ; xj /. k
attaching map'is given in terms of the standard generators of 1 the so-called surface-word. 2.8. Examples W n 1 S 1 57 by In order to save space we refer to [44, p. 83–87] for the discussion of the fundamental group of surfaces in general. We mention at least some results. They will not be used in this text. (2.8.14) Theorem..1/ The fundamental group of a closed connected orientable surface Fg of genus g 1 has the presentation 1.Fg / D ha1; b1; : : : ; ag ; bg j a1b1a1 1 b1 1 : : : ag bg a1 g b1 g i:.2/ The fundamental group of a closed connected non-orientable surface Ng of genus g has the presentation 1.Ng / D ha1; : : : ; ag j a2 1a2 2 : : : a2 g i:.3/ The fundamental group of a compact connected surface with non-empty boundary is a free group. The number of generators is the ﬁnite number 1.Fg / where.Fg / is the so-called Euler characteristic..4/ A simply connected surface is homeomorphic to R2 or S 2. There are many different deﬁnitions of the genus. We mention a geometric property: The genus of a closed connected orientable surface is the maximal number g of disjointly embedded circles such that their complement is connected. The genus of a closed connected non-orientable surface is the maximal number g of disjointly embedded Möbius bands such that their complement is connected. The sphere has Þ genus zero by the Jordan separation theorem. Problems 1. Let S 1 R2 0 R3 be the standard circle. Let D D f.0; 0; t/ j 2 t 2g and S 2.2/ D fx 2 R3 j kxk D2g. Then S 2.2/ [ D is a deformation retract of X D R3 X S 1. The space X is h-equivalent to S 2 _ S 1. 2. Consider the loop based at.0; 0/ in the plane as shown in Figure 2.5. Determine which 1 C1 Figure 2.5. 58 Chapter 2. The Fundamental Group element in 1.R2 X ˙1
/ this loop represents in terms of the generators u; v in 2.8.7. Determine the winding number about points in each of the six complementary regions. 3. Let.Xj j j 2 J / be a family of subspaces of X such that the interiors X ı j cover X. Then each morphism in ….X/ is a composition of morphisms in the Xj. If the intersections Xi \ Xj are path connected and 2X i \ Xj, then 1.X; / is generated by loops in the Xj. 4. Let i0 in (2.6.2) be an isomorphism. Then j1 is an isomorphism. This statement is a general formal property of pushouts. If i0 is surjective, then j1 is surjective. 5. Projective plane. The real projective plane P 2 is deﬁned as the quotient of S 2 by the relation x x. Let Œx0; x1; x2 denote the equivalence class of x D.x0; x1; x2/. We can also obtain P 2 from S 1 by attaching a 2-cell S 1 j D2'ˆ P 1 J P 2. Here P 1 D fŒx0; x1; 0g P 2 and '.x0; x1/ D Œx0; x1; 0. The space P 1 is homeomorphic to S 1 via Œx0; x1; 0 7! z2; z D x0 C ix1; and'corresponds to the standard map of degree p 1 kxk2. As an application of 2.8.10 we 2. The map ˆ is x D.x0; x1/ 7! Œx0; x1; obtain 1.P 2/ Š Z=2. Another interpretation of the pushout: P 2 is obtained from D2 by identifying opposite points of the boundary S 1. The subspace f.x0; x1/ j kxk 1=2g becomes in P 2 a Möbius band M. Thus P 2 is obtainable from a Möbius band M and a 2-disk D by identiﬁcation of the boundary circles by a homeomorphism. The projective plane cannot be embedded into R3, as we will prove in (18.
3.7). There exist models in R3 with self-intersections (technically, the image of a smooth immersion.) The projective plane is a non-orientable surface. 6. Klein bottle. The Klein bottle K can be obtained from two Möbius bands M by an identiﬁcation of their boundary curves with a homeomorphism, K D M [@M M. Apply the theorem of Seifert and van Kampen and obtain the presentation 1.K/ D h a; b j a2 D b2 i. The elements a2; ab generate a free abelian subgroup of rank 2 and of index 2 in the fundamental group. The element a2 generates the center of this group, it is represented by the central loop @M. The quotient by the center is isomorphic to Z=2 Z=2. The space M=@M is homeomorphic to the projective plane P 2. If we identify the central @M to a point, we obtain a map q W K D M [@M M! P 2 _ P 2. The induced map on the fundamental group is the homomorphism onto Z=2 Z=2. 2.9 Homotopy Groupoids The homotopy category does not have good categorical properties. Therefore we consider “homotopy” as an additional structure on the category TOP of topological spaces. The category TOP will be enriched: The set of morphisms ….X; Y / between two objects carries the additional structure of a groupoid. The fundamental groupoid is the special case in which X is a point. 2.9. Homotopy Groupoids 59 Recall that a category has the data: objects, morphisms, identities, and composition of morphisms. The data satisfy the axioms: composition of morphisms is associative; identities are right and left neutral with respect to composition. Let X and Y be topological spaces. We deﬁne a category ….X; Y / and begin with the data. The objects are the continuous maps X! Y. A morphism from f W X! Y to g W X! Y is represented by a homotopy K W f'g. Two such homotopies K and L deﬁne the same morphism if they are homotopic relative to X @I with @I D f0; 1g the boundary of I. Let us use a second symbol
J D Œ0; 1 for the unit interval. This means: A map ˆ W.X I / J! Y is a homotopy relative to X @I, if ˆ.x; 0; t/ is independent of t and ˆ.x; 1; t/ is also independent of t. Therefore ˆt W X I! Y;.x; s/ 7! ˆ.x; s; t/ is for each t 2 J a homotopy from f to g. For this sort of relative homotopy one has, as before, the notion of a product and an inverse, now with respect to the J -variable. Hence we obtain an equivalence relation on the set of homotopies from f to g. We now deﬁne: A morphism W f! g in ….X; Y / is an equivalence class of homotopies relative to X @I from f to g. Composition of morphisms, denoted ~, is deﬁned by the product of homotopies K W f'g; L W g'h; ŒL ~ ŒK D ŒK L W f! h: This is easily seen to be well-deﬁned (use ˆt I ‰t ). The identity of f in ….X; Y / is represented by the constant homotopy kf W f'f. The veriﬁcation of the category axioms is based on the fact that a reparametriza- tion of a homotopy does not change its class. (2.9.1) Lemma. Let ˛ W I! I be a continuous map with ˛.0/ D 0 and ˛.1/ D 1. Then K and K ı.id ˛/ are homotopic relative to X @I. Proof. ˆ.x; s; t/ D K.x;.1 t/s C t˛.s// is a suitable homotopy. (2.9.2) Proposition. The data for ….X; Y / satisfy the axioms of a category. The category is a groupoid. Proof. The associativity of the composition follows, because.K L/ M D K.L M / ı.id �
�/; 4, ˛.t/ D t C 1 4 for 1 4 t 1 2, ˛.t/ D t 2 C 1 2 with ˛ deﬁned by ˛.t/ D 2t for t 1 for 1 2 t 1. Similarly, for each K W f'g the homotopies kf K, K, and K kg differ by a parameter change. Therefore the constant homotopies represent the identities in the category. The inverse homotopy K represents an inverse of the morphism deﬁned by K. Hence each morphism is an isomorphism. Proof: The assignments.x; s; t/ 7! K.x; 2s.1 t // for 0 s 1 2 s 1 yield a homotopy relative to X @I from K K to the constant homotopy. 2 and.x; s; t/ 7! K.x; 2.1 s/.1 t// for 1 60 Chapter 2. The Fundamental Group The endomorphism set of an object in a groupoid is a group with respect to composition as group law. We thus see, from this view point, that the notion of homotopy directly leads to algebraic objects. This fact is a general and systematic approach to algebraic topology. The homotopy categories of Section 2.2 have a similar enriched structure. If we work, e.g., with pointed spaces and pointed homotopies, then we obtain for pointed spaces X and Y a category …0.X; Y /. The objects are pointed maps. Morphisms are represented by pointed homotopies, and the equivalence is deﬁned by homotopies ˆ rel X @I such that each ˆt is a pointed homotopy. The remainder of this section can be skipped on a ﬁrst reading. We study the dependence of the groupoids ….X; Y / on X and Y. The formal structure of this dependence can be codiﬁed in the notion of a 2-category. Suppose given ˛ W U! X and ˇ W Y! V. Composition with ˛ and ˇ yield a functor ˇ# D …#.ˇ/ W ….X; Y /! ….X; V /; which sends f to ˇf and
ŒK to ŒˇK and a functor ˛# D …#.˛/ W ….X; Y /! ….U; Y /; which sends f to f ˛ and ŒK to ŒK.˛ id/. They satisfy.ˇ1ˇ2/# D ˇ1.˛1˛2/# D ˛# 1. These functors are compatible in the following sense: 2˛# # ˇ2 # and (2.9.3) Proposition. Suppose K W f'g W X! Y and L W u'v W Y! Z are given. Then ŒL ˘ K D v#ŒK ~ f #ŒL D g#ŒL ~ u#ŒK. Here L ˘ K W uf'vg W X I! Z;.x; t/ 7! L.K.x; t/; t/. Proof. We use the bi-homotopy L ı.K id/ W X I I! Z. Restriction to the diagonal of I I deﬁnes L ˘ K. Along the boundary of the square we have the following situation. vf L.f id/ vK t s L˘K vg L.g id/.s; t/ 2 I I uf uK ug g#ŒL ~ u#ŒK is represented by uK L.g id/. If we compose the bi-homotopy 2 and.t/ D.1; 2t 1/ for t 1 with id.X/, where.t/ D.2t; 0/ for t 1 2, 2.9. Homotopy Groupoids 61 we obtain uK L.g id/. In the same manner we obtain L.f id/ vK if we compose the bi-homotopy with id.x/ ı, where ı.t / D.0; 2t / for t 1 2 and ı.t/ D.2t 1; 1/ for t 1 2. The maps and ı are homotopic relative to @I by a linear homotopy in the square. They are also homotopic to the diagonal t 7!.t; t/ of the square. (2.9.4) Corollary.
The homotopy L induces a natural transformation L# W u#! v# W ….X; Y /! ….X; Z/: The value of L# at f is f #ŒL. The homotopy K induces a natural transformation K# W f #! g# W ….Y; Z/! ….X; Z/: The value of K# at u is u#ŒK. (2.9.5) Corollary. If u W Y! Z is an h-equivalence, then u# is an equivalence of categories. Similarly in the contravariant case. The data and assertions that we have obtained so far deﬁne on TOP the structure of a 2-category. In this context, the ordinary morphisms f W X! Y are called 1-morphisms and the morphisms ŒK W f'g are called 2-morphisms. The composition ~ of 2-morphisms is called vertical composition. We also have a horizontal composition of 2-morphisms deﬁned as ŒL ˘ ŒK D ŒL ˘ K. Because of (2.9.3) we need not deﬁne ˘ via the diagonal homotopy; we can use instead (2.9.3) as a deﬁnition ŒL ˘ ŒK D v#ŒK ~ f #ŒL D g#ŒL ~ u#ŒK. (2.9.6) Note. From this deﬁnition one veriﬁes the commutation rule of a 2-category.ı ~ / ˘.ˇ ~ ˛/ D.ı ˘ ˇ/ ~. ˘ ˛/. The following ﬁgure organizes the data (horizontal – vertical). Conversely, one can derive (2.9.3) from the commutation rule (2.9.6). With the constant homotopy ku of u we have ku ˘ ˛ D u#˛; ˘ kf D f #; kv ˘ ˛ D v#˛; ˘ kg D g# and this yields ˘ ˛ D. ~ ku/ ˘.kg ~
˛/ D. ˘ kg / ~.ku ˘ ˛/ D g# ~ u#˛: In a similar manner one obtains ˘ ˛ D v#˛ ~ f #. Chapter 3 Covering Spaces A covering space is a locally trivial map with discrete ﬁbres. Objects of this type can be classiﬁed by algebraic data related to the fundamental group. The reduction of geometric properties to algebraic data is one of the aims of algebraic topology. The main result of this chapter has some formal similarity with Galois theory. A concise formulation of the classiﬁcation states the equivalence of two categories. We denote by COVB the category of covering spaces of B; it is the full subcategory of TOPB of spaces over B with objects the coverings of B. Under some restrictions on the topology of B this category is equivalent to the category TRAB D Œ….B/; SET of functors ….B/! SET and natural transformations between them. We call it the transport category. It is a natural idea that, when you move from one place to another, you carry something along with you. This transport of “information” is codiﬁed in moving along the ﬁbres of a map (here: of a covering). We will show that the transport category is equivalent to something more familiar: group actions on sets. The second important aspect of covering space theory is the existence of a universal covering of a space. The automorphism group of the universal covering is the fundamental group of the space – and in this manner the fundamental group appears as a symmetry group. Moreover, the whole category of covering spaces is obtainable by a simple construction (associated covering of bundle theory) from the universal covering. In this chapter we study coverings from the view-point of the fundamental group. Another aspect belongs to bundle theory. In the chapter devoted to bundles we show for instance that isomorphism classes of n-fold coverings over a paracompact space B correspond to homotopy classes B! BS.n/ into a so-called classifying space BS.n/. 3.1 Locally Trivial Maps. Covering Spaces Let p W E! B be continuous and U B open. We assume that p is surjective to avoid empty ﬁbres. A trivialization of p over U is
a homeomorphism'W p1.U /! U F over U, i.e., a homeomorphism which satisﬁes pr1 ı' D p. This condition determines the space F up to homeomorphism, since'induces a homeomorphism of p1.u/ with fug F. The map p is locally trivial if there exists an open covering U of B such that p has a trivialization over each U 2 U. A locally trivial map is also called a bundle or ﬁbre bundle, and a local trivialization a bundle chart. We say, p is trivial over U, if there exists a bundle chart over U. If 3.1. Locally Trivial Maps. Covering Spaces 63 p is locally trivial, then the set of those b 2 B for which p1.b/ is homeomorphic to a ﬁxed space F is open and closed in B. Therefore it sufﬁces for most purposes to ﬁx the homeomorphism type of the ﬁbres. If the ﬁbres are homeomorphic to F, we call F the typical ﬁbre. A locally trivial map is open, hence a quotient map. A covering space or a covering1 of B is a locally trivial map p W E! B with discrete ﬁbres. If F is discrete (D all subsets are open and closed), then U F is homeomorphic to the topological sum qx2F U fxg. The summands U fxg are canonically homeomorphic to U. If'W p1.U /! U F is a trivialization, then p yields via restriction a homeomorphism of '1.U fxg/ with U. A covering is therefore a local homeomorphism. The summands '1.U fxg/ D Ux are the sheets of the covering over U ; the pre-image p1.U / is therefore the topological sum of the sheets Ux; the sheets are open in E and mapped homeomorphically onto U under p. If jF j Dn 2 N, we talk about an n-fold covering. The trivial covering with typical ﬁbre F is the projection pr W B F! B. We say, U is admissible or evenly covered if there exists a trivialization over U. (3.1.1) Example. The exponential
function p W R! S 1, t 7! exp.2 it/ is a covering with typical ﬁbre Z. For each t 2 R and p.t/ D z we have a homeomorphism p1.S 1 X z/ D ` n2Z t C n; t C n C 1Œ Š t; t C 1Œ Z; and p maps each summand homeomorphically. Þ (3.1.2) Proposition. Let p W E! B be a covering. Then the diagonal D of E E is open and closed in Z D f.x; y/ 2 E E j p.x/ D p.y/g. Proof. Let Ux be an open neighbourhood of x which is mapped homeomorphically under p. Then Z \.Ux Ux/ D Wx is contained in D, and Wx is an open neighbourhood of.x; x/ in Z. This shows that D is open. Let x 6D y and p.x/ D p.y/. Let x 2 Ux and y 2 Uy be the sheets of p over the open set U B. Since x 6D y, the intersection Ux \ Uy is empty. Hence Z \.Ux Uy/ is an open neighbourhood of.x; y/ in Z and disjoint to D. This shows that also the complement Z X D is open. Let p W E! B and f W X! B be maps; then F W X! E is a lifting of f along p, ifpF D f. (3.1.3) Proposition (Uniqueness of liftings). Let p W E! B be a covering. Let F0; F1 W X! E be liftings of f W X! B. Suppose F0 and F1 agree somewhere. If X is connected, then F0 D F1. Proof..F0; F1/ yield a map F W X! Z. By assumption, F 1.D/ is not empty, and hence, by (3.1.2), open and closed. If X is connected, then F 1.D/ D X, i.e., F0 D F1. 1Observe that the term “covering” has two quite different meanings in topology. 64 Chapter 3. Covering Spaces (3.1.4) Proposition. Let q W E!
B Œ0; 1 be locally trivial with typical ﬁbre F. Then B has an open cover U such that q is trivial over each set U Œ0; 1, U 2 U. Proof. If q is trivial over U Œa; b and over U Œb; c, then q is trivial over U Œa; c. Two trivializations over U fbg differ by an automorphism, and this automorphism can be extended over U Œb; c. Use this extended automorphism to change the trivialization over U Œb; c, and then glue the trivializations. By compactness of I there exist 0 D t0 < t1 < < tn D 1 and an open set U such that q is trivial over U Œti ; tiC1. For the classiﬁcation of covering spaces we need spaces with suitable local properties. A space X is called locally connected (locally path connected) if for each x 2 X and each neighbourhood U of x there exists a connected (path connected) neighbourhood V of x which is contained in U. Both properties are inherited by open subspaces. (3.1.5) Proposition. The components of a locally connected space are open. The path components of a locally path connected space Y are open and coincide with the components. Proof. Let K be the component of x. Let V be a connected neighbourhood of x. Then K [ V is connected and therefore contained in K. This shows that K is open. Let U be a component of Y and K a path component of U. Then U X K is a union of path components, hence open. In the case that U 6D K we would obtain a decomposition of U. We see that each point in a locally path connected space has a neighbourhood basis of open path connected sets. (3.1.6) Remark. Let B be path connected and locally path connected. Since a covering is a local homeomorphism, the total space E of a covering of B is locally path connected. Let E0 be a component of E and p0 W E 0! B the restriction of p. Then p0 is also a covering: The sets U, over which p is trivial, can be taken as path connected, and then a sheet over U is either contained in E 0 or disjoint to E0. Since B is path connected, we see by path lifting (3
.2.9) that p0 is surjective. By Þ (3.1.5), E is the topological sum of its components. A left action G E! E,.g; x/ 7! gx of a discrete group G on E is called properly discontinuous if each x 2 E has an open neighbourhood U such that U \ gU D ; for g 6D e. A properly discontinuous action is free. For more details about this notion see the chapter on bundle theory, in particular (14.1.12). A left G -principal covering consists of a covering p W E! B and a properly discontinuous action of G on E such that p.gx/ D p.x/ for.g; x/ 2 G E and such that the induced action on each ﬁbre is transitive. 3.1. Locally Trivial Maps. Covering Spaces 65 (3.1.7) Example. A left G-principal covering p W E! B induces a homeomorphism of the orbit space E=G with B. The orbit map E! E=G of a properly Þ discontinuous action is a G-principal covering. A covering p W E! B has an automorphism group Aut.p/. An automorphism is a homeomorphism ˛ W E! E such that p ı ˛ D p. Maps of this type are also called deck transformations of p. If p is a left G-principal covering, then each left translation lg W E! E, x 7! gx is an automorphism of p. We thus obtain a homomorphism l W G! Aut.p/. Let E be connected. Then an automorphism ˛ is determined by its value at a single point x 2 E, and ˛.x/ is a point in the ﬁbre p1.p.x//. Since G acts transitively on each ﬁbre, the map l is an isomorphism. Thus the connected principal coverings are the connected coverings with the largest possible automorphism group. Conversely, we can try to ﬁnd principal coverings by studying the action of the automorphism group. (3.1.8) Proposition. Let p W E! B be a covering. (1) If E is connected, then the action of
ant maps. We call a G-principal covering p W E! B over the path connected space B universal if the functor A.p/ is an equivalence of categories. 3.2 Fibre Transport. Exact Sequence The relation of a covering space to the fundamental groupoid is obtained via path lifting. For this purpose we now introduce the notion of a ﬁbration which will be studied later in detail. A map p W E! B has the homotopy lifting property (HLP) for the space X if the following holds: For each homotopy h W X I! B and each map a W X! E such that pa.x/ D hi.x/, i.x/ D.x; 0/ there exists a homotopy H W X I! E with pH D h and H i D a. We call H a lifting of h with initial condition a. The map p is called a ﬁbration if it has the HLP for all spaces. (3.2.1) Example. A projection p W B F! B is a ﬁbration. Let a.x/ D.a1.x/; a2.x//. The condition pa D hi says a1.x/ D h.x; 0/. If we set H.x; t/ D Þ.h.x; t/; a2.x//, then H is a lifting of h with initial condition a. (3.2.2) Theorem. A covering p W E! B is a ﬁbration. Proof. Let the homotopy h W X I! B and the initial condition a be given. Since I is connected, a lifting with given initial condition is uniquely determined (see (3.1.3)). Therefore it sufﬁces to ﬁnd for each x 2 X an open neighbourhood Vx such that hjVx I admits a lifting with initial condition ajVx. By uniqueness (3.1.3), these partial liftings combine to a well-deﬁned continuous map. By (3.2.3) there exists for each x 2 B an open neighbourhood Vx and an n 2 N such that h maps Vx Œi=n;.i C 1/=n into a set U over which p is trivial. Since p W p1.U /! U
this purpose assume given: (1) u W I! Fb; (2) h W I I! B a homotopy of paths from b to c; (3) V0; V1 W I! E liftings of h0, h1 with initial points u.0/; u.1/. These data yield a map a W I @I [ 0 I! E, deﬁned by a.s; "/ D V".s/ and a.0; t/ D u.t /. The lifting H of h with initial condition a, according to (3.2.4), yields a path t 7! H.1; t/ in Fc from V0.1/ to V1.1/. This shows that the map v# is well-deﬁned and depends only on the morphism Œv in the fundamental groupoid. The rule w#v# D.v w/# is easily veriﬁed from the deﬁnitions. Thus we have shown: (3.2.5) Proposition. The assignments b 7! 0.Fb/ and Œv 7! v# yield a functor Tp W ….B/! SET. We call Tp D T.p/ the transport functor associated to p. The functor Tp provides us with 0.Fb/ 1.B; b/! 0.Fb/;.x; Œv/ 7! v#.x/ D x Œv; a right action of the fundamental group on the set 0.Fb/. We write 0.F; x/ if Œx is chosen as base point of the set 0.F /. We use the action to deﬁne @x W 1.B; b/! 0.Fb; x/; Œv 7! x Œv: The map @x is 1.B; b/-equivariant, i.e., @xŒv w D.@xŒv/ Œw. 68 Chapter 3. Covering Spaces Recall that a sequence A ˇ! C of pointed maps is exact at B if the image of ˛ equals the kernel ˇ1./ of ˇ. Similarly for longer sequences. In this context a group is pointed by its neutral element. ˛! B (3.2
.6) Theorem. Let p.x/ D b and i W Fb E. The sequence 1.Fb; x/ i 1.E; x/ p 1.B; b/ @x 0.Fb; x/ i 0.E; x/ p 0.B; b/ is exact. Proof. It is easily veriﬁed from the deﬁnitions that the composition of two maps is the constant map. We consider the remaining four cases: kernel image. Let Œu 2 1.E; x/ and h W I I! B a null homotopy of pu. Consider the lifting problem for h with initial condition a W I 0 [ @I I! E with a.s; 0/ D u.s/ and a."; t/ D x. The lifting H of h is then a homotopy of loops from u to a loop in the image of i. Let @xŒv D Œx. This means: There exists a lifting V of v from x to V.1/ 2 Œx. Choose a path U W I! Fb from V.1/ to x. Then V U is a loop in E, and its class maps under 1.p/ to Œv, since pU is constant. Let 0.i/Œy D Œx. There exists a path w W I! E from x to y. The projection v D pw is a loop and @xŒv D Œy, by deﬁnition of @x. Let 0.p/Œy D Œb. Thus there exists a path v W I! B from p.y/ to x. Let V W I! E be a lifting of v with initial point y. Then V.1/ 2 Fb, and V shows 0.i/.ŒV.1/ D Œy. There is more algebraic structure in the sequence. (3.2.7) Proposition. The pre-images of elements under @x are the left cosets of 1.B; b/ with respect to p1.E; x/. The pre-images of 0.i/ are the orbits of the 1.B; b/-action on 0.Fb; x/. Proof. Let @xŒu D @xŒv. Choose liftings U, V
of u, v which start in x, and let W W I! Fb be a path from U.1/ to V.1/. Then U W V is a loop in E, and p.U W V / v'u, i.e., the elements Œu and Œv are contained in the same left coset. Conversely, elements in the same coset have the same image under @x. (A similar assertion holds for right cosets.) Suppose 0.i/Œa D 0.i/Œb. Then there exists a path w W I! E from a to b. Set v D pw. Then Œa Œv D Œb. Conversely, elements in the same orbit have equal image under 0.i/. We can apply (3.2.6) to a covering p W E! B. The ﬁbres are discrete. Therefore 1.Fb; x/ is the trivial group 1. Hence p W 1.E; x/! 1.B; b/ is injective. We state (3.2.6) for a covering: 3.2. Fibre Transport. Exact Sequence 69 (3.2.8) Proposition. Let p W E! B be a covering over a path connected space B. Then the sequence 1! 1.E; x/ p! 1.B; b/ @x! 0.Fb; x/ i! 0.E; x/ is exact and i is surjective. (The sets Fb D 0.Fb/ and 0.E/ have x as base point, and i W Fb E.) Thus E is path connected if and only if 1.B; b/ acts transitively on Fb. The isotropy group of x 2 Fb is the image of p W 1.E; x/! 1.B; b/. (3.2.9) Proposition (Path lifting). Let p W E! B be a covering. Let w W I! B be a path which begins at p.e/ D w.0/. Then there exists a unique lifting of w which begins in e. Two paths in E which start in the same point are homotopic if and only if their images in B are homotopic. Proof. The existence of the lifting follows from (3.2.2), applied to a point X, and the uniqueness holds
by (3.1.3). Let h W I I! B be a homotopy of paths and H W I I! E a lifting of h. Since t 7! H."; t/ are continuous maps into a discrete ﬁbre, they are constant (" D 0; 1). Hence H is a homotopy of paths. Let u0; u1 W I! E be paths which start in x, and suppose that pu0 and pu1 are homotopic. If we lift a homotopy between them with constant initial condition, then the result is a homotopy between u0 and u1. Let p W E! B be a right G-principal covering. Each ﬁbre Fb carries a free right transitive G-action. From the construction of the transport functor it is immediate that the ﬁbre transport TpŒw W Fb! Fc is G-equivariant. The left action.a; x/ 7! a x D a#.x/ of b D ….B/.b; b/ on Fb commutes with the right G-action; we say in this case that Fb is a.b; G/-set. Fix x 2 Fb. For each a 2 b there exists a unique x.a/ 2 G such that a x D x x.a/, since the action of G is free and transitive. The assignment a 7! x.a/ is a homomorphism x W b! G. Since 1.B; b/ is the opposite group to b, we set ıx.a/ D x.a/1. Then ıx W 1.B; b/! G is a homomorphism. Recall the map @x W 1.B; p.x//! Fb. If we compose it with the bijection x W G! Fb, g 7! xg, we obtain xıx D @x. Then (3.2.8) yields: (3.2.10) Proposition. Let p W E! B be a right G-principal covering with path connected total space. Then the sequence of groups and homomorphisms 1! 1.E; x/ p! 1.B; p.x// ıx! G! 1 is exact ( for each x 2 E). The image of p is a normal subgroup. The
space E is simply connected if and only if ıx is an isomorphism. Thus, if E is simply connected, then G is isomorphic to the fundamental group of B. 70 Chapter 3. Covering Spaces If we apply this to the exponential covering R! S 1, a Z-principal covering, we again obtain 1.S 1/ Š Z. The transport functor Tp has an additional property, it is locally trivial in the following sense. Let p be trivial over U, and let b; c 2 U. Then TpŒw W Fb! Fc is independent of the path w W I! U from b to c. This is due to the fact that lifts of paths inside U stay within a sheet over U. 3.3 Classiﬁcation of Coverings Let TRAB D Œ….B/; SET denote the category of functors ….B/! SET (objects) and natural transformations between them (morphisms). We call this category the transport category. Let p W E! B be a covering. We have constructed the associated transport functor Tp D T.p/ W ….B/! SET. For a morphism ˛ W p! q between coverings the restrictions ˛b W p1.b/! q1.b/ of ˛ to the ﬁbres yield a natural transformation T.˛/ W T.p/! T.q/ between the corresponding transport functors. So we have obtained a functor T W COVB! TRAB : A path connected space B is called a transport space if T is an equivalence of categories. The main theorem of this section gives conditions under which the transport functor T is an equivalence. (3.3.1) Note. Let p W E! B be a covering with simply connected E. Then B is path connected. If p is trivial over U, then two paths in U between the same points are homotopic in B. Proof. The space B is path connected, since p is assumed to be surjective and E is path connected. Let u0; u1 be paths in U between the same points. By (3.2.9) they have liftings v0; v1 which connect the same points. Since E is simply connected, v0; v1 are homotopic in E and hence u0; u1 are homot
opic in B. A set U B is transport-simple if two paths in U between the same points are homotopic within B. A space B is semi-locally simply connected if it has an open covering by transport-simple sets. We have just seen that this condition is implied by the existence of a simply connected covering. We call B transport-local, if B is path connected, locally path connected and semi-locally simply connected. (3.3.2) Theorem (Classiﬁcation I). Let B be path connected, locally path connected and semi-locally simply connected. Then B is a transport space, i.e., T is an equivalence of categories. 3.3. Classiﬁcation of Coverings 71 Proof. We begin by constructing a functor X W TRAB! COVB in the opposite direction. Let ˆ W ….B/! SET be a functor. We construct an b2B ˆ.b/, and associated covering p D p.ˆ/ W X.ˆ/! B. As a set, X.ˆ/ D p.ˆ/ sends ˆ.b/ to b. Let U be the set of open, path connected and transportsimple subsets of B. We deﬁne bundle charts over sets U 2 U. For b 2 U we deﬁne ` 'U;b W U ˆ.b/! p1.U /;.u; z/ 7! ˆ.w/z with some path w in U from b to u. By our assumption on U, the map 'U;b is welldeﬁned, i.e., the choice of w does not matter. By construction, 'U;b is bijective. We claim: There exists a unique topology on X.ˆ/ such that the 'U;b of this type are homeomorphisms onto open subsets. By general principles of gluing, we have to verify that the transition maps '1 V;c ı 'U;b W.U \ V / ˆ.b/!.U \ V / ˆ.c/ are homeomorphisms. Let x 2 U \ V and let W U \ V be an open, path connected neighbourhood of x. Let ux be a path from b to x inside U, and v
x a path from c to x inside V. Then for all y 2 W V;c ı 'U;b.y; z/ D '1 '1 V;c ı 'U;b.x; z/; because in order to deﬁne 'U;b.y; z/ we can take the product of ux with a path wy in W from x to y, and similarly for 'V;c, so that the contribution of the piece wy cancels. This shows that the second component of '1 V;c ı 'U;b is on W ˆ.b/ independent of x 2 W. The continuity of the transition map is a consequence. If ˛ W ˆ1! ˆ2 is a natural transformation, then the morphism X.˛/ W X.ˆ1/! X.ˆ2/; x 2 ˆ1.b/ 7! ˛.x/ 2 ˆ2.b/; induced by ˛, is continuous with respect to the topologies just constructed and hence a morphism of coverings. The continuity of X.˛/ follows from the fact, that bundle charts ˆU;b for X.ˆ1/ and X.ˆ2/ transform X.˛/ into id ˛.b/ W U ˆ1.b/! U ˆ2.b/: This ﬁnishes the construction of the functor X W TRAB! COVB. We now show that the functors T and X are mutually inverse equivalences of categories, i.e., that XT and TX are naturally isomorphic to the identity functor. From our constructions we see immediately a canonical homeomorphism W X.T.p// Š E.p/ over B for each covering p W E.p/! B, namely set-theoretically the identity. We have to show that is continuous. Let v 2 Fb X.T.p//. Let W be a 72 Chapter 3. Covering Spaces neighbourhood of v in E.p/. Then there exists an open neighbourhood V W of v in E.p/ such that V is path connected and U D p.V / 2 U. Let b D p.v/. Then we have the bundle charts 'U;b, and W D 'U
;b.U v/ is a neighbourhood of v in X.T.p//. From the construction of 'U;v we see that.W / V. This shows the continuity at v. It is easily veriﬁed that the homeomorphisms constitute a natural transformation. Conversely, we have to verify that the transport functor of p.ˆ/ is ˆ. Using the bundle charts 'U;b this is ﬁrst veriﬁed for paths in U. But each morphism in ….B/ is a composition of morphisms represented by paths in such sets U. Fix b 2 B. A canonical functor is the Hom-functor ….b; / of the category ….B/. Let pb W E b! B denote the associated covering. We still assume that B is transport-local. The automorphism group b D ….b; b/ of b in ….B/, the opposite fundamental group 1.B; b/, acts on E b ﬁbrewise from the right by composition of morphisms. The action is free and transitive on each ﬁbre. Via our bundle charts it is easily veriﬁed that the action on Eb is continuous. Thus pb is a right b-principal covering. From (3.2.8) we see that Eb is simply connected. Thus we have shown: (3.3.3) Theorem. The canonical covering pb W E b! B associated to the Homfunctor ….b; / has a simply connected total space. The right action of ….b; b/ on the ﬁbres by composition of morphisms is the structure of a right principal covering on pb. Problems 1. Let S be the pseudo-circle. The space S is simply connected. But S has non-trivial connected principal coverings. They can be obtained by a pullback along suitable maps S! S 1 from Z=n-principal or Z-principal coverings of S 1. In this sense S behaves like S 1. We see that certain local properties of B are necessary in order that T is an equivalence. 2. Let f W B! C be a continuous map. The pullback p W X! B of a covering q W Y! C along f is a covering. Pulling back morphisms yields a functor f D
COV.f / W COVC! COVB. The map f induces a functor ….f / W ….B/! ….C /, and composition with functors ….C /! SET yields a functor TRA.f / W TRAC! TRAB. These functors are compatible TB ı COV.f / D TRA.f / ı TC W COVC! TRAB. 3.4 Connected Groupoids In this section the space B is assumed to be path connected. A functor ….B/! SET is an algebraic object. The category of these functors has an equivalent description in terms of more familiar algebraic objects, namely group actions. We explain this equivalence. Let … be a connected groupoid (i.e. there exists at least one morphism between any two objects) with object set B, e.g., … D ….B/ for a path connected space B. 3.4. Connected Groupoids 73 Let ….x; y/ denote the set of morphisms x! y and D b D ….b; b/, the automorphism group of b with respect to composition of morphisms. A functor F W …! SET has an associated set F.b/ with left b-action b F.b/! F.b/;.˛; x/ 7! F.˛/.x/: A natural transformation ˛ W F! G yields a map ˛.b/ W F.b/! G.b/ which is b-equivariant. In this manner we obtain a functor "b W Œ…; SET! b-SET from the functor category of functors …! SET into the category of left b-sets and equivariant maps. We construct a functor b in the opposite direction. So let A be a b-set. The Hom-functor ….b; ‹/ is a functor into the right b-sets, namely b acts on ….b; x/ by composition of morphisms. These data yield the functor ˆ.A/ D ….b; ‹/ A. (Here again A B denotes the quotient of A B by the equivalence relation.ag; b/.a; gb/;.a; g; b/ 2 A B for left -sets A and right -sets B.) A
b-map f W A! B induces a natural transformation ˆ.f / W ˆ.A/! ˆ.B/. This ﬁnishes the deﬁnition of b. (3.4.1) Proposition. The functors "b and b are mutually inverse equivalences of categories. Proof. The composition "bb associates to a b-set A the b-set ….b; b/ A, with b-action g.f; z/. The isomorphisms A W ….b; b/ A! A;.f; z/ 7! f z form a natural equivalence W "bb'Id. The composition b"b associates to a functor F W …! SET the functor ….b; / F.b/. The maps ˇF.x/ W ….b; x/ F.b/! F.x/;.f; z/ 7! F.f /z form a natural transformation, i.e., a morphism ˇF W b"b.F /! F in Œ…; SET. Since … is a connected groupoid, the ˇF.x/ are bijective, and therefore constitute an isomorphism in the functor category. The ˇF are a natural equivalence ˇ W b"b'Id. In our previous notation TRAB D Œ….B/; SET. From (3.3.2) and (3.4.1) we obtain for each transport-local space an equivalence of categories COVB! b-SET, the composition of the transport functor T with "b. It associates to a covering p W E! B the b-set Fb. The inverse equivalence associates to a b-set A the covering ` X.bA/ D x2B ….B/.b; x/ A! B: 74 Chapter 3. Covering Spaces It is the covering Eb A! B associated to the b-principal covering (3.3.3). Let p W E! B be a right G-principal covering with path connected B. Then we have the functors G- SET A.p/ COVB T TRAB "b'b- SET: The composition associates to a G-set F the b-set Fb G F, where the b-action is induced from
the left b-action on Fb. Now suppose in addition that E is simply connected. Then we have a bijection 'F x W F! Fb G F, z 7! Œx; z for a ﬁxedx 2 Fb as well as the isomorphism x W b! G, see (3.2.10). The relation x.a/ z D a Œx; z holds. So if we view G-sets via x as b-sets, then the above composition of functors is the identity. Thus we have shown: (3.4.2) Proposition. Let p W E! B be a simply connected G-principal covering. Then A.p/ is an equivalence of categories if and only if T is an equivalence of categories. (3.4.3) Theorem. The following properties of B are equivalent: (1) B is a transport space, i.e., T is an equivalence of categories. (2) B has a universal right G-principal covering p W E! B with simply con- nected total space E. Proof..1/ ).2/. Since "b ı T is an equivalence of categories, each object of b- SET is isomorphic to an object in the image of "b ı T. Thus there exists a covering p W E! B such that its b-set Fb is isomorphic to the b-set b. By another property of an equivalence of categories, the morphisms p! p correspond under "b ı T bijectively to the b-maps Fb! Fb. The b-morphisms b! b are the right translations by elements of b. Thus b acts simply and transitively on E. From (3.2.8) we see that E is simply connected. The left action of the automorphism group Aut.p/ on E is properly disconIf we rewrite this as tinuous and the induced action on each ﬁbre is transitive. a right action of the opposite group G, we obtain a right G-principal covering. Proposition (3.4.2) now says that p is universal..2/ ).1/ is a consequence of (3.4.2). From a geometric view point the interesting coverings are those with connected total space. Let p W E! B be a universal right
G-principal covering. A left G-set A is the disjoint sum of its orbits. We have a corresponding sum decomposition of the total space E G A into the sum of E G C, where C runs through the orbits of A. An orbit is a transitive G-set and isomorphic to a homogeneous set G=H for some subgroup H of G. The homeomorphism E G G=H Š E=H shows 3.4. Connected Groupoids 75 that the summands E G C are path connected. The action of H on E is properly discontinuous and therefore E! E=H an H -principal covering. Also the induced map pH W E=H! B is a covering. The category of homogeneous G-sets and G-maps is the orbit category Or.G/ of G. The sets G=K and G=L are isomorphic if and only if the subgroups K and L are conjugate in G. The isotropy groups of G=H are conjugate to H. The inclusion of the subcategory Or.G/ into the category of transitive G-sets is an equivalence. Let p W E! B be a universal right G-principal covering with simply connected E. Then the functor A.p/ induces an equivalence of Or.G/ with the category of connected coverings of B. Each covering is thus isomorphic to a covering of the form pH W E=H! B for a subgroup H of G. We ﬁx z 2 p1.b/ E and obtain an isomorphism ız W G! 1.B; b/. It sends g 2 G to the loop Œp ı wg where wg W I! E is a path from z to zg1. Let q W X! B be a connected covering. We know that the induced homomorphism p W 1.X; x/! 1.B; b/ is injective. The image is called the characteristic subgroup C.p; x/ of p with respect to x. Let u W I! X be a path from x to y 2 p1.b/. Then w D pu is a loop and C.p; y/ D ŒwC.p; x/Œw1, thus different base points in p1.b/ yield conjugate characteristic subgroups.
Conversely, each subgroup conjugate to C.p; x/ arises this way. We apply this to the covering pH with Nz D zH 2 E=H. Then.pH /.1.E=H; Nz/ D ız.H / D C.pH ; Nz/: We collect the results in the next theorem. (3.4.4) Theorem (Classiﬁcation II). Let B be a transport space. The category of connected coverings of B is equivalent to the orbit category Or.1.B; b//. The isomorphism class of a connected covering q W X! B corresponds under this equivalence to the isomorphism class of 1.B; b/=C.q; x/ for any x 2 p1.b/. The isomorphism class of a connected covering is determined by the conjugacy class of its characteristic subgroup. Problems 1. The automorphism group of pH W E=H! B is NH=H, where NH denotes the normalizer of H in 1.B; b/. The covering is a principal covering (also called regular covering), if and only if H is a normal subgroup of 1.B; b/. 2. The connected coverings of S 1 are, up to isomorphism, the maps pn W z 7! zn for n 2 N and p W R! S 1, t 7! exp.2 it/. These coverings are principal coverings. 3. Let B be a contractible space. Is the identity id W B! B a universal G-principal covering for the trivial group G? 76 Chapter 3. Covering Spaces 3.5 Existence of Liftings The following theorem (3.5.2) is interesting and important, because it asserts the existence of liftings under only the necessary algebraic conditions on the fundamental groups. (3.5.1) Lemma. Let w0 and w1 be paths in E beginning in x. Let ui D pwi. Then w0.1/ D w1.1/ if and only if u0.1/ D u1.1/ and Œu0 u 1 is contained in p1.E; x/. Proof. If w0.1/ D w1.1/, then pŒw0 w 1. Conversely: We lift u0 u 1 with initial point x. Since
Œu0 u 1 2 p1.E; x/ there exists a loop which is homotopic to u0 u 1, and which has a lifting with initial point x. By (3.2.9), u0 u 1 itself has a lifting as a loop. Therefore u0 and u1 have liftings with initial point x and the same end point. These liftings are then necessarily w0 and w1. 1 D Œu0 u (3.5.2) Theorem. Let p W E! B be a covering. Suppose Z is path connected and locally path connected. Let f W Z! B be a map with f.z/ D p.x/. Then there exists a lifting ˆ W Z! E of f with ˆ.z/ D x if and only if f1.Z; z/ is contained in p1.E; x/. Proof. If a lifting exists, then the inclusion of groups holds by functoriality of 1. Suppose f1.Z; z/ p1.E; x/. We begin by constructing ˆ as a set map. Then we show its continuity. Let z0 2 Z. There exists a path w from z to z0. Let v W I! E be a lifting of f w starting in x. We want to deﬁne ˆ by ˆ.z/ D v.1/. Let w1 be another path from z to z0 and v1 a lifting of f w1 starting in x. Then pv.1/ D f w.1/ D f.z0/ D f w1.1/ D pv1.1/I moreover Œpv pv 1 D fŒw w 1 2 p1.E; x/: By (3.5.1) we havev.1/ D v1.1/; this shows that ˆ is well-deﬁned if we set ˆ.z/ D v.1/. Continuity of ˆ. Let U be an open neighbourhood of ˆ.z0/, such that p is trivial over p.U / D V, and let p W U! V have the inverse homeomorphism q W V! U. Let W be a path connected neighbourhood of z0 such that f.W / V. We claim ˆ.W / U.
Let z1 2 W and let w1 be a path in W from z0 to z1. Then w w1 is a path from z to z1, and v1 D v qf w1 a path with pv1 D f ı.w w1/ and v1.0/ D x. Thus v1.1/ 2 U. (3.5.3) Theorem. Let X be a topological group with neutral element x and let p W E! X be a covering with path connected and locally path connected E. For each e 2 p1.x/ there exists a unique group structure on E which makes E into a topological group with neutral element e and such that p is a homomorphism. 3.5. Existence of Liftings 77 Proof. Construction of a group structure on E. Let m W X X! X be the group multiplication. We try to ﬁnd M W E E! E as a lift m.p p/ along p with M.e; e/ D e. This can be done, by (3.5.2), if m.p p/1.E E/ p1.E/. This inclusion holds, since (using (2.7.3)) m.pp/Œ.w1; w2/ D Œpw1pw2 D Œpw1pw2 D Œp.w1w2/ D pŒw1w2: From the uniqueness of liftings one shows that M is associative. In a similar manner we see that (passage to) the inverse in X has a lifting to E, and uniqueness of liftings shows that the result is an inverse for the structure M. A well-known result of Hermann Weyl is that a compact, connected, semi-simple Lie group has a ﬁnite simply connected covering. See [29, V.7] about fundamental groups of compact Lie groups. The group O.n/ has two different two-fold coverings which are non-trivial over SO.n/. They are distinguished by the property that the elements over the reﬂections at hyperplanes have order 2 or 4 (n 1). We will see that 1.SO.n// Š Z=2 for n 3. The corresponding simply connected covering groups are the spinor groups Spin.n/; see e.g., [
29, I.6]. We repeat an earlier result in a different context. We do not assume that B has a universal covering. (3.5.4) Proposition. Let B be path connected and locally path connected. Coverings pi W.Xi ; xi /!.B; b/ with path connected total space are isomorphic if and only if their characteristic subgroups are conjugate in 1.B; b/. Proof. Since C.p1; x1/ and C.p2; x2/ are conjugate we can change the base point x2 such that the groups are equal. By (3.5.2), there exist morphisms f1 W.X1; x1/!.X2; x2/ and f2 W.X2; x2/!.X1; x1/, and since f2f1.x1/ D x1; f1f2.x2/ D x2 both compositions are the identity. By functoriality of 1 we see that isomorphic coverings have conjugate charac- teristic subgroups. Problems 1. We have given a direct proof of (3.5.2), although it can also be derived from our previous classiﬁcation results. The existence of a lift ˆ is equivalent to the existence of a section in the covering which is obtained by pullback along f. The 1.Z; z/-action on the ﬁbre Eb of the pullback is obtained from the 1.B; b/-action via f W 1.Z; z/! 1.B; b/. The existence of a section is equivalent to Eb having a ﬁxed point under the 1.Z; z/-action. If the inclusion of groups holds as in the statement of the theorem, then a ﬁxed point exists because the image of p is the isotropy group of the 1.B; b/-action. 2. Let p W E! B be a covering with path connected and locally path connected total space. The following are equivalent: (1) Aut.p/ acts transitively on each ﬁbre of p. (2) Aut.p/ acts transitively on some ﬁbre of p. (3) The characteristic subgroup is normal in 1.B; b/. (4) p is an Aut.p/-principal covering. 78 Chapter
3. Covering Spaces 3.6 The Universal Covering We collect some of our results for the standard situation that B is path connected, locally path connected and semi-locally simply connected space. Let us now call a covering p W E! B a universal covering if E is simply connected. (3.6.1) Theorem (Universal covering). Let B be as above. (1) There exists up to isomorphism a unique universal covering p W E! B. (2) The action of the automorphism group Aut.p/ on E furnishes p with the structure of a left Aut.p/-principal covering. (3) The group Aut.p/ is isomorphic to 1.B; b/. Given x 2 p1.b/, an isomorphism x W Aut.p/! 1.B; b/ is obtained, if we assign to ˛ 2 Aut.p/ the class of the loop pw for a path w from x to ˛.x/. (4) The space Eb is simply connected. Proof. (1) Existence is shown in (3.3.3). Since B is locally path connected, the total space of each covering has the same property. Let pi W Ei! B be simply connected coverings with base points xi 2 p1.b/. By (3.5.2), there exist morphisms ˛ W p1! p2 and ˇ W p2! p1 such that ˛.x1/ D x2 and ˇ.x2/ D x1. By uniqueness of liftings, ˛ˇ and ˇ˛ are the identity, i.e., ˛ and ˇ are isomorphisms. This shows uniqueness. i (2) By (3.1.8), the action of Aut.p/ on E is properly discontinuous. As in (1) one shows that Aut.p/ acts transitively on each ﬁbre of p. The map Aut.p/nE! B, induced by p, is therefore a homeomorphism. Since E! Aut.p/nE is a principal covering, so is p. (3) Since E is simply connected, there exists a unique homotopy class of paths w from x to ˛.x/. Since x and ˛.x/ are contained in the same
ﬁbre, pw is a loop. Therefore x is well-deﬁned. If we lift a loop u based at b to a path w beginning in x, then there exists ˛ 2 Aut.p/ such that ˛.x/ D w.1/. Hence x is surjective. Two paths starting in x have the same end point if and only if their images in B are homotopic. Hence x is injective. If v is a path from x to ˛.x/ and w a path from x to ˇ.x/, then v ˇw is a path from x to ˛ˇ.x/. Hence x is a homomorphism. (4) is shown in (3.3.3). (3.6.2) Theorem (Classiﬁcation III). Suppose that B has a universal covering p W E! B. Then p is a 1.B; b/-principal covering. Each connected covering of B is isomorphic to a covering of the form E=H! B, H 1.B; b/ a subgroup. This covering has H as a characteristic subgroup. Two such coverings are isomorphic if and only if the corresponding subgroups of 1.B; b/ are conjugate. 3.6. The Universal Covering 79 Problems Q1 1 S 1 is not semi-locally simply connected. 1. The product 2. Is the product of a countably inﬁnite number of the universal covering of S 1 a covering? 3. Identify in S 1 the open upper and the open lower hemi-sphere to a point. The resulting space X has four points. Show 1.X/ Š Z. Does X have a universal covering? 4. The quotient map p W Rn! Rn=Zn is a universal covering. The map q W Rn! T n,.xj / 7!.exp 2 ixj / is a universal covering of the n-dimensional torus T n D S 1 S 1. Let f W T n! T n be a continuous automorphism, and let F W Rn! Rn be a lifting of f q along q with F.0/ D 0. The assignments x 7! F.x/ C F.y/ and x 7! F.x C y/ are lift
ings of the same map with the same value for x D 0. Hence F.x C y/ D F.x/ C F.y/. From this relation one deduces that F is a linear map. Since F.Zn/ Zn, the map F is given by a matrix A 2 GLn.Z/. Conversely, each matrix in GLn.Z/ gives us an automorphism of T n. The group of continuous automorphisms of T n is therefore isomorphic to GLn.Z/. 5. Classify the 2-fold coverings of S 1 _ S 1 and of S 1 _ S 1 _ S 1. (Note that a subgroup of index 2 is normal.) 6. The k-fold (k 2 N) coverings of S 1 _ S 1 correspond to isomorphism classes of D 1.S 1 _ S 1/ D h u i h v i-sets of cardinality k. An action of on f1; : : : ; kg is determined by the action of u and v, and these actions can be arbitrary permutations of f1; : : : ; kg. Figure 3.1. The 3-fold regular coverings of S 1 _ S 1. Hence these actions correspond bijectively to the elements of Sk Sk (Sk the symmetric group). A bijection ˛ of f1; : : : ; kg is an isomorphism of the actions corresponding to.u; v/ and.u0; v0/ if and only if ˛1u0˛ D u; ˛1v0˛ D v. The isomorphism classes of k-fold coverings correspond therefore to the orbit set of the action Sk.Sk Sk/! Sk Sk;.˛; u; v/ 7!.˛u˛1; ˛v˛1/: Consider the case k D 3 and S3 D h A; B j A3 D 1; B 2 D 1; BAB 1 D A1g. The three conjugacy classes are represented by 1; A; B. We can normalize the ﬁrst component of each orbit correspondingly. If we ﬁx u, then the centralizer Z.u/ of u acts on the second component. We have Z.1/ D S3, Z.B/ D f1; Bg, and Z.A/ D
f1; A; A2g. This yields the following representing pairs for the orbits:.1; 1/;.1; A/cn;.1; B/;.A; 1/cn;.A; A/cn;.B; 1/;.B; A/c;.A; A2/cn;.B; B/;.B; AB/c:.A; B/c; 80 Chapter 3. Covering Spaces The transitive actions (which yield connected coverings) have the addition c, the normal subgroups (which yield regular coverings) have the addition n. Draw ﬁgures for the connected coverings. For this purpose study the restrictions of the coverings to the two summands S 1; note that under restriction a connected covering may become disconnected. Over each summand one has a 3-fold covering of S 1; there are three of them. 7. Classify the regular 4-fold coverings of S 1 _ S 1. 8. The Klein bottle has three 2-fold connected coverings. One of them is a torus, the other two are Klein bottles. 9. Let X be path connected and set Y D X I =X @I. Show 1.Y / Š Z. Show that Y has a simply connected universal Z-principal covering. Is Y always locally path connected? 10. Construct a transport space which is not locally path connected. 11. The space Rn with two origins is obtained from Rn C Rn by identifying x 6D 0 in the ﬁrst summand with the same element in the second summand. Let M be the line with two origins. Construct a universal covering of M and determine 1.M /. What can you say about 1 of Rn with two origins for n > 1? 12. Make the fundamental groupoid ….B/ into a topological groupoid with object space B. (Hypothesis (3.6.1). Use (14.1.17).) 13. Let X be a compact Hausdorff space and H.X/ the group of homeomorphism. Then H.X/ together with the CO-topology is a topological group and H.X/ X! X,.f; x/ 7! f.x/ a continuous group action. 14. The space C.S 1; S 1/ with CO-topology becomes a
topological group under pointwise multiplication of maps. 15. There are two continuous homomorphisms e W C.S 1; S 1/! S 1, f 7! f.1/ and d W C.S 1; S 1/! Z, f 7! degree.f /. Let M 0.S 1/ be the kernel of.e; d /. Let further fn W S 1! S 1, z 7! zn. The homomorphism s W S 1 Z! C.S 1; S 1/,.˛; n/ 7! ˛fn is continuous. The map M 0.S 1/.S 1 Z/! C.S 1; S 1/;.f;.˛; n// 7! f s.˛; n/ is an isomorphism of topological groups. The space M 0.S 1/ is isomorphic to the space V of continuous functions W R! R with '.0/ D 0 and '.x C 2/ D '.x/ or, equivalently, to the space of continuous functions ˛ W S 1! R with ˛.1/ D 0. The space V carries the sup-norm and the induced CO-topology. 16. Let M.S 1/ be the group of homeomorphisms S 1! S 1 of degree 1 with CO-topology. Each 2 S 1 yields a homeomorphism f W z 7! x. In this way S 1 becomes a subgroup of M.S 1/. Let M1.S 1/ be the subgroup of homeomorphisms f with f.1/ D 1. Then S 1 M1.S 1/! M.S 1/;.; h/ 7! f ı h is a homomorphism. The space M1.S 1/ is homeomorphic to the space H of homeomorphism f W Œ0; 1! Œ0; 1 with f.0/ D 0. The space H is contractible; a contraction is ft.x/ D.1 t/f.x/ C tx. Therefore the inclusion S 1! M.S 1/ is an h-equivalence. The space H.S 1/ of homeomorphisms of S 1 is h-equivalent to O.2/. Chapter 4 Elementary Homotopy Theory Further analysis and applications of the homotopy notion require a certain amount
of formal consideration. We deal with several related topics. (1) The construction of auxiliary spaces from the basic “homotopy cylinder” X I : mapping cylinders, mapping cones, suspensions; and dual constructions based on the “path space” X I. These elementary constructions are related to the general problem of deﬁning homotopy limits and homotopy colimits. (2) Natural group structures on Hom-functors in TOP0. By category theory they arise from group and cogroup objects in this category. But we mainly work with the explicit constructions: suspension and loop space. (3) Exact sequences involving homotopy functors based on “exact sequences” among pointed spaces (“space level”). These so-called coﬁbre and ﬁbre sequences are a fundamental contribution of D. Puppe to homotopy theory [155]. The exact sequences have a three-periodic structure, and it has by now become clear that data of this type are an important structure in categories with (formal) homotopy (triangulated categories). As an application, we use a theorem about homotopy equivalences of mapping cylinders to prove a gluing theorem for homotopy equivalences. The reader may have seen partitions of unity. In homotopy theory they are used to reduce homotopy colimits to ordinary colimits. Here we treat the simplest case: pushouts. 4.1 The Mapping Cylinder Let f W X! Y be a map. We construct the mapping cylinder Z D Z.f / of f via the pushout X C X id Cf X C Y h i0;i1 i h j;J i X I a Z.f / Z.f / D X I C Y =f.x/.x; 1/; J.y/ D y; j.x/ D.x; 0/: Here it.x/ D.x; t/. Since hi0; i1 i is a closed embedding, the maps hj; J i, j and J are closed embeddings. We also have the projection q W Z.f /! Y,.x; t/ 7! f.x/, y 7! y. The relations qj D f and qJ D id hold. We denote elements in Z.f / by 82 Chapter 4. Elementary Homotopy Theory
their representatives in x/ I The map J q is homotopic to the identity relative to Y. The homotopy is the identity on Y and contracts I relative 1 to 1. We thus have a decomposition of f into a closed embedding J and a homotopy equivalence q. From the pushout property we see: Continuous maps ˇ W Z.f /! B correspond bijectively to pairs h W X I! B and ˛ W Y! B such that h.x; 1/ D ˛f.x/. In the following we consider Z.f / as a space under X C Y via the embedding (inclusion) hj; J i. We now study homotopy commutative diagrams together with homotopies ˆ W f 0˛'ˇf. In the case that the diagram is commutative, the pair.˛; ˇ/ is a morphism from f to f 0 in the category of arrows in TOP. We consider the data.˛; ˇ; ˆ/ as a generalized morphism. These data induce a map D Z.˛; ˇ; ˆ/ W Z.f /! Z.f 0/ deﬁned by (.y/ D ˇ.y/; y 2 Y;.x; s/ D.˛.x/; 2s/; x 2 X; s 1=2; x 2 X; s 1=2: ˆ2s1.x/; The diagram X C Y Z.f / ˛Cˇ Z.˛;ˇ;ˆ/ X 0 C Y 0 Z.f 0/ is commutative. The composition of two such morphisms between mapping cylinders is homotopic to a morphism of the same type. Suppose we are given f 00 W X 00! Y 00, ˛0 W X 0! X 00, ˇ0 W Y 0! Y 00, and a homotopy ˆ0 W f 00˛0'ˇ0f 0. These data yield a composed homotopy ˆ0 ˘ ˆ W f 00˛0˛'ˇ0ˇf deﬁned by (.ˆ0 ˘ ˆ/t D
ˆ0 2t ı ˛; ˇ0 ı ˆ2t1; t 1=2; t 1=2: (This is the product of the homotopies ˆ0 t ˛ and ˇ0ˆt.) 4.1. The Mapping Cylinder 83 (4.1.1) Lemma. There exists a homotopy Z.˛0; ˇ0; ˆ0/ ı Z.˛; ˇ; ˆ/'Z.˛0˛; ˇ0ˇ; ˆ0 ˘ ˆ/ which is constant on X C Y. Proof. Both maps coincide on Y and differ on X I by a parameter transformation of I. We also change ˛ and ˇ by a homotopy. Suppose given homotopies At W X! X 0 and B t W Y! Y 0 and a homotopy t W f 0At'B t f. We assume, of course, that t D. t s / is continuous on X I I. We get a homotopy Z.At ; B t ; t / W Z.f /! Z.f 0/ which equals At on these subspaces. We use the fact that At ; B t ; ˆ induce a homotopy t. (4.1.2) Lemma. Suppose At ; B t with A0 D ˛; B 0 D ˇ and ˆ W f 0A0'B 0f are given.Then there exists t with 0 D ˆ. Proof. One applies a retraction X I I! X.@I I [ I 0/ to the map W X.@I I [I 0/! Y 0 deﬁned by.x; s; 0/ D ˆ.x; s/,.x; 0; t/ D f 0At.x/ and.x; 1; t/ D B t f.x/. Suppose now that X 00 D X, Y 00 D Y, f 00 D f and ˛0˛'id, ˇ0ˇ'id, f ˛0'ˇ0f 0. We choose homotopies At W ˛0˛'id; B t
W ˇ0ˇ'id; ˆ0 W f ˛0'ˇ0f 0: As before, we have the composition ‰ D ˆ0 ˘ˆ. We use (4.1.2) to ﬁnd a homotopy t with 0 D ‰ and t W fAt'B t f. Let 1 be the inverse homotopy of 1. Let D Z.1X ; 1Y ; 1 / ı Z.˛0; ˇ0; ˆ0/ W Z.f 0/! Z.f /; this morphism restricts to ˛0 C ˇ0 4.1.3) Proposition. There exists a homotopy from ı Z.˛; ˇ; ˆ/ to the identity which equals..k At / k C.k B t / k/ on X C Y ; here k denotes a constant homotopy. Proof. By (4.1.1) there exists a homotopy relative to X C Y of the composition in question to Z.1X ; 1Y ; 1 / ı Z.˛0˛; ˇ0ˇ; ‰/. By (4.1.2) we have a further homo/ ı Z.1X ; 1Y ; 1/, which equals At C B t on X C Y, and then topy to Z.1X ; 1Y ; 1 by (4.1.1) a homotopy to Z.1X ; 1Y ; 1 ˘ 1/, which is constant on X C Y. The homotopy 1 ˘ 1 W f'f is homotopic relative to X @I to the constant homotopy kf of f. We thus have an induced homotopy relative X C Y to Z.1X ; 1Y ; kf / and ﬁnally a homotopy to the identity (Problems 1 and 2). (4.1.4) Theorem. Suppose ˛ and ˇ are homotopy equivalences. Then the map Z.˛; ˇ; ˆ/ is a homotopy equivalence. 84 Chapter 4. Elementary Homotopy Theory Proof. The morphism in (4.1.3) has a right homotopy inverse.
We can apply (4.1.1) and (4.1.3) to and see that also has a left homotopy inverse. Hence is a homotopy equivalence. From ı Z.˛; ˇ; ˆ/'id we now conclude that Z.˛; ˇ; ˆ/ is a homotopy equivalence. Problems 1. Suppose ˆ and ‰ are homotopic relative to X @I. Then Z.˛; ˇ; ˆ/ and Z.˛; ˇ; ‰/ are homotopic relative to X C Y. 2. In the case that f 0˛ D ˇf we have the map Z.˛; ˇ/ W Z.f /! Z.f 0/ induced by ˛ id Cˇ. Let k be the constant homotopy. Then Z.˛; ˇ; k/'Z.˛; ˇ/ relative to X C Y. 3. Let Œˆ denote the morphism in ….X; Y 0/ represented by ˆ. We think of.˛; ˇ; Œˆ/ as a morphism from ˛ to ˇ. The composition is deﬁned by.˛0; ˇ0; Œˆ0/ ı.˛; ˇ; Œˆ/ D.˛0˛; ˇ0ˇ; Œˆ0 ˘ ˆ/. Show that we obtain in this manner a well-deﬁned category. (This deﬁnition works in any 2-category.) 4.2 The Double Mapping Cylinder Given a pair of maps f W A! B and g W A! C. The double mapping cylinder f A g! C / is the quotient of B C A I C C with respect to Z.f; g/ D Z.B the relations f.a/.a; 0/ and.a; 1/ g.a/ for each a 2 A. We can also deﬁne it via a pushout A C A f Cg B C C h i0;i1 i h j0;j1 i A I Z.f; g/.
The map hj0; j1 i is a closed embedding. In the case that f D id.A/, we can identify Z.id.A/; g/ D Z.g/. We can also glue Z.f / and Z.g/ along the common subspace A and obtain essentially Z.f; g/ (up to I [f0g I Š I ). A commutative diagram ˛ A0 C C 0 induces Z.ˇ; ˛; / W Z.f; g/! Z.f 0; g0/, the quotient of ˇ C ˛ id C. We can also generalize to an h-commutative diagram as in the previous section. (4.2.1) Theorem. Suppose ˇ, ˛, are h-equivalences. Then Z.ˇ; ˛; / is an h-equivalence. 4.2. The Double Mapping Cylinder 85 In order to use the results about the mapping cylinder, we present Z.f; g/, up to canonical homeomorphism, also as the pushout of j A W A! Z.f / and j B W A! Z.g/. Here the subspace Z.f / corresponds to the image of B C A Œ0; 1=2 in Z.f; g/ and Z.g/ to the image of A Œ1=2; 1 C C. We view Z.f; g/ If we are given homotopies ˆB W f 0˛'ˇf, as a space under B C A C C. ˆC W g0˛'g, we obtain an induced map ‰ D Z.˛; ˇ; ˆB / [A Z.˛; ; ˆC / W Z.f; g/! Z.f 0; g0/ which extends ˇ C ˛ C. (4.2.2) Theorem. Let ˛ be an h-equivalence with h-inverse ˛0 and suppose ˇ and have left h-inverses ˇ0; 0. Choose homotopies At W ˛0˛'id, B t W ˇ0ˇ'id, C t W 0
'id. Then there exists W Z.f 0; g0/! Z.f; g/ and a homotopy from ı ‰ to the identity which extends..k B t / k C.k At / k C.k C t / k/. Proof. The hypotheses imply ˇ0f 0'f ˛ and 0g0'g˛. We can apply (4.1.3) and ﬁnd left homotopy inverses B of Z.˛; ˇ; ˆB / and C of Z.˛; ; ˆC /. Then D B [A C has the desired properties. Theorem (4.2.1) is now a consequence of (4.2.2). The reasoning is as for (4.1.4). In general, the ordinary pushout of a pair of maps f; g does not have good homotopy properties. One cannot expect to have a pushout in the homotopy category. A pushout is a colimit, in the terminology of category theory. In homotopy theory one replaces colimits by so-called homotopy colimits. We discuss this in the simplest case of pushouts. Given a diagram X0 f X fC j XC jC X (1) and a homotopy h W jf'jCfC. We obtain an induced map'W Z.f; fC/! X which is the quotient of hj; h; jC i W X C X0 I C XC! X. We deﬁne: The diagram (1) together with the homotopy h is called a homotopy pushout or homotopy cocartesian if the map'is a homotopy equivalence. This deﬁnition is in particular important if the diagram is commutative and h the constant homotopy. Suppose we have inclusions f˙ W X0 X˙ and j˙ W X˙ X such that X D X [ XC. In the case that the interiors X ı ˙ cover X, the space X is a pushout in the category TOP. In many cases it is also the homotopy pushout; the next proposition is implied by (4.2.4) and (4.2.5). (4.2
.3) Proposition. Suppose the covering X˙ of X is numerable (deﬁned below). Then X is the homotopy pushout of f˙ W X0 X˙. 86 Chapter 4. Elementary Homotopy Theory For the proof we ﬁrst compare Z.f; fC/ with the subspace N.X; XC/ D X 0 [ X0 I [ XC 1 of X I. We have a canonical bijective map ˛ W Z.f; fC/! N.X; XC/. Both spaces have a canonical projection to X (denoted pZ; pN ), and ˛ is a map over X with respect to these projections. (4.2.4) Lemma. The map ˛ is an h-equivalence over X and under X˙. Proof. Let W I! I be deﬁned by.t/ D 0 for t 1=3,.t/ D 1 for t 2=3 and.t/ D 3t 1 for 1=3 t 2=3. We deﬁne ˇ W N.XXC/! Z.f; fC/ as id.X0/ on X0 I and the identity otherwise. Homotopies ˛ˇ'id and ˇ˛'id are induced by a linear homotopy in the I -coordinate. The reader should verify that ˇ and the homotopies are continuous. The covering X˙ of X is numerable if the projection pN has a section. A section is determined by its second component s W X! Œ0; 1, and a function of this type deﬁnes a section if and only if X X X s1.0/; X X XC s1.1/. (4.2.5) Lemma. Suppose pN has a section. Then pN is shrinkable. Proof. A homotopy ı pN'id over X is given by a linear homotopy in the I -coordinate. (4.2.6) Corollary. Suppose the covering X˙ is numerable. Then pZ is shrinkable. (4.2.7) Theorem. Let.X; X˙/ and.Y; Y˙/ be numerable coverings
. Suppose that F W X! Y is a map with F.X˙/ Y˙. Assume that the induced partial maps F˙ W X˙! Y˙ and F0 W X0! Y0 are h-equivalences. Then F is an h-equivalence. Proof. This is a consequence of (4.2.1) and (4.2.6). The double mapping cylinder of the projections X X Y! Y is called the join X? Y of X and Y. It is the quotient space of X I Y under the relations.x; 0; y/.x; 0; y0/ and.x; 1; y/.x0; 1; y/. Intuitively it says that each point of X is connected with each point of Y by a unit interval. The reader should verify S m? S n Š S mCnC1. One can also think of the join as CX Y [XY X C Y where CX denotes the cone on X. 4.3 Suspension. Homotopy Groups We work with pointed spaces. Each object in the homotopy groupoid …0.X; Y / for TOP0 has an automorphism group. We describe the automorphism group of the constant map in a different manner. 4.3. Suspension. Homotopy Groups 87 A map K W X I! Y is a pointed homotopy from the constant map to itself if and only if it sends the subspace X @I [ fxg I to the base point y of Y. The quotient space †X D X I =.X @I [ fxg I / is called the suspension of the pointed space.X; x/. The base point of the suspension is the set which we identiﬁed to a point. I.................... X 1 0 q!.................... †X 1 1 2 0 A homotopy K W X I! Y from the constant map to itself thus corresponds to a pointed map xK W †X! Y, and homotopies relative to X @I correspond to pointed homotopies †X! Y. This leads us to the homotopy set Œ†X; Y 0. This set carries a group structure (written additively) which is deﬁned for representing maps by ( f C g W.x; t/
7! f.x; 2t/; g.x; 2t 1/; t 1 2 1 2 t: (Again we consider the group opposite to the categorically deﬁned group.) The inverse of Œf is represented by.x; t/ 7! f.x; 1 t/. For this deﬁnition we do not need the categorical considerations, but we have veriﬁed the group axioms. If f W X! Y is a pointed map, then f id.I / is compatible with passing to the suspensions and induces †f W †X! †Y,.x; t/ 7!.f.x/; t/. In this manner the suspension becomes a functor † W TOP0! TOP0. This functor is compatible with homotopies: a pointed homotopy Ht induces a pointed homotopy †.Ht /. There exists a canonical homeomorphism I kCl =@I kCl D I k=@I k ^ I l =@I l which is the identity on representing elements in I kCl D I k I l. We have for each pointed space X canonical homeomorphisms.X ^ I k=@I k/ ^ I l =@I l Š X ^ I kCl =@I kCl ; †l.†kX/ Š †kCl X: We deﬁne the k-fold suspension by †kX D X ^.I k=@I k/. Note that †nX is canonically homeomorphic to X I n=X @I n [ fxg @I n. In the homotopy set Œ†kX; Y 0 we have k composition laws, depending on which of the I -coordinates we use: (.f Ci g/.x; t/ D f.x; t1; : : : ; ti1; 2ti ; tiC1; : : : /; g.x; t1; : : : ; ti1; 2ti 1; tiC1; : : : /; ti 1 2 ; 1 2 ti : 88 Chapter 4. Elementary Homotopy Theory We show in a moment that all these group structures coincide and that they are abelian (n 2). For the purpose of the proof one veriﬁes directly the commutation
rule (unravel the deﬁnitions).a C1 b/ C2.c C1 d / D.a C2 c/ C1.b C2 d /: t2 " c a d b! t1 (4.3.1) Proposition. Suppose the set M carries two composition laws C1 and C2 with neutral elements ei. Suppose further that the commutation rule holds. Then C1 D C2 D C, e1 D e2 D e, and the composition C is associative and commutative. Proof. The chain of equalities a D a C2 e2 D.a C1 e1/ C2.e1 C1 e2/ D.a C2 e1/ C1.e1 C2 e2/ D.a C2 e1/ C1 e1 D a C2 e1 shows that e1 is a right unit for C2. In a similar manner one shows that e1 is a left unit and that e2 is a left and right unit for C1. Therefore e1 D e1 C2 e2 D e2. The equalities a C2 b D.a C1 e/ C2.e C1 b/ D.a C2 e/ C1.e C2 b/ D a C1 b show C1 D C2 D C. From b C c D.e C b/ C.c C e/ D.e C c/ C.b C e/ D c C b we obtain the commutativity. Finally a C.b C c/ D.a C e/ C.b C c/ D.a C b/ C.e C c/ D.a C b/ C c shows associativity. The suspension induces a map † W ŒA; Y 0! Œ†A; †Y 0, Œf 7! Œ†f, also called suspension. If A D †X, then † is a homomorphism, because the addition in Œ†X; Y 0 is transformed by † into C1. Suppose X D S 0 D f˙e1g with base point e1. We have a canonical homeo- morphism I n=@I n Š †nS 0 Š S 0 I n=S 0 @I n [ e1 I n which sends x 2 I n
to.e1; x/. The classical homotopy groups of a pointed space are important algebraic in- variants. The n-th homotopy group is n.X/ D n.X; x/ D ŒI n=@I n; X0 D Œ.I n; @I n/;.X; x/; n 1: These groups are abelian for n 2. We can use each of the n coordinates to deﬁne the group structure. 4.4. Loop Space 89 4.4 Loop Space We now dualize the concepts of the previous section. Let.Y; y/ be a pointed space. The loop space Y of Y is the subspace of the path space Y I (with compact-open topology) consisting of the loops in Y with base point y, i.e., Y D fw 2 Y I j w.0/ D w.1/ D yg: The constant loop k is the base point. A pointed map f W Y! Z induces a pointed map f W Y! Z, w 7! f ı w. This yields the functor W TOP0! TOP0. It is compatible with homotopies: A pointed homotopy Ht yields a pointed homotopy Ht. We can also deﬁne the loop space as the space of pointed maps F 0.I =@I; Y /. The quotient map p W I! I =@I induces Y p W Y I =@I! Y I and a homeomorphism F 0.I =@I; Y /! Y of the corresponding subspaces. (4.4.1) Proposition. The product of loops deﬁnes a multiplication m W Y Y! Y;.u; v/ 7! u v: It has the following properties: (1) m is continuous. (2) The maps u 7! k u and u 7! u k are pointed homotopic to the identity. (3) m.m id/ and m.id m/ are pointed homotopic. (4) The maps u 7! u u and u 7! u u are pointed homotopic to the constant map. Proof. (1) By (2.4.3) it sufﬁces to prove continuity of the adjoint map Y Y I! Y;.u; v; t/ 7
Œ; M 0 is a contravariant functor into the category of monoids. A monoid is a set together with a composition law with neutral element. An H -space is associative if m.m id/'m.id m/ and commutative if m'm with the interchange.x; y/ D.y; x/. An inverse for an H -space is a map W M! M such that m. id/d and m.id /d are homotopic to the constant map (d the diagonal). An associative H -space with inverse is a group object in h- TOP0. By a general principle we have spelled out the deﬁnition in TOP0. The axioms of a group are satisﬁed up to homotopy. A homomorphism (up to homotopy) between H -spaces.M; m/ and.N; n/ is a map W M! N such that n. /'m. A subtle point in this context is the problem of “coherence”, e.g., can a homotopy-associative H -space be h-equivalent to a strictly associative one (by a homomorphism up to homotopy)? The loop space..X/; m/ is a group object in h- TOP0. One can try other algebraic notions “up to homotopy”. Let.M; m/ be an associative H -space and X a space. A left action of M on X in h- TOP0 is a map r W M X! X such that r.m id/'r.id r/ and x 7! r.; x/ is homotopic to the identity. A comonoid in h-TOP0 is a pointed space C together with a pointed map (comultiplication) W C! C _C such that pr1 and pr2 are pointed homotopic to the identity. In ŒC; Y 0 we have the composition law C deﬁned as Œf C Œg D Œı ı.f _ g/ ı with the codiagonal (also called the folding map) ı D hid; id i W Y _ Y! Y. The functor ŒC; 0 is a covariant functor into the category of monoids. The comult
iplication is coassociative up to homotopy if.id.C / _ / ı and. _ id.C // ı are pointed homotopic; it is cocommutative up to homotopy if and are pointed homotopic, with W C _C! C _C the interchange map. Let.C; / and.D; / be 4.5. Groups and Cogroups 91 monoids in h-TOP0; a cohomomorphism up to homotopy'W C! D is a pointed map such that.' _ '/ and'are pointed homotopic. A coinverse W C! C for the comonoid C is a map such that ı.id _/ and ı. _ id/ are both pointed homotopic to the constant map. A coassociative comonoid with coinverse in h-TOP0 is called a cogroup in h-TOP0. Let.C; / be a coassociative comonoid and Y a space. A left coaction of C on Y (up to homotopy) is a map W Y! C _ Y such that.id _/ '. _ id/ and prY'id. The suspension †X is such a cogroup. We deﬁne the comultiplication W †X! †X _ †X; D i1 C i2 as the sum of the two injections i1; i2 W †X! †X _ †X. Explicitly,.x; t/ D.x; 2t/ in the ﬁrst summand for t 1 2, and.x; t/ D.x; 2t 1/ in the second summand for 1 2 t. 1 1 2 0 2.................... 1! 2 1 The previously deﬁned group structure on Œ†X; Y 0 is the -sum. (4.5.1) Proposition. Let.C; / be a comonoid and.M; m/ a monoid in h-TOP0. The composition laws C and Cm in ŒC; M 0 coincide and are associative and commutative. Proof. We work in h-TOP0, as we should; thus morphisms are pointed homotopy classes. We have the projections pk W M M! M and the injections il W C
! C _C. Given f W C _C! M M we set fkl D pkf il. Then mf D p1f Cmp2f and f D f i1 C f i2. From these relations we derive the commutation rule.mf / D.f11 C f12/ Cm.f21 C f22/; m.f/ D.f11 Cm f21/ C.f12 Cm f22/: Now apply (4.3.1). Problems 1. Let X be a pointed space and suppose that the Hom-functor Œ; X0 takes values in the category of monoids. Then X carries, up to homotopy, a unique H -space structure m which 92 Chapter 4. Elementary Homotopy Theory induces the monoid structures on ŒA; X0 as Cm. There is a similar result for Hom-functors ŒC; 0 and comonoid structures on C. 2. Let S.k/ D I k=@I k. We have canonical homeomorphisms k.Y / D F 0.S.k/; Y / Š F..I k; @I k/;.Y; // and kl.Y / Š kCl.Y /: 3. The space F..I; 0/;.Y; // Y I is pointed contractible. 4. Let F2.I; X/ D f.u; v/ 2 X I X I j u.1/ D v.0/g. The map 2 W F2.I; X/! F.I; X/,.u; v/ 7! u v is continuous. 5. Let F3.I; X/ D f.u; v; w/ j u.1/ D v.0/, v.1/ D w.0/g. The two maps 3; 0 3 W F3.I; X/! F.I; X/;.u; v; w/ 7!.u v/ w; u.v w/ are homotopic over X X where the projection onto X X is given by.u; v; w/ 7!.u.0/; w.1//. 6. Verify the homeomorphism F 0.I =@I; Y / Š Y. 4.6 The Coﬁbre Sequence A pointed map
embedding f1 W Y! C.f /, and CX CX CY induces j W CX! C.f /. The pushout property says: The pairs ˛ W Y! B, h W CX! B with ˛f D hi1, i.e., the pairs of ˛ and null homotopies of ˛f, correspond to maps ˇ W C.f /! B with ˇj D h. If Œ˛ is contained in the kernel of f, then there exists ˇ W C.f /! B with ˇf1 D ˛, i.e., Œ˛ is contained in the image of f 1. Moreover, f1f W X! C.f / is null homotopic with null homotopy j. This shows that the sequence X f! Y f1! C.f / is h-coexact. We iterate the passage from f to f1 and obtain the h-coexact sequence f X Y f1 C.f / f2 C.f1/ f3 C.f2/ f4 : : : : The further investigations replace the iterated mapping cones with homotopy equivalent spaces which are more appealing. This uses the suspension. It will be important that the suspension of a space arises in several ways as a quotient space; certain canonical bijections have to be proved to be homeomorphisms. In the next diagram the left squares are pushout squares and p; p.f /; q.f / are quotient maps. The right vertical maps are homeomorphisms, see Problem 1. Now †X Š CX= i1X, by the identity on representatives. Therefore we view p, p.f /, 94 Chapter 4. Elementary Homotopy Theory and q.f / as morphisms to †X. i1 f1 X Y f i1 p CX=i1X D †X CX j C.f / p.f / C.f /=f1Y D †X f2 C Y j1 C.f1/ q.f / C.f1/=j1C Y D †X (4.6.1) Note. The quotient map q.f / is a homotopy equivalence. Proof. We deﬁne a hom
otopy ht of the identity of C.f1/ which contracts C Y along the cone lines to the cone point and drags CX correspondingly ( ht.x; s/ D.x;.1 C t/s/;.f.x/; 2.1 C t/s/;.1 C t/s 1;.1 C t/s 1; ht.y; s/ D.y;.1 t/s/: In order to verify continuity, one checks that the deﬁnition is compatible with the equivalence relation needed to deﬁne C.f1/. The end h1 of the homotopy has the form s.f / ı q.f / with s.f / W †X! C.f1/,.x; s/ 7! h1.x; s/. The composition q.f / ı s.f / W.x; s/ 7!.x; min.2s; 1// is also homotopic to the identity, as we know from the discussion of the suspension. This shows that s.f / is h-inverse to q.f /. We treat the next step in the same manner: C.f / C.f1/ C.f2/ f2 p.f / q.f / f3 p.f1/ †.f /ı †X q.f1/ †Y with an h-equivalence q.f1/. Let W †X! †X,.x; t/ 7!.x; 1 t/ be the inverse of the cogroup †X. The last diagram is not commutative if we add the morphism †f to it. Rather the following holds: (4.6.2) Note. †.f / ı ı q.f /'p.f1/. Proof. By (4.6.1) it sufﬁces to study the composition with s.f /. We know already that p.f1/s.f / W.x; s/ 7!.f.x/; min.1; 2.1s// is homotopic to †f ı W.x; s/ 7!.f.x/; 1 s/. 4.6. The Coﬁbre
Sequence 95 An h-coexact sequence remains h-coexact if we replace some of its spaces by h-equivalent ones. Since the homeomorphism does not destroy exactness of a sequence, we obtain from the preceding discussion that the sequence X f Y f1 C.f / p.f / †X †f †Y is h-coexact. We can continue this coexact sequence if we apply the procedure above to †f instead of f. The next step is then.†f /1 W †Y! C.†f /. But it turns out that we can also use the suspension of the original map †.f1/ W †Y! †C.f / in order to continue with an h-coexact sequence. This is due to the next lemma. (4.6.3) Lemma. There exists a homeomorphism 1 W C.†f /! †C.f / which satisﬁes 1 ı.†f /1 D †.f1/. Interchange of I Proof. C †X and †CX are both quotients of X I I. coordinates induces a homeomorphism W C †X! †CX which satisﬁes ı i †X 1 D †.i X 1 /. We insert this into the pushout diagrams for C.†f / and †C.f / and obtain an induced 1. We use that a pushout in TOP0 becomes a pushout again if we apply † (use -†-adjunction). We now continue in this manner and obtain altogether an inﬁnite h-coexact sequence. (4.6.4) Theorem. The sequence X f Y f1 C.f / p.f / †X †f †Y †f1 †C.f / †p.f / †2X †2f : : : is h-coexact. We call it the Puppe-sequence or the coﬁbre sequence of f ([155]). The functor Œ; B0 applied to the Puppe-sequence yields an exact sequence of pointed sets; it consists from the fourth place onwards of groups and homomorphisms and from the seventh place onwards of abelian groups. See [49]
for an introduction to some other aspects of the coﬁbre sequence. The derivation of the coﬁbre sequence uses only formal properties of the homotopy notion. There exist several generalizations in an axiomatic context; for an introduction see [69], [101], [18]. Let f W X! Y be a pointed map. Deﬁne W C.f /! †X _ C.f / by.x; t / D..x; 2t/; / for 2t 1,.x; t/ D.;.x; 2t 1// for 2t 1, and.y/ D y. This map is called the h-coaction of the h-cogroup †X on C.f /. This terminology is justiﬁed by the next proposition. 96 Chapter 4. Elementary Homotopy Theory 0 1 2 1...........! C.f / †X _ C.f / (4.6.5) Proposition. The map induces a left action Œ†X; B0 ŒC.f /; B0 Š Œ†X _ C.f /; B0! ŒC.f /; B0;.˛; ˇ/ 7! ˛ ˇ ˇ: This action satisﬁes ˛1ˇp.f /˛2 D p.f /.˛1˛2/. Moreover, f if and only if there exists ˛ such that ˛ ˇ ˇ1 D ˇ2. Thus f map of the orbits of the action. 1.ˇ1/ D f 1.ˇ2/ 1 induces an injective Proof. That satisﬁes the axioms of a group action is proved as for the group axioms involving †X. Also the property involving p.f / is proved in the same manner. It remains to verify the last statement. Assume that f; g W C.f /! B are maps which become homotopic when restricted to Y. Consider the subspaces C0 D f.x; t/ j 2t 1g C.f / and C1 D f.x; t/ j 2t 1g [Y C.f /. These inclusions are coﬁbrations (
correspond to pointed homotopies from the constant map to e1h, if we pass from h to the adjoint map B I! Y. We deﬁne F.f / via a pullback q f F.f / f 1 X F Y e1 Y F.f / D f.x; w/ 2 X F Y j f.x/ D w.1/g f 1.x; w/ D x; q.x; w/ D w: The maps ˇ W B! F.f / correspond to pairs ˛ D f 1ˇ W B! X together with the null homotopies qˇ W B! F Y of f ˛. This shows that F.f / h-exact. We now iterate the passage from f to f 1 f 1! X f! Y is : : : f 4 F.f 2/ f 3 F.f 1/ f 2 F.f / f 1 X f Y and show that F.f 1/ and F.f 2/ can be replaced, up to h-equivalence, by Y and X. We begin with the remark that.f 1/1./ D f.x; w/ j w.0/ D ; w.1/ D f.x/; x D g can be identiﬁed with Y, via w 7!.; w/. Let i.f / W Y! F.f / be the associated inclusion of this ﬁbre of f 1. The space (by its pullback deﬁnition) F.f 1/ D f.x; w; v/ j w.0/ D ; w.1/ D f.x/; x D v.1/; v.0/ D g 98 Chapter 4. Elementary Homotopy Theory can be replaced by the homeomorphic space F.f 1/ D f.w; v/ 2 F Y FX j w.1/ D f v.1/g: Then f 2 becomes f 2 W F.f 1/! F.f /,.w; v/ 7!.v.1/; w/. The map j.f / W Y! F.f 1/; w 7!.w; k/ satisﬁes f 2 ı j.f / D i.f /. (4.7
.1) Note. The injection j.f / is a homotopy equivalence. Proof. We construct a homotopy ht of the identity of F.f 1/ which shrinks the path v to its beginning point and drags behind the path w correspondingly. We write t.w; v/; h2 ht.w; v/ D.h1 t.w; v// 2 F Y FX and deﬁne ( h1 t.s/ D w.s.1 C t//; f v.2.1 C t/s/; s.1 C t/ 1; s.1 C t/ 1; h2 t.s/ D v.s.1 t//: The end h1 of the homotopy has the form j.f / ı r.f / with r.f / W F.f 1/! Y;.w; v/ 7! w.f v/: The relation.r.f / ı j.f //.w/ D w k shows that also r.f / ı j.f / is homotopic to the identity. The continuity of ht is proved by passing to the adjoint maps. We treat the next step in a similar manner. F.f 2/ f 3 F.f 1/ i.f 1/ j.f 1/ j.f / i.f 1/.v/ D.k; v/: X f Y The upper triangle is commutative, and (4.7.2) applies to the lower one. The map i.f 1/ is the embedding of the ﬁbre over the base point. Let W Y! Y, w 7! w be the inverse. (4.7.2) Note. j.f / ı ı f'i.f 1/. Proof. We compose both sides with the h-equivalence r.f / from the proof of (4.7.1). Then r.f /ıi.f 1/ equals v 7! k.f v/, and this is obviously homotopic to ı f W v 7!.f v/. As a consequence of the preceding discussion we see that the sequence X f Y i.f / F.f / f 1 X f Y is h-exact. 4.7. The Fibre Sequence 99 (4.7.3
) Lemma. There exists a homeomorphism 1 W F.f /! F.f / such that.f /1 D.f 1/ ı 1. Proof. From the deﬁnitions and standard properties of mapping spaces we have F.f / X F Y and F.f / X F Y. We use the exponential law for mapping spaces and consider F Y and F Y as subspaces of Y I I. In the ﬁrst case we have to use all maps which send @I I [ I 0 to the base point, in the second case all maps which send I @I [ 0 I to the base point. Interchanging the I -coordinates yields a homeomorphism and it induces 1. We now continue as in the previous section. (4.7.4) Theorem. The sequence : : : 2f 2Y i.f / F.f / f 1 X f Y i.f / F.f / f 1 X f Y is h-exact. We call it the ﬁbre sequence of f. When we apply the functor ŒB; 0 to the ﬁbre sequence we obtain an exact sequence of pointed sets which consists from the fourth place onwards of groups and homomorphisms and from the seventh place onwards of abelian groups ([147]). Problems 1. Work out the dual of (4.6.5). 2. Describe what happens to the ﬁbre sequence under adjunction. A map a W T! F.f / D f.x; w/ 2 X F Y j f.x/ D w.1/g has two components b W T! X and ˇ W T! F Y. Under adjunction, ˇ corresponds to a map B W C T! Y from the cone over T. The condition f.x/ D w.1/ is equivalent to the commutativity f b D Bi1. This transition is also compatible with pointed homotopies, and therefore we obtain a bijection Œa 2 ŒT; F.f /0 Š Œi1; f 0 3 ŒB; b. This bijection transforms f 1 into the restriction ŒB; b 2 Œi1; f 0! ŒT; X0 3 Œb. In the next step we have Š Œ†T; Y 0 Œ �
�T; Y 0 i.f / ŒT; F.f /0 Š Œi1; f 0, Œ ı p; c. The image of is obtained in the following manner: With the quotient map p W C T! †T we have B D ı p, and b is the constant map c. 3. There exist several relations between ﬁbre and coﬁbre sequences. The adjunction.†; / yields in TOP0 the maps W X! †X (unit of the adjunction) and " W †X! X (counit of the adjunction). These are natural in the variable X. For each f W X! Y we also have natural maps W F.f /! C.f /; " W †F.f /! C.f / 100 Chapter 4. Elementary Homotopy Theory deﬁned by.x; w/.t/ D Œx; 2t; w.2 2t/; t 1=2; t 1=2; and " adjoint to. Verify the following assertions from the deﬁnitions. (1) The next diagram is homotopy commutative Y Y i.f / F.f / f 1 f1 C.f / p.f / †X. X (2) Let i W X! Z.f / be the inclusion and r W Z.f /! Y the retraction. A path in Z.f / that starts in and ends in X 0 yields under the projection to C.f / a loop. This gives a map W F.i/! C.f /. The commutativity ı F.r/'ı holds. (3) The next diagram is homotopy commutative i.f1/ F.f1/ Y ı q.f1/ı †X †f †Y. Chapter 5 Coﬁbrations and Fibrations This chapter is also devoted to mostly formal homotopy theory. In it we study the homotopy extension and lifting property. An extension of f W A! Y along i W A! X is a map F W X! Y such that F i D f. If i W A X is an inclusion, then this is an extension in the ordinary sense. Many topological problems can be given the form
of an extension problem. It is important to ﬁnd conditions on i under which the extendibility of f only depends on the homotopy class of f. If this is the case, then f is called a coﬁbration. The dual of the extension problem is the lifting problem. Suppose given maps p W E! B and f W X! B. A lifting of f along p is a map F W X! E such that pF D f. We ask for conditions on p such that the existence of a lifting only depends on the homotopy class of f. If this is the case, then f is called a ﬁbration Each map is the composition of a coﬁbration and a homotopy equivalence and (dually) the composition of a homotopy equivalence and a ﬁbration. The notions are then used to deﬁne homotopy ﬁbres (“homotopy kernels”) and homotopy coﬁbres (“homotopy cokernels”). Axiomatizations of certain parts of homotopy theory (“model categories”) are based on these notions. The notions also have many practical applications, e.g., to showing that maps are homotopy equivalences with additional properties like ﬁbrewise homotopy equivalences. Another simple typical example: A base point x 2 X is only good for homotopy theory if the inclusion fxg X is a coﬁbration (or the homotopy invariant weakening, a so-called h-coﬁbration). This is then used to study the interrelation between pointed and unpointed homotopy constructions, like pointed and unpointed suspensions. 5.1 The Homotopy Extension Property A map i W A! X has the homotopy extension property (HEP) for the space Y if for each homotopy h W A I! Y and each map f W X! Y with f i.a/ D h.a; 0/ there exists a homotopy H W X I! Y with H.x; 0/ D f.x/ and H.i.a/; t/ D h.a; t /. We call H an extension of h with initial condition f. The map i W A! X 102 Chapter 5. Co
ﬁbrations and Fibrations is a coﬁbration if it has the HEP for all spaces. The data of the HEP are displayed in the next diagram. We set!.x; t/ and e0.w/ D w.0/. h H f Y I e0 iid f X I H Y A I h For a coﬁbration i W A! X, the extendibility of f only depends on its homotopy class. From this deﬁnition one cannot prove directly that a map is a coﬁbration, but it sufﬁces to test the HEP for a universal space Y, the mapping cylinder Z.i/ of i. Recall that Z.i/ is deﬁned by a pushout.i/: Pairs of maps f W X! Y and h W A I! Y with hi A 0 D f i then correspond to maps W Z.i/! Y with b D f and k D h. We apply this to the pair i X 0 W X! X I and i id W A I! X I and obtain s W Z.i/! X I such that sb D i X 0 and sk D i id. Now suppose that i is a coﬁbration. We use the HEP for the space Z.i/, the initial condition b and the homotopy k. The HEP then provides us with a map r W X I! Z.i/ such that ri X 0 D b and r.i id/ D k. We conclude from rsb D ri X 0 D b, rsk D r.i id/ D k and the pushout property that rs D id.Z.i//, i.e., s is an embedding and r a retraction. Let r be a retraction of s. Given f and h, ﬁnd as above and set H D r. Then H extends h with initial condition f. Altogether we have shown: (5.1.1) Proposition. The following statements about i W A! X are equivalent: (1) i is a coﬁbration. (2) i has the HEP for the mapping cylinder Z.i/. (3) s W Z.i/! X I has a retraction. A coﬁbration i W A!
X is an embedding; and i.A/ is closed in X, if X is a Hausdorff space (Problem 1). Therefore we restrict attention to closed coﬁbrations whenever this simpliﬁes the exposition. A pointed space.X; x/ is called wellpointed and the base point nondegenerate if fxg X is a closed coﬁbration. 5.1. The Homotopy Extension Property 103 (5.1.2) Proposition. If i W A X is a coﬁbration, then there exists a retraction. If A is closed in X and if there exists a retraction r, then i is a coﬁbration. Proof. Let x/ D.x; 0/, and h.a; t/ D.a; t/. Apply the HEP to obtain a retraction r D H. If A is closed in X, then g W X 0[AI! Y, g.x; 0/ D f.x/,.a; t/ D h.a; t/ is continuous. A suitable extension H is given by gr. (5.1.3) Example. Let be a retraction. Set r.x; t/ D.r1.x; t/; r2.x; t//. Then H W X I I! X I;.x; t; s/ 7!.r1.x; t.1 s//; st C.1 s/r2.x; t// is a homotopy relative to X 0 [ A I of r to the identity, i.e., a deformation retraction. (5.1.4) Example. The inclusions S n1 Dn and @I n I n are coﬁbrations. A Þ retraction r W Dn! S n1 I [ Dn 0 was constructed in (2.3.5). It is an interesting fact that one need not assume A to be closed. Strøm [180, Theorem 2] proved that an inclusion A X is a coﬁbration if and only if the subspace X 0 [ A I is a retract of X I. If we multiply a retraction by id.Y / we obtain again a retraction. Hence AY! X Y is a (closed) coﬁbration

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