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for each Y, if i W A! X is a (closed) coﬁbration. Since we have proved (5.1.2) only for closed coﬁbrations, we mention another special case, to be used in a moment. Let Y be locally compact and i W A! X a coﬁbration. Then i id W A Y! X Y is a coﬁbration. For a proof use the fact that via adjunction and the exponential law for mapping spaces the HEP of i id for Z corresponds to the HEP of i for ZY. (5.1.5) Proposition. Let A X and assume that A I X I has the HEP for Y. Given maps'W A I I! Y; H W X I! Y; f " W X I! Y such that '.a; s; 0/ D H.a; s/; f ".x; 0/ D H.x; "/; f ".a; t/ D '.a; "; t/ " 2 f0; 1g, a 2 A, x 2 X, s; t 2 I. Then there exists ˆ W X I I! Y such that ˆ.a; s; t/ D '.a; s; t/; ˆ.x; s; 0/ D H.x; s/; ˆ.x; "; t/ D f ".x; t/: Proof. H and f " together yield a map ˛ W X.I 0 [ @I I /! Y deﬁned by ˛.x; s; 0/ D H.x; s/ and ˛.x; "; t/ D f ".x; t/. By our assumptions, ˛ and'coincide on A.I 0 [ @I I /. Let k W.I I; I 0 [ @I I /!.I I; I 0/ be a homeomorphism of pairs. Since A I! X I has the HEP for Y, there exists ‰ W X I I! Y which extends'ı.1 k1/ and ˛ ı.1 k1/. The map ˆ D ‰ ı.1 k/ solves the extension problem. 104 Chapter 5. Coﬁbrations and Fibrations (5.
1.6) Proposition. Let i W A X be a coﬁbration. Then X @I [ A I X I is a coﬁbration. Proof. Given h W.AI [X @I /I! Y and an initial condition H W X I! Y, we set'D hjA I I and f ".x; t/ D h.x; "; t/. Then we apply (5.1.5). For A D ; we obtain from (5.1.5) that X @I X I is a coﬁbration, in particular @I I and f0g I are coﬁbrations. Induction over n shows again that @I n I n is a coﬁbration. We list some special cases of (5.1.5) for a coﬁbration A X. (5.1.7) Corollary..1/ Let ˆ W X I! Y be a homotopy. Suppose'D ˆjA I is homotopic rel A @I to. Then ˆ is homotopic rel X @I to ‰ W X I! Y such that ‰jA I D..2/ Let ˆ solve the extension problem for.'; f / and ‰ the extension problem for. ; g/. Suppose f'g rel A and'' rel A @I. Then ˆ1'‰1 rel A..3/ Let ˆ; ‰ W X I! Y solve the extension problem for.h; f /. Then there exists a homotopy W ˆ'‰ rel X 0 [ A I. (5.1.8) Proposition. Let a pushout diagram in TOP be given. A j X f F B J Y If j has the HEP for Z, then J has the HEP for Z. If j is a coﬁbration, then J is a coﬁbration. Proof. Suppose h W B I! Z and'W Y! Z are given such that h.b; 0/ D fJ.b/ for b 2 B. We use the fact that the product with I of a pushout is again a pushout. Since j is a coﬁbration, there exists Kt W X! Z such that
K0 D 'f and Kt j D ht f. By the pushout property, there exists Ht W Y! Z such that Ht F D Kt and Ht J D ht. The uniqueness property shows H0 D ', since both maps have the same composition with Fj and Jf. We call J the coﬁbration induced from j via cobase change along f. Example. If A X is a coﬁbration, then fAg X=A is a coﬁbration. S n1 Dn is a coﬁbration, hence fS n1g D n=S n1 is a coﬁbration. The space Dn=S n1 Þ is homeomorphic to S n; therefore.S n; / is well-pointed. Example. If.Xj / is a family of well-pointed spaces, then the wedge well-pointed. W j Xj is Þ 5.1. The Homotopy Extension Property 105 Our next result, the homotopy theorem for coﬁbration says, among other things, that homotopic maps induce h-equivalent coﬁbrations from a given coﬁbration under a cobase change. Let j W A! X be a coﬁbration and'homotopy. We consider two pushout diagrams. A j X f F B jf Yf A j X g G B jg Yg Since j is a coﬁbration, there exists a homotopy ˆt W X! Yf with initial condition ˆ0 D F and ˆt j D jf 't. The pushout property of the Yg -diagram provides us with a unique map D'such that jg D jf and G D ˆ1. (We use the notation'although the map depends on ˆ1.) Thus'is a morphism of coﬁbrations W jg! jf between objects in TOPB. Moreover G'F. We now verify that the homotopy class of is independent of some of the choices involved. Let t be another homotopy from f to g which is homotopic to 't relative to A @I. Let ‰t W X! Yf be an extension of jf t with initial
condition ‰0 D F. Let W A I I! B be a homotopy rel A @I from'to. These data give us on X 0 I [ X I @I a map into Yf such that.x; 0; t/ D F.x/;.x; s; 0/ D ˆ.x; s/;.x; s; 1/ D ‰.x; s/: By (5.1.5) there exists an extension, still denoted, to X I I such that jf D.j id id/. We multiply the Yg diagram by I and obtain again a pushout. It provides us with a unique homotopy K W Yg I! Yf such that K ı.G id/ D 1 and K ı.jg id/ D jf ı pr where 1 W X I! Yf,.x; t/ 7!.x; 1; t /. By construction, K is a homotopy under B from'to a corresponding map obtained from t and ‰t. We thus have shown that the homotopy class ŒB under B of only depends on the morphism Œ' from f to g in the groupoid ….A; B/. Let us write Œ D ˇŒ'. We verify that ˇ is a functor ˇ.Œ ~ Œ'/ D ˇŒ' ı ˇŒ. Let W g'h W A! B. Choose a homotopy ‰t W X! Yg with ‰0 D G and ‰t j D jg t. Then W Yh! Yg is determined by jh D jg and H D ‰1. (Here.H; jh/ are the pushout data for.j; h/.) Since '‰0 D 'G D ˆ1, we can form the product homotopy ˆ '‰. It has the initial condition F and satisﬁes.ˆ '‰/.j id/ D jf''jg D jf.' /. Hence ', constructed with this homotopy, is determined by'H D '‰1 D'H and'jh D jf D 'jg D'jh. Therefore'represents ˇ.�
� ~ Œ'/. Let h-COFB denote the full subcategory of h-TOPB with objects the coﬁbrations under B. Then we have shown above: 106 Chapter 5. Coﬁbrations and Fibrations (5.1.9) Theorem. Let j W A! X be a coﬁbration. We assign to the object f W A! B in ….A; B/ the induced coﬁbration jf W B! Yf and to the morphism Œ' W f! g in ….A; B/ the morphism Œ' W jg! jf. These assignments yield a contravariant functor ˇj W ….A; B/! h-COFB. Since ….A; B/ is a groupoid, Œ' is always an isomorphism in h-TOPB. We refer to this fact as the homotopy theorem for coﬁbrations. (5.1.10) Proposition. In the pushout (5.1.8) let j be a coﬁbration and f a homotopy equivalence. Then F is a homotopy equivalence. Proof. With an h-inverse g W B! A of f we form a pushout B J Y g G A i Z. Since gf'id, there exists, by (5.1.9), an h-equivalence W Z! X under A such that GF'id. Hence F has a left h-inverse and G a right h-inverse. Now interchange the roles of F and G. Problems 1. A coﬁbration is an embedding. For the proof use that i1 W A! Z.i /, a 7!.a; 1/ is an embedding. From i1 D rsi1 D ri X 1 i then conclude that i is an embedding. Consider a coﬁbration as an inclusion i W A X. The image of s W Z.i /! X I is the subset X 0 [ A I. Since s is an embedding, this subset equals the mapping cylinder, i.e., one can deﬁne a continuous map X 0 [ A I by specifying its restrictions to X 0 and A I. (This is always so if A is closed in
X, and is a special property of i W A X if i is a coﬁbration.) Let X be a Hausdorff space. Then a coﬁbration i W A! X is a closed embedding. Let be a retraction. Then x 2 A is equivalent to r.x; 1/ D.x; 1/. Hence A is the coincidence set of the maps X! X I, x 7!.x; 1/, x 7! r.x; 1/ into a Hausdorff space and therefore closed. 2. If i W K! L, j W L! M have the HEP for Y, then j i has the HEP for Y. A homeomorphism is a coﬁbration. ; X is a coﬁbration. The sum qij W q Aj! qXj of coﬁbrations ij W Aj! Xj is a coﬁbration. 3. Let p W P! Q be an h-equivalence and i W A B a coﬁbration. Then f W A! P has an extension to B if and only if pf has an extension to B. Suppose f0; f1 W B! P agree on A. If pf0 and pf1 are homotopic rel A so are f0; f1. 4. Compression. Let A X be a coﬁbration and f W.X; A/!.Y; B/ a map which is homotopic as a map of pairs to k W.X; A/!.B; B/. Then f is homotopic relative to A to a map g such that g.X/ B. 5. Let A X be a coﬁbration and A contractible. Then the quotient map X! X=A is a homotopy equivalence. 6. The space C 0X D X I =X 1 is called the unpointed cone on X. We have the closed inclusions j W X! C 0X, x 7!.x; 0/ and b W fg! C 0X, 7! fX 1g. Both maps are coﬁbrations. 7. Let f W A X be an inclusion. We have a pushout diagram 5.2. Transport 107 j J A f X C 0A
F X [ C 0A. Since j is a coﬁbration, so is J. If f is a coﬁbration, then F is a coﬁbration. There exists a canonical homeomorphism X [ C 0A=C 0A Š X=A; it is induced by J. Since C 0A is contractible, we obtain a homotopy equivalence X [ C 0A! X [ C 0A=C 0A Š X=A. 8. The unpointed suspension †0X of a space X is obtained from X I if we identify each of the sets X 0 and X 1 to a point. If is a basepoint of X, we have the embedding j W I! †0X, t 7!.; t/. If fg X is a closed coﬁbration, then j is a closed (induced) coﬁbration. The quotient map †0X! †X is a homotopy equivalence. 5.2 Transport Let i W K! A be a coﬁbration and'W K I! X a homotopy. We deﬁne a map '# W Œ.A; i/;.X; '0/K! Œ.A; i/;.X; '1/K; called transport along ', as follows: Let f W A! X with f i D '0 be given. Choose a homotopy ˆt W A! X with ˆ0 D f and ˆt i D 't. We deﬁne '#Œf D Œˆ1. Then (5.1.5) shows that '# is well deﬁned and only depends on the homotopy class of'rel K @I, i.e., on the morphism Œ' 2 ….K; X/. From the construction we see.' /# D #'#. Altogether we obtain: (5.2.1) Proposition. Let i W K! A be a coﬁbration. The assignments '0 7! Œi; '0K and Œ' 7! '# yield a transport functor from ….K; X/ to sets. Since ….K; X/ is a groupoid,
'# is always bijective. The transport functor measures the difference between “homotopic” in TOPK and in TOP. The following is a direct consequence of the deﬁnitions. (5.2.2) Proposition. Let i W K! A be a coﬁbration. Let f W.A; i/!.X; g/ and f 0 W.A; i/!.X; g0/ be morphisms in TOPK. Then Œf D Œf 0, if and only if there exists Œ' 2 ….K; X/ from.X; g/ to.X; g0/ with Œf 0K D '#Œf K. (5.2.3) Proposition. Let i W K! A be a coﬁbration, g W K! X a map, and W X I! Y a homotopy. Then. ı.g id//# ı 0 D 1, if we set iŒf D Œ i f. Proof. Use that t f is an extension of t g and apply the deﬁnition. 108 Chapter 5. Coﬁbrations and Fibrations (5.2.4) Proposition. Let f W X! Y be an ordinary homotopy equivalence and i W K! A a coﬁbration. Then f W Œ.A; i/;.X; g/K! Œ.A; i/;.Y; fg/K is bijective. Proof. Let g be h-inverse to f and choose'W id'gf. Consider Œi; vK f Œi; f vK g Œi; gf vK f Œi; fgf vK: Since gf D.gf / D Œ'.v id/# id, we conclude from (5.2.1) and (5.2.3) that gf is bijective, hence g is surjective. The bijectivity of fg shows that g is also injective. Therefore g is bijective and hence f is bijective too. (5.2.5) Proposition. Let i W K! X and j W K! Y be coﬁbrations and f W X! Y
an h-equivalence such that f i D j. Then f is an h-equivalence under K. Proof. By (5.2.4), we have a bijective map f W Œ.Y; j /;.X; i/K! Œ.Y; j /;.Y; j /K: Hence there exists Œg with fŒgK D ŒfgK D ŒidK. Since f is an h-equivalence, so is g. Since also g is bijective, g has a homotopy right inverse under K. Hence g and f are h-equivalences under K. (5.2.6) Proposition. Let i W A! X be a coﬁbration and an h-equivalence. Then i is a deformation retract. Proof. The map i is a morphism from id.A/ to i. By (5.2.5), i is an h-equivalence under A. This means: There exists a homotopy X I! X rel A from the identity to a map r W X! A such that ri D id.A/, and this is what was claimed. (5.2.7) Proposition. Given a commutative diagram u u0 A i A0 0 Y g Y 0 X f X 0 with a coﬁbration i and h-equivalences and 0. Given v W A! X and'W v'u. Then there exists v0 W A0! X 0 and '0 W 0v0'u0 such that v0i D f v and '0 t i D g't. Proof. We have bijective maps (note 0f v D gv'gu D u0i).g'/# ı 0 W Œ.A0; i/;.X 0; f v/A! Œ.A0; i/;.Y 0; 0f v/A! Œ.A0; i/;.Y 0; u0i/A: Œv0A D Œu0A. This means: v0i D Let v0 W A0! X 0 be chosen such that.g'/# 0 f v; and 0v0 has a transport along g' to a map which is homotopic under A to u0.
5.2. Transport 109 This yields a homotopy '00 W 0v0'u0 such that '00.i.a/; t/ D g'.a; min.2t; 1//. The homotopy'k W.a; t/ 7! '.a; min.2t; 1// is homotopic rel A @I to '. We now use (5.1.6) in order to change this '00 into another homotopy '0 with the desired properties. If we apply (5.2.7) in the case that u and u0 are the identity we obtain the next result (in different notation). It generalizes (5.2.5). (5.2.8) Proposition. Given a commutative diagram A i X f F B j Y with coﬁbrations i, j and h-equivalences f and F. Given g W B! A and'W gf'id, there exists G W Y! X and ˆ W GF'id such that Gj D ig and ˆt i D i't. In particular:.F; f / is an h-equivalence of pairs, and there exists a homotopy inverse of the form.G; g/ W j! i. (5.2.9) Proposition. Suppose a commutative diagram a1 b1 X0 f0 Y0 a2 b2 X1 f1 Y2 X2 f2 Y2 is given with coﬁbration aj, bj and h-equivalences fj. Let X be the colimit of the aj and Y the colimit of the bj. Then the map f W X! Y induced by the fj is a homotopy equivalence. Proof. We choose inductively h-equivalences Fn W Yn! Xn such that anFn1 D Fnbn and homotopies 'n W Xn I! Xn from Fnfn to id.Xn/ such that an'n1 D 'n.an id/. This is possible by (5.2.7). The Fn and 'n induce F W Y! X and'W X I! X; Ff'id. Hence F is a left homotopy inverse of f. Problems 1. Let i W K! A and j W K! B be co
ﬁbrations. Let ˛ W.B; j /!.A; i/ be a morphism under K, W X! Y a continuous map, and'W K I! X a homotopy. Then Œ.A; i /;.X; '0/K '# Œ.A; i/;.X; '1/K Œ˛;K Œ˛;K Œ.B; j /;.Y; '0/K.'/# Œ.B; j /;.Y; '1/K 110 Chapter 5. Coﬁbrations and Fibrations commutes; here Œ˛; K Œf D Œf ˛. 2. Apply the transport functor to pointed homotopy sets. Assume that the inclusion fg A is a coﬁbration. For each path w W I! X we have the transport w# W ŒA;.X; w.0//0! ŒA;.X; w.1//0: As a special case we obtain a right action of the fundamental group (transport along loops) ŒA; X0 1.X; /! ŒA; X0;.x; ˛/ 7! x ˛ D ˛#.x/: Let v W ŒA; X0! ŒA; X denote the forgetful map which disregards the base point. The map v induces an injective map from the orbits of the 1-action into ŒA; X. This map is bijective, if X is path connected. A space is said to be A-simple if for each path w the transport w# only depends on the endpoints of w; equivalently, if for each x 2 X the fundamental group 1.X; x/ acts trivially on ŒA;.X; x/0. If A D S n, then we say n-simple instead of A-simple. We call X simple if it is A-simple for each well-pointed A. 3. The action on ŒI =@I; X0 D 1.X/ is given by conjugation. Hence this action is trivial if and only if the fundamental group is abelian. 4. Let ŒA; X0 carry a composition law induced by
a comultiplication on A. Then w# is a homomorphism. In particular 1.X/ acts by homomorphisms. (Thus, if the composition law on ŒA; X0 is an abelian group, then this action makes this group into a right module over the integral group ring Z1.X/.) 5. Write S.1/ D I =@I and 1.X/ D ŒS.1/; X0. Then we can identify ŒA; X01.X; / Š ŒA _ S.1/; X0. The action of the previous problem is induced by a map W A! A _ S.1/ which can be obtained as follows. Extend the homotopy I! A _ S.1/, t 7! t 2 S.1/ to a homotopy'W A I! A _ S.1/ with the initial condition A A _ S.1/ and set D '1. Express in terms of and the comultiplication of S.1/ the fact that the induced map is a group action ( is a coaction up to homotopy). 6. Let.X; e/ be a path connected monoid in h-TOP0. Then the 1.X; e/-action on ŒA; X0 is trivial. 5.3 Replacing a Map by a Coﬁbration We recall from Section 4.1 the construction of 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 f Cid Y C X h i0;i1 i h s;j i X I a Z.f / Z.f / D X I C Y =f.x/.x; 0/; s.y/ D y; j.x/ D.x; 1/: Since hi0; i1 i is a closed coﬁbration, the maps hs; j i, s and j are closed coﬁbrations. We also have the projection q W Z.f /! Y,.x; t/ 7! f.x/, y 7! y. In the case that f W X Y, let p W Y! Y =X be the quotient map. We also have the quotient 5.3. Replacing a Map by
a Coﬁbration 111 map P W Z.f /! C.f / D Z.f /=j.X/ onto the mapping cone C.f /. (Now we consider the unpointed situation. The “direction” of the unit interval is different from the one in the previous chapter.) We display the data in the next diagram. The map r is induced by q. Y =.f / s P Z.f / C.f / qj D f; qs D id r P s D c.f /; pq D rP (5.3.1) Proposition. The following assertions hold: (1) j and s are coﬁbrations. (2) sq is homotopic to the identity relative to Y. Hence s is a deformation retraction with h-inverse q. (3) If f is a coﬁbration, then q is a homotopy equivalence under X and r the induced homotopy equivalence. (4) c.f / W Y! C.f / is a coﬁbration. Proof. (1) was already shown. (2) The homotopy contracts the cylinder X I to X 0 and leaves Y ﬁxed, ht.x; c/ D.x; tc C 1 t/, ht.y/ D y. (3) is a consequence of (5.2.5). (4) c.f / is induced from the coﬁbration i0 W X! X I =X 1 via cobase change along f. We have constructed a factorization f D qj into a (closed) coﬁbration and a homotopy equivalence q. Factorizations of this type are unique in the following sense. Suppose f D q0j 0 W X! Z0! Y is another such factorization. Then iq0 W Z0! Z satisﬁes iq0j 0'i. Since j 0 is a coﬁbration, we can change iq0'k such that kj 0 D j. Since iq0 is an h-equivalence, the map k is an h-equivalence under X, by (5.2.5). Also qk'q0. This expresses a uniqueness of the factorization. If f D q
j W X! Z! Y is a factorization into a coﬁbration j and a homotopy equivalence q, then Z=j.X/ is called the (homotopical) coﬁbre of f. The uniqueness of the factorization implies uniqueness up to homotopy equivalence of the coﬁbre. If f W X Y is already a coﬁbration, then Y! Y =X is the projection onto the coﬁbre; in this case q W Z! Y is an h-equivalence under X. The factorization of a map into a coﬁbration and a homotopy equivalence is a useful technical tool. The proof of the next proposition is a good example. 112 Chapter 5. Coﬁbrations and Fibrations (5.3.2) Proposition. Let a pushout diagram A j X f F B J Y with a coﬁbration j be given. Then the diagram is a homotopy pushout. Proof. Let qi W A! Z.j /! X be the factorization of j. Since q is a homotopy equivalence under A, it induces a homotopy equivalence q [A id W Z.f / [A B D Z.f; j /! X [A B D Y of the adjunction spaces. (5.3.3) Proposition. Let a commutative diagram A0 ˛ l 0 ˇ B 0 A l B k0 k C 0 C L L0 K D K0 ı D0 be given. Suppose the inner and the outer square are homotopy cocartesian. If ˛, ˇ, are homotopy equivalences, then ı is a homotopy equivalence. Proof. From the data of the diagram we obtain a commutative diagram Z.k; l/ Z.ˇ;˛;/ Z.k0; l 0/'D ı '0 D0 where'and '0 are the canonical maps. By hypothesis,'and '0 are homotopy equivalences. By (4.2.1) the map Z.ˇ; ˛; / is a homotopy equivalence. (5.3.4) Proposition. Given a commutative diagram as in the previous proposition. Assume that the
squares are pushout diagrams. Then ı is induced by ˛, ˇ,. Suppose that ˛, ˇ, are homotopy equivalences and that one of the maps k, l and one of the maps k0, l 0 is a coﬁbration. Then ı is a homotopy equivalence. Proof. From (5.3.2) we see that the squares are homotopy cocartesian. Thus we can apply (5.3.3). 5.4. Characterization of Coﬁbrations 113 Problems 1. A map f W X! Y has a left homotopy inverse if and only if j W X! Z.f / has a retraction r W Z.f /! X. The map f is a homotopy equivalence if and only if j is a deformation retract. 2. In the case of a pointed map f W.X; /!.Y; / one has analogous factorizations into a coﬁbration and a homotopy equivalence. One replaces the mapping cylinder Z.f / with the pointed mapping cylinder Z0.f / deﬁned by the pushout X _ X h i0;i1 i XI f _id Y _ X h s;j i Z0.f / with the pointed cylinder XI D X I =fg I. The maps h i0; i1 i, h s; j i, s and j are pointed coﬁbrations. We have a diagram as for (5.3.1) with pointed homotopy equivalences s; q and C 0.f / D Z0.f /=j.X/ the pointed mapping cone, the pointed coﬁbre of f. 3. h i0; i1 i W X _ X! XI is an embedding. 4. Let f W X! Y and g W Y! Z be pointed maps. We have canonical maps ˛ W C.f /! C.gf / and ˇ W C.gf /! C.g/; ˛ is the identity on the cone and maps Y by g, and ˇ is the identity on Z and maps the cone by f id. Show that ˇ is the pointed homotopy coﬁbre of ˛. 5.4 Characterization of Co
ﬁbrations We look for conditions on A X which imply that this inclusion is a coﬁbration. We begin by reformulating the existence of a retraction (5.1.2). (5.4.1) Proposition. There exists a retraction if and only if the following holds: There exists a map u W X! Œ0; 1Œ and a homotopy'W X I! X such that: (1) A u1.0/ (2) '.x; 0/ D x for x 2 X (3) '.a; t/ D a for.a; t/ 2 A I (4) '.x; t/ 2 A for t > u.x/. Proof. Suppose we are given a retraction r. We set '.x; t / D pr1 ı r.x; t/ and u.x/ D maxft pr2 ı r.x; t/ j t 2 I g. For (4) note the following implications: t > u.x/, pr2 r.x; t / > 0, r.x; t/ 2 A I, '.x; t / 2 A. The other properties are immediate from the deﬁnition. Conversely, given u and ', then r.x; t/ D.'.x; t/; max.t u.x/; 0// is a retraction. (5.4.2) Note. Let tn > u.x/ be a sequence which converges to u.x/. Then (4) implies '.x; u.x// 2 xA. If u.x/ D 0, then x D '.x; 0/ D '.x; u.x// 2 xA. Thus 114 Chapter 5. Coﬁbrations and Fibrations xA D u1.0/. Therefore in a closed coﬁbration A X the subspace A has the remarkable property of being the zero-set of a continuous real-valued function. Þ (5.4.3) Lemma. Let u W X! I and A D u1.0/. Let ˆ W f'g W X! Z rel A. Then there exists Qˆ W f'g rel A such that Qˆ.x; t/ D Qˆ.x; u.x// D Qˆ.x;
1/ for t u.x/. Proof. We deﬁne Qˆ by Qˆ.x; t/ D ˆ.x; 1/ for t u.x/ and by ˆ.x; tu.x/1/ for t < u.x/. For the continuity of Qˆ on C D f.x; t/ j t u.x/g see Problem 1. We call.X; A/ a neighbourhood deformation retract (NDR ), if there exist a homotopy W X I! X and a function v W X! I such that: (1) A D v1.0/ (2).x; 0/ D x for x 2 X (3).a; t/ D a for.a; t/ 2 A I (4).x; 1/ 2 A for 1 > v.x/. The pair. ; v/ is said to be an NDR-presentation of.X; A/. (5.4.4) Proposition..X; A/ is a closed coﬁbration if and only if it is an NDR. Proof. If A X is a closed coﬁbration, then an NDR-presentation is obtained from (5.4.1) and (5.4.2). For the converse, we modify an NDR-presentation. ; u/ by (5.4.3) and apply (5.4.1) to the result. Q ; u/. (5.4.5) Theorem (Union Theorem). Let A X, B X, and A \ B X be closed coﬁbrations. Then A [ B X is a coﬁbration. Proof ([112]). Let'W.A [ B/ I! Z be a homotopy and f W X! Z an initial condition. There exist extensions ˆA W X I! Z of 'jAI and ˆB W B I! Z of 'jB I with initial condition f. The homotopies ˆA and ˆB coincide on.A \ B/ I. Therefore there exists ‰ W ˆA'ˆB rel.A \ B/ I [ X 0. Let p W X I! X I = be the quotient map which identiﬁ
es each interval fcg I, c 2 A \ B to a point. Let T W I I! I I switch the factors. Then ‰ ı.id T factors over p id and yields W.X I = / I! Z. Let u W X! I and v W X! I be functions such that A D u1.0/ and B D v1.0/. Deﬁne j W X! X I = by j.x/ D.x; u.x/=.u.x/ C v.x/// for x … A \ B and by j.x/ D.x; 0/ D.x; t/ for x 2 A \ B. Using the compactness of I one shows the continuity of j. An extension of'and f is now given by ı.j id/. (5.4.6) Theorem (Product Theorem). Let A X and B Y be closed coﬁbrations. Then the inclusion X B [ A Y X Y is a coﬁbration. Proof, and A B D.A Y / \.X B/ X B X Y are coﬁbrations. Now apply (5.4.5). Problems 5.5. The Homotopy Lifting Property 115 1. Let C D f.x; t/ j t u.x/g and q W X I! C,.x; t/ 7!.x; tu.x//. Then Qˆq D ˆ. It sufﬁces to show that q is a quotient map. The map W X I! X I I,.x; t/ 7!.x; t; u.x// is an embedding onto a closed subspace D. The map m W I I! I,.a; b/ 7! ab is proper, hence M D id m is closed. The restriction of M to D is closed, hence M D q is closed and therefore a quotient map. 2. The inclusion 0 [ fn1 j n 2 Ng Œ0; 1 is not a coﬁbration. The inclusions Aj D f0; j 1g I are coﬁbrations. Hence (5.4.5) does not hold for an inﬁnite number of coﬁbrations. 3. Set X
D fa; bg with open sets ;; fag; X for its topology. Then A D fag X is a non-closed coﬁbration. The product X A [ A X X X is not a coﬁbration. 4. Let Aj X be closed coﬁbrations (1 j n). For all f1; : : : ; ng let A D T S j 2 Aj X be a coﬁbration. Then n 1 Aj X is a coﬁbration. 5. Let A and B be well-pointed spaces. Then A ^ B is well-pointed. 5.5 The Homotopy Lifting Property 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 h.x; 0/ there exists a homotopy H W X I! E with pH D h and H.x; 0/ D a.x/. We call H a lifting of h with initial condition a. The map p is called a ﬁbration (sometimes Hurewicz ﬁbration) if it has the HLP for all spaces. It is called a Serre ﬁbration if it has the HLP for all cubes I n, n 2 N0. Serre ﬁbrations sufﬁce for the investigation of homotopy groups. In order to see the duality we can use the dual deﬁnition of homotopy and specify the data in the right diagram. It uses the evaluation e0 E W E I! E, w 7! w.0/: e0 B E e0 E pI a EI h X H We begin by introducing the dual W.p/ of the mapping cylinder. It is deﬁned B I by the pullback E p B b W.p/ e0 B k B I W.p/ D f.x; w/ 2 E B I j p.x/ D w.0/g; k.x; w/ D w; b.x; w/ D x: 116 Chapter 5. Coﬁbrations and Fibrations If we apply the pullback property to e0 E, p
I, we obtain a unique map r W EI! W.p/, v 7!.v.0/; pv/ such that br D e0 E and kr D pI. If we apply the HLP to.W.p/; b; k/, we obtain a map s W W.p/! EI such that e0 E s D b and pI s D k. The relations brs D e0 E s D b and krs D pI s D k imply rs D id, by uniqueness. Therefore s is a section of r. Conversely, given data.a; h/ for a homotopy lifting problem. They combine to a map W X! W.p/. The composition H D s with a section s is a solution of the lifting problem. Therefore we have shown: (5.5.1) Proposition. The following statements about p W E! B are equivalent: (1) p is a ﬁbration. (2) p has the HLP for W.p/. (3) r W EI! W.p/ has a section. (5.5.2) Proposition. Let p W E! B have the HLP for X. Let i W A X be a closed coﬁbration and an h-equivalence. Let f W X! B be given and a W A! E a lifting of f over A, i.e., pa D f i. Then there exists a lifting F of f which extends a. Proof. By (5.2.6) and (5.4.2) we know: There exists u W X! I and'W X I! X rel A such that A D u1.0/, '1 D id.X/, '0.X/ A. Set r W X! A, x 7! '0.x/. Deﬁne a new homotopy ˆ W X I! X by ˆ.x; t / D '.x; t u.x/1/ for t < u.x/ and ˆ.x; t/ D '.x; 1/ D x for t u.x/. We have seen in (5.4.3) that ˆ is continuous. Apply the HLP to h D f ˆ with initial condition b D ar W X! E. The veriﬁ
cation h.x; 0/ D f ˆ.x; 0/ D f '.x; 0/ D f r.x/ D par.x/ D pb.x/ shows that b is indeed an initial condition. Let H W X I! E solve the lifting problem for h; b. Then one veriﬁes that F W X! E, x 7! H.x; u.x// has the desired properties. (5.5.3) Corollary. Let p W E! B have the HLP for X I and let i W A X be a closed coﬁbration. Then each homotopy h W X I! B with initial condition given on A I [ X 0 has a lifting H W X I! E with this initial condition. Proof. This is a consequence of (5.1.3) and (5.5.2). (5.5.4) Proposition. Let i W A B be a (closed) coﬁbration of locally compact spaces. The restriction from B to A yields a ﬁbration p W ZB! ZA. Let p W X! B be a ﬁbration. Then pZ W X Z! B Z is a ﬁbration for locally compact Z. Proof. Use adjunction and the fact that X A! X B is a coﬁbration for each X. 5.5. The Homotopy Lifting Property 117 (5.5.5) Proposition. Let p W E! B be a ﬁbration. Then r W EI! W.p/, v 7!.v.0/; pv/ is a ﬁbration. Proof. A homotopy lifting problem for X and r is transformed via adjunction into a lifting problem for p and X I with initial condition given on the subspace X.I 0 [ 0 I /. (5.5.6) Proposition. Let p W E! B be a ﬁbration, B0 B and E0 D p1.B0/. If B0 B is a closed coﬁbration, then E0 E is a closed coﬁbration. Proof. Let u W B! I and h W B I! B be an NDR-presentation of B0 B. Let H W X I!
X solve the homotopy lifting problem for h.p id/ with initial condition id.X/. Deﬁne K W X! X by K.x; t/ D H.x; min.t; up.x///. Then.K; up/ is an NDR-presentation for X0 X. The proof of the next formal proposition is again left to the reader. (5.5.7) Proposition. Let a pullback in TOP be given. Y q C F f X p B If q has the HLP for Z, then so also has p. If p is a ﬁbration, then q is a ﬁbration. We call q the ﬁbration induced from the ﬁbration p via base change along f. In the case that f W C B the restriction p W p1.C /! C can be taken as the induced ﬁbration. (5.5.8) Example. X I! X @I Š X X W w 7!.w.0/; w.1// is a ﬁbration (5.5.4). The evaluation e1 W F Y! Y, w 7! w.1/ is a ﬁbration (restriction to Y ). Þ Hence we have the induced ﬁbration f 1 W F.f /! Y. The homotopy theorem for ﬁbrations says, among other things, that homotopic maps induce h-equivalent ﬁbrations. Let p W X! B be a ﬁbration and'W f'g W C! B a homotopy. We consider two pullback diagrams. Yf pf C F f X p B Yg pg C G g X p B There exists a homotopy ˆt W Yf! X such that ˆ0 D F and pˆt D 't pf. The pullback property of the right square yields a map D'W Yf! Yg such that G D ˆ1 and pg D pf. Let t W f'g be homotopic to 't by a homotopy 118 Chapter 5. Coﬁbrations and Fibrations W C I I! B relative to C @I. We obtain in a similar manner a map from a lifting ‰t
of t pf. Claim: The maps'and are homotopic over C. In order to verify this, we lift the homotopy ı.pf id id/ W Yf I I! B to a homotopy with initial data.y; s; 0/ D ˆ.y; s/,.y; s; 1/ D ‰.y; s/, and.y; 0; t / D F.t/ by an application of (5.1.5). The homotopy H W.y; t/ 7!.y; 1; t / yields, by the pullback property of the right square, a homotopy K W Yf I! Yg such that GK D H and pg K D pr ıpf. By construction, K is a homotopy over C from f to g. The reader should now verify the functoriality Œ' D Œ Œ'. Let h-FIBC be the full subcategory of h-TOPC with objects the ﬁbrations over C. (5.5.9) Proposition. Let p W X! B be a ﬁbration. We assign to f W C! B the induced ﬁbration pf W Yf! C and to the morphism Œ' W f! g in ….C; B/ the morphism Œ'. This yields a functor ….C; B/! h-FIBC. Since ….C; B/ is a groupoid, Œ' is always an isomorphism in h-TOPB. This fact we call the homotopy theorem for ﬁbrations. As a special case of (5.5.9) we obtain the ﬁbre transport. It generalizes the ﬁbre transport in coverings. Let p W E! B be a ﬁbration and w W I! B a path from b to c. We obtain a homotopy equivalence TpŒw W Fb! Fc which only depends on the homotopy class Œw of w, and TpŒu v D TpŒvTpŒu. This yields a functor Tp W ….B/! h-TOP. In particular the ﬁbres over points in the same path component of B are h-equ
ivalent. (5.5.10) Proposition. In the pullback (5.5.7) let p be a ﬁbration and f a homotopy equivalence. Then F is a homotopy equivalence. Proof. The proof is based on (5.5.9) and follows the pattern of (5.1.10). (5.5.11) Remark. The notion of ﬁbration and coﬁbration are not homotopy invariant. The projection I 0 [ 0 I! I,.x; t/ 7! x is not a ﬁbration, but the map is over I h-equivalent to id. One deﬁnition of an h-ﬁbration p W E! B is that homotopies X I! B which are constant on X Œ0; "; " > 0 can be lifted with a given initial condition; a similar deﬁnition for homotopy extensions gives the notion on an h-coﬁbration. In [46] you can ﬁnd details about these notions. Problems 1. A composition of ﬁbrations is a ﬁbration. A product of ﬁbrations is a ﬁbration. ;!B is a ﬁbration. 2. Suppose p W E! B has the HLP for Y I n. Then each homotopy h W Y I n I! B has a lifting to E with initial condition given on Y.I n 0 [ @I n I /. 3. Let p W E! B I be a ﬁbration and p0 W E0! B its restriction to B 0 D B. Then there exists a ﬁbrewise h-equivalence from p0 id.I / to p which is over B 0 the inclusion E0! E. 5.6. Transport 119 4. Go through the proof of (5.5.9) and verify a relative version. Let.C; D/ be a closed coﬁbration. Consider only maps C! B with a ﬁxed restriction d W D! B and homotopies relative to D. Let pD W YD! B be the pullback of p along d. Then the maps pf have the form.pf ; pD
/ W.Yf ; YD/!.C; D/. By (5.5.6),.Yf ; YD/ is a closed coﬁbration, and by (5.5.3) the homotopies ˆt can be chosen constant on YD. The maps'W Yf! Yg are then the identity on YD. The homotopy class of'is unique as a map over C and under YD. 5. Let.B; C / be a closed deformation retract with retraction r W B! C. Let p W X! B be a ﬁbration and pC W XC! C its restriction to C. Then there exists a retraction R W X! XC such that pC R D rp. 5.6 Transport We construct a dual transport functor. Let p W E! B be a ﬁbration,'W Y I! B a homotopy and ˆ W Y I! E a lifting along p with initial condition f. We deﬁne '# W Œ.Y; '0/;.E; p/B! Œ.Y; '1/;.E; p/B ; Œf 7! Œˆ1: One shows that this map is well deﬁned and depends only on the homotopy class of'relative to Y @I (see the analogous situation for coﬁbrations). Moreover,.' /# D #'#. (5.6.1) Proposition. The assignments f 7! Œf; pB and Œ' 7! '# are a functor, called transport functor, from ….Y; B/ into the category of sets. Since ….Y; B/ is a groupoid, '# is always bijective. (5.6.2) Note. Let p W E! B be a ﬁbration and W X I! Y be a homotopy. 0 W Œg; p! Œg 0; p is the composition with 0. Then 1 D Œg t # 0 ; here (5.6.3) Theorem. Let f W X! Y be an h-equivalence and p W E! B be a ﬁbration. Then f W Œv; p
B! Œvf; pB is bijective for each v W Y! B. Proof. The proof is based on (5.6.1) and (5.6.2) and formally similar to the proof of (5.2.4). (5.6.4) Theorem. Let p W X! B and q W Y! B be ﬁbrations. Let h W X! Y be an h-equivalence and a map over B. Then f is an h-equivalence over B. Proof. The proof is based on (5.6.3) and formally similar to the proof of (5.2.5). (5.6.5) Corollary. Let q W Y! C be a ﬁbration and a homotopy equivalence. Then q is shrinkable. Let p W E! B be a ﬁbration. Then the canonical map r W EI! W.p/ is Þ shrinkable (see (5.5.5)). 120 Chapter 5. Coﬁbrations and Fibrations 5.7 Replacing a Map by a Fibration Let f W X! Y be a map. Consider the pullback W.f / Y I.q;p/.e0;e1/ X Y Y Y f id W.f / D f.x; w/ 2 X Y I j f.x/ D w.0/g; q.x; w/ D x; p.x; w/ D w.1/: Since.e0; e1/ is a ﬁbration (see (5.5.8)), the maps.q; p/, q and p are ﬁbrations. Let s W X! W.f /, x 7!.x; kf.x//, with ky the constant path with value y. Then qs D id and ps D f. (The “direction” of the unit interval is again different from the one in the previous chapter.) We display the data and some other to be explained below in a diagramf / W.f / p J D Y Y The map s is a shrinking of q; a homotopy ht W sq'id is given by ht.x; w/ D.x; wt /, wt.s/ D w..1
t/s/. We therefore have a factorization f D ps into a homotopy equivalence s and a ﬁbration p. If f D p0s0 is another factorization of this type, then there exists a ﬁbrewise homotopy equivalence k W W.f /! W 0 such that p0k D p and ks's0. This expresses the uniqueness of the factorization. Now suppose f is a pointed map with base points. Then W.f / is given the base point.; k/. The maps p; q; s become pointed maps, and the homotopy ht is pointed too. One veriﬁes that q and p are pointed ﬁbrations. Let F.f / D p1./ and F D f 1./ be the ﬁbres over the base point, with j and J the inclusions. The map q induces r. We call F.f / the homotopy ﬁbre of f. We use the same notion for the ﬁbre of any replacement of f by a ﬁbration as above. If f is already a ﬁbration, then q is a ﬁbrewise homotopy equivalence (5.6.4) and r the induced homotopy equivalence; hence the actual ﬁbre is also the homotopy ﬁbre. A map f W X! Y has a right homotopy inverse if and only if p W W.f /! X has a section. It is a homotopy equivalence if and only if p is shrinkable. Chapter 6 Homotopy Groups The ﬁrst fundamental theorem of algebraic topology is the Brouwer–Hopf degree theorem. It says that the homotopy set ŒS n; S n has for n 1 a homotopically deﬁned ring structure. The ring is isomorphic to Z, the identity map corresponds to 1 2 Z and the constant map to 0 2 Z. The integer associated to a map f W S n! S n is called the degree of f. We have proved this already for n D 1. Also in the general case “degree n” roughly means that f winds S n n-times around itself. In order to give precision to this statement, one has to count the number of pre-images of a �
�regular” value with signs. This is related to a geometric interpretation of the degree in terms of differential topology. Our homotopical proof of the degree theorem is embedded into a more general investigation of homotopy groups. It will be a simple formal consequence of the so-called excision theorem of Blakers and Massey. The elegant elementary proof of this theorem is due to Dieter Puppe. It uses only elementary concepts of homotopy theory, it does not even use the group structure. (The excision isomorphism is the basic property of the homology groups introduced later where it holds without any restrictions on the dimensions.) Another consequence of the excision theorem is the famous suspension theorem of Freudenthal. There is a simple geometric construction (the suspension) which leads from ŒS m; S n to ŒS mC1; S nC1. Freudenthal’s theorem says that this process after a while is “stable”, i.e., induces a bijection of homotopy sets. This is the origin of the so-called stable homotopy theory – a theory which has developed into a highly technical mathematical ﬁeld of independent interest and where homotopy theory has better formal and algebraic properties. (Homology theory belongs to stable homotopy.) The degree theorem contains the weaker statement that the identity of S n is not null homotopic. It has the following interpretation: If you extend the inclusion S n1 Rn to a continuous map f W Dn! Rn, then there exists a point x with f.x/ D 0. For n D 1 this is the intermediate value theorem of calculus; the higher dimensional analogue has other interesting consequences which we discuss under the heading of the Brouwer ﬁxed point theorem. This chapter contains the fundamental non-formal results of homotopy theory. Based on these results, one can develop algebraic topology from the view-point of homotopy theory. The chapter is essentially independent of the three previous chapters. But in the last section we refer to the deﬁnition of a coﬁbration and a suspension. 122 Chapter 6. Homotopy Groups 6.1 The Exact Sequence of Homotopy Groups Let I n be the Cartesian product of n copies of the unit interval I D Œ0; 1, and @I n D f.t1; : : : ; t
n/ 2 I n j ti 2 f0; 1g for at least one ig its combinatorial boundary (n 1). We set I 0 D fzg, a singleton, and @I 0 D ;. In I n=@I n we use @I n as base point. (For n D 0 this yields I 0=@I 0 D fzg C fg, an additional disjoint base point.) The n-th homotopy group of a pointed space.X; / is n.X; / D Œ.I n; @I n/;.X; fg/ Š ŒI n=@I n; X0 with the group structure deﬁned below. For n D 1 it is the fundamental group. The deﬁnition of the set n.X; / also makes sense for n D 0, and it can be identiﬁed with the set 0.X/ of path components of X with Œ as a base point. The composition law on n.X; / for n 1 is deﬁned as follows. Suppose Œf and Œg in n.X; / are given. Then Œf C Œg is represented by f Ci g: (.1/.f Ci g/.t1; : : : ; tn/ D f.t1; : : : ; ti1; 2ti ; : : : ; tn/ g.t1; : : : ; ti1; 2ti 1; : : : ; tn/ for ti 1 2 ; for 1 2 ti : As in the case of the fundamental group one shows that this composition law is a group structure. The next result is a consequence of (4.3.1); a direct veriﬁcation along the same lines is easy. See also (2.7.3) and the isomorphism (2) below. (6.1.1) Proposition. For n 2 the group n.X; / is abelian, and the equality C1 D Ci holds for i 2. We now deﬁne relative homotopy groups (sets) k.X; A; / for a pointed pair.X; A/. For n 1, let J n D @I n I [ I n f0g @I nC1 I n I and set J 0
D f0g I. We denote by nC1.X; A; / the set of homotopy classes of maps of triples f W.I nC1; @I nC1; J n/!.X; A; /. (Recall that this means f.@I nC1/ A; f.J n/ fg, and for homotopies H we require Ht for each t 2 I to be a map of triples.) Thus, with notation introduced earlier, nC1.X; A; / D Œ.I nC1; @I nC1; J n/;.X; A; /: A group structure Ci, 1 i n is deﬁned again by the formula (1) above. There is no group structure in the case n D 0. We now consider n as a functor. Composition with f W.X; A; /!.Y; B; / induces f W n.X; A; /! n.Y; B; /; this is a homomorphism for n 2. Similarly, f W.X; /!.Y; / induces for n 1 a homomorphism f W n.X; /! n.Y; /. The functor properties.gf / D gf and id D id are clear. The morphism j W n.X; /! n.X; A; / is obtained by interpreting the ﬁrst group as n.X; fg; / and then using the map induced by the inclusion.X; fg; /.X; A; /. Maps which are pointed homotopic induce the same homomorphisms. The group n.X; A; / is commutative for n 3. 6.1. The Exact Sequence of Homotopy Groups 123 Let h W.I nC1; @I nC1; J n/!.X; A; / be given. We restrict to I n D I n f1g and obtain a map @h W.I n; @I n/!.A; /. Passage to homotopy classes then yields the boundary operator @ W nC1.X; A; /! n.A; /. The boundary operator is a homomorphism for n 1. For n D 0 we have @Œh D Œh.
1/. We rewrite the homotopy groups in terms of mapping spaces. This is not strictly necessary for the following investigations but sometimes technically convenient. Let k.X; / be the space of maps I k! X which send @I k to the base point; the constant map is the base point. The space 1.X/ D.X/ is the loop space of X. Given a map.I n; @I n/!.X; / we have the induced map xf W I nk! k.X; / which sends u 2 I nk to I k! X,.t1; : : : ; tk/ 7! f.t1; : : : ; tk; u1; : : : ; unk/. This adjunction is compatible with homotopies and induces a bijection.2/ n.X; / Š nk.k.X; /; /: Adjunction as above also yields a bijection.3/ nC1.X; A; / Š nC1k.k.X/;.A/k; /: These isomorphisms are natural in.X; A; /, compatible with the boundary operators, and the group structures. (6.1.2) Theorem (Exact homotopy sequence). The sequence! n.A; / i! n.X; / j! n.X; A; / @!! 1.X; A; / @! 0.A; / i! 0.X; / is exact. The maps i and j are induced by the inclusions. Proof. We prove the exactness for the portion involving 0 and 1 in an elementary manner. Exactness at 0.A; / and the relations @j D 0 and ji D 0 are left to the reader. Let w W I! X represent an element in 1.X; A; / with @Œw D 0. This means: There exists a path u W I! A with u.0/ D w.1/ and u.1/ D. The product w u is then a loop in X. The homotopy H which is deﬁned by w ( u Ht.s/ D w.2s=.1 C t//; u.t C 2.1 s//; 2s 1 C t; 2s 1 C t;.................... H
t w u shows jŒw u D Œw. Thus we have shown exactness at 1.X; A; /. 124 Chapter 6. Homotopy Groups Given a loop w W I! X. Let H W.I; @I; 0/ I!.X; A; / be a homotopy from w to a constant path. Then u W s 7! H.1; s/ is a loop in A. We restrict H to the boundary of the square and compose it with a linear homotopy to prove k w'k u. w H u k w'k u; u.t / D H.1; t/:........................................ Hence iŒu D iŒk u D Œk w D Œw. We now apply this part of the exact sequence to the pairs.n.X/; n.A// and obtain the other pieces of the sequence via adjunction. (6.1.3) Remark. We previously introduced the mapping space F./ D f.a; w/ 2 A X I j w.0/ D ; w.1/ D ag; with base point.; k/, k W I! fg the constant path. This space is homeomorphic to F.X; A/ D fw 2 X I j w.0/ D ; w.1/ 2 Ag; i./ becomes the inclusion.X/ F.X; A/, and 1 the evaluation F.X; A/! A, w 7! w.1/. For n 1 we assign to f W.I nC1; @I nC1; J n/!.X; A; / the adjoint map f ^ W I n! F.X; A/, deﬁned by f ^.t1; : : : ; tn/.t/ D f.t1; : : : ; tn; t/. It sends @I n to the base point and induces a pointed map xf W I n=@I n! F.X; A/. By standard properties of adjunction we see that the assignment Œf 7! Œ xf is a well-deﬁned bijection.4/ nC1.X; A; / Š n.F.X; A/; /; and in fact a homomorphism with respect to the composition laws Ci for 1
i n. These considerations also make sense for n D 0. In the case that A D fg, the space F.X; A/ is the loop space.X/. The exact sequence is also obtained from the ﬁbre sequence of. Under the identiﬁcations (4) the boundary operator is transformed into W ŒI n=@I n; F.X; A/0! ŒI n=@I n; A0; 1 and nC1.X; /! nC1.X; A; / is transformed into i./ W ŒI n=@I n;.X/0! ŒI n=@I n; F.X; A;/ 0: Now apply B D I n=@I n to the ﬁbre sequence (4.7.4) of W A X to see the Þ exactness of a typical portion of the homotopy sequence. 6.1. The Exact Sequence of Homotopy Groups 125 The sequence (6.1.2) is compatible with maps f W.X; A; /!.Y; B; /. In particular f@ D @f. (6.1.4) Remark. In the sequel it will be useful to have different interpretations for elements in homotopy groups. (See also the discussion in Section 2.3.) We set S.n/ D I n=@I n and D.n C 1/ D CS.n/, the pointed cone on S.n/. We have homeomorphisms S.n/! @I nC1=J n; D.n C 1/! I nC1=J n; the ﬁrst one x 7!.x; 1/, the second one the identity on representatives in I n I ; moreover we have the embedding S.n/! D.nC1/, x 7!.x; 1/ which we consider as an inclusion. These homeomorphisms allow us to write n.X; / Š ŒI n=@I n; X0 D ŒS.n/; X0; nC1.X; A; / Š Œ.I nC1=J n; @I nC1=J n/;.X; A/0 Š Œ.D.nC1/; S.n//
;.X; A/0; and @ W nC1.X; A; /! n.A; / is induced by the restriction from D.n C 1/ to S.n/. The pointed cone on S n is DnC1: We have a homeomorphism S n I =.S n 0 [ enC1 I /! DnC1;.x; t/ 7!.1 t/enC1 C tx: Therefore we can also represent elements in nC1.X; A; / by pointed maps.DnC1; S n/!.X; A/ and elements in n.X; / by pointed maps S n! X. In comparing these different models for the homotopy groups it is important to remember the homeomorphism between the standard objects (disks and spheres), Þ since there are two homotopy classes of homeomorphisms. S Problems 1. n.A; A; a/ D 0. Given f W.I n; @I n; J n1/!.A; A; a/. Then a null homotopy is ft.x1; : : : ; xn/ D f.x1; x2; : : : ;.1 t/xn/. 2. Let 2X 0 X1 X2 be a sequence of T1-spaces (i.e., points are closed). Give X D n1 Xn the colimit topology. Then a compact subset K X is contained in some Xn. Use this to show that the canonical maps n.Xi ; /! n.X; / induce an isomorphism colimi n.Xi ; / Š n.X; /. 3. Let.X; A; B; b/ be a pointed triple. Deﬁne the boundary operator @ W n.X; A; b/! n1.A; b/! n1.A; B; b/ as the composition of the previously deﬁned operator with the map induced by the inclusion. Show that the sequence! n.A; B; b/! n.X; B; b/! n.X; A; b/ @! n1.A; B; b/! is exact. The sequence ends with 1.X; A; b/. 4. The group structure in nC1.X; A; / is
induced by an h-cogroup structure on.D.n C 1/; S.n// in the category of pointed pairs. 126 Chapter 6. Homotopy Groups 5. Let f W.X; x/!.Y; y/ be a pointed map. One can embed the induced morphism f W n.X; x/! n.Y; y/ into an exact sequence which generalizes the case of an inclusion f. Let Z.f / be the pointed mapping cylinder of f and f D pi W X! Z.f /! Y the standard factorization into an inclusion and a homotopy equivalence, as explained in (5.3.1). We can now insert the isomorphism p W n.Z.f /; /! n.Y; / into the exact sequence of the pair and obtain an exact sequence! n.X; / f! n.Y; /! n.Z.f /; X/; /! : One can deﬁne the groups n.Z.f /; X; / without using the mapping cylinder. Consider commutative diagrams with pointed maps'and ˆ. @I n=J n1'X \j I n=J n1 f ˆ Y We consider.'; ˆ/ W j! f as a morphism in the category of pointed arrows. Let n.f / denote the set of homotopy classes of such morphisms. For f W X Y we obtain the previously deﬁned n.Y; X; /. The projection p W i! f induces an isomorphism n.Z.f /; X; / D n.i/! n.f /. One can also use the ﬁbre sequence of f. 6.2 The Role of the Base Point We have to discuss the role of the base point. This uses the transport along paths. Let a path v W I! X and f W.I n; @I n/!.X; v.0// be given. We consider v as a homotopy Ov of the constant map @I n! fv.0/g. We extend the homotopy vt to a homotopy Vt W I n! X with initial condition f D V0. An extension exists because @I n I n is a coﬁbration. The next proposition is a special case
of (5.2.1) and problems in that section. In order to be independent of that section, we also repeat a proof in the present context. (6.2.1) Proposition. The assignment ŒV0 7! ŒV1 is a well-deﬁned map v# W n.X; v.0//! n.X; v.1// which only depends on the morphism Œv in the fundamental groupoid ….X/. The relation.vw/# D w# ıv# holds, and thus we obtain a transport functor from ….X/ which assigns to x0 2 X the group n.X; x0/ and to a path v the morphism v#. The map v# is a homomorphism. Proof. Let'W f'g be a homotopy of maps.I n; @I n/!.X; x0/ and W v'w a homotopy of paths from x0 to x1. Let Vt W I n! X be a homotopy which extends.f; Ov/ and Wt a homotopy which extends.g; Ow/. These data combine to a map on T D I n 0 I [ @I n I I [ I n I @I I nC2 as follows: On I n 0 I 6.2. The Role of the Base Point 127 we use ', on @I n I I we use O, on I n I 0 we use V, and on I n I 1 we use W. If we interchange the last two coordinates then T is transformed into J nC1. Therefore our map has an extension to I nC2, and its restriction to I n 1 I is a homotopy from V1 to W1. This shows the independence of the representatives f and v. The other properties are clear from the construction. There is a similar transport functor in the relative case. We start with a function f W.I n; @I n; J n1/!.X; A; a0/ and a path v W I! A from a0 to a1. We consider the path as a homotopy of the constant map J n1! fa0g. Then we extend this homotopy to a homotopy Vt W.I n; @I n/!.X; A/. An extension exists because J n1
@I n and @I n I n are coﬁbrations. (6.2.2) Proposition. The assignment ŒV0 7! ŒV1 is a well-deﬁned map v# W n.X; A; a0/! n.X; A; a1/ which only depends on the morphism Œv in the fundamental groupoid ….A/. For n 2 the map v# is a homomorphism. As above we have a transport functor from ….A/. Since v# is always bijective, homotopy groups associated to base points in the same path component are isomorphic. We list some naturality properties of the transport functors. As a special case of the functor property we obtain right actions of the fundamental groups: n.X; x/ 1.X; x/! n.X; x/; n.X; A; a/ 1.A; a/! n.X; A; a/;.˛; ˇ/ 7! ˛ ˇ D ˇ#.˛/;.˛; ˇ/ 7! ˛ ˇ D ˇ#.˛/: We also have an action of 1.A; a/ on n.X; a/ via the natural homomorphism; more generally, we can make the n.X; a/ into a functor on ….A/ by viewing a path in A as a path in X. From the constructions we see: (6.2.3) Proposition. The morphisms in the exact homotopy sequence of the pair.X; A/ are natural transformations of transport functors on ….A/. In particular, they are 1.A; /-equivariant with respect to the actions above. Continuous maps f W.X; A/!.Y; B/ are compatible with the transport func- tors f.w#.˛// D.f w/#.f.˛//: Let ft W.X; A/!.Y; B/ be a homotopy and set w W t 7! f.a; t/. Then the diagram n.X; A; a/ f0 f1 n.Y; B; f0a/ w# n.Y; B; f1a/ is commut
ative. As in the proof of (2.5.5) one uses this fact to show: 128 Chapter 6. Homotopy Groups (6.2.4) Proposition. Let f W.X; A/!.Y; B/ be an h-equivalence. Then the induced map f W n.X; A; a/! n.Y; B; f a/ is bijective. Suppose that f that induces isomorphisms j.A/! j.B/ and j.X/! j.Y / for j 2 fn; n C 1g, n 1. Then the Five Lemma (11.1.4) implies that f W nC1.X; A; /! nC1.Y; B; / is an isomorphism. With some care, this also holds for n D 0, see Problem 3. Let f W.X; A/!.Y; B/ be a map of pairs such that the individual maps X! Y and A! B induce for each base point in A isomorphism for all n, then f W n.X; A; a/! n.Y; B; f.a// is bijective for each n 1 and each a 2 A. For the case n D 1 see Problem 3. The transport functors have special properties in low dimensions. (6.2.5) Proposition. Let v W I! X be given. Then v# W 1.X; v.0//! 1.X; v.1// is the map Œw 7! ŒvŒwŒv. In particular, the right action of 1.X; x/ on itself is given by conjugation ˛ ˇ D ˇ1˛ˇ. (6.2.6) Proposition. Let x1; x2 2 2.X; A; / be given. Let z D @x2 2 1.A; /. Then x1 z D.x2/1x1x2 (multiplicative notation for 2). Proof. We ﬁrst prove the claim in a universal situation and then transport it by naturality to the general case. Set D D D.2/; S D S.1/. Let 1; 2 2 2.D _ D; S _ S/ be the elements represented by the inclusions of the summands
.D; S/!.D _ D; S _ S/. Set D @.2/ 2 1.S _ S/. From (6.2.3) and (6.2.5) we compute @.1 / D.@1/ D 1.@1/ D.@2/1.@1/.@2/ D @.1 2 12/: Since D _ D is contractible, @ is an isomorphism, hence 1 2 D 1 2 12. Let now h W.D _ D; S _ S/!.X; A/ be a map such that hik represents xk, i.e., h.k/ D xk. The computation x1 z D.h1/.@h2/ D h.1 / D h.1 2 12/ D x1 2 x1x2 proves the assertion in the general case. (6.2.7) Corollary. The image of the natural map 2.X; /! 2.X; A; / is con- tained in the center. The actions of the fundamental group also explain the difference between pointed and free (D unpointed) homotopy classes. (6.2.8) Proposition. Let ŒS.n/; X0=./ denote the orbit set of the 1.X; /-action on ŒS.n/; X0. The map ŒS.n/; X0! ŒS.n/; X which forgets the base point induces an injective map v W ŒS.n/; X0=./! ŒS.n/; X. For path connected X the map v is bijective. The forgetful map n.X; A; / D Œ.D.n/; S.n 1//;.X; A/0! Œ.D.n/; S.n 1//;.X; A/ 6.3. Serre Fibrations 129 induces an injective map of the orbits of the 1.A; /-action; this map is bijective if A is path connected.n 2/. Problems 1. Let A be path connected. Each element of 1.X; A; a/ is represented by a loop in.X; a/. The map j W 1.X; a/! 1.X; A; a/ induces
a bijection of 1.X; A; a/ with the right (or left) cosets of 1.X; a/ modulo the image of i W 1.A; a/! 1.X; a/. 2. Let x 2 1.X; A; a/ be represented by v W I! X with v.1/ 2 A and v.0/ D a. Let w W I! X be a loop in.X; a/. The assignment.Œw; Œv/ 7! Œw v D Œw Œv deﬁnes a left action of the group 1.X; a/ on the set 1.X; A; a/. The orbits of this action are the pre-images of elements under @ W 1.X; A; a/! 0.A; a/. Let.F; f / W.X; A/!.Y; B/ be a map of pairs. Then F W 1.X; A; a/! 1.Y; B; f.a// is equivariant with respect to the homomorphism F W 1.X; a/! 1.Y; f.a//. Let Œv 2 1.X; A; a/ with v.1/ D u 2 A. The isotropy group of Œv is the image of 1.A; u/ in 1.X; a/ with respect to Œw 7! Œv w v. Find an example ˛0; ˛1 2 1.X; A; a/ such that ˛0 has trivial and ˛1 non-trivial isotropy group. It is in general impossible to deﬁne a group structure on 1.X; A; a/ such that 1.X; a/! 1.X; A; a/ becomes a homomorphism. 3. Although there is only a restricted algebraic structure at the beginning of the exact sequence we still have a Five Lemma type result. Let f W.X; A/!.Y; B/ be a map of pairs. If f W 0.A/! 0.B/ and f W 1.X; a/! 1.Y; f.a// are surjective and f W 0.X/! 0.Y / is injective, then f W 1.X
; A; a/! 1.Y; B; f.a// is surjective. Suppose that for each c 2 A the maps f W 1.X; c/! 1.Y; f.c// and f W 0.A/! 0.B/ are injective and f W 1.A; c/! 1.B; f.c// is surjective, then f W 1.X; A; a/! 1.Y; B; f.a// is injective for each a 2 A. 4. Let.X; A/ be a pair such that X is contractible. Then @ W qC1.X; A; a/! q.A; a/ is for each q 0 and each a 2 A a bijection. 5. Let A X be an h-equivalence. Then n.X; A; a/ D 0 for n 1 and a 2 A. 6. Let X carry the structure of an h-monoid. Then 1.X/ is abelian and the action of the fundamental group on n.X; / is trivial. 7. Give a proof of (6.2.8). 8. The 1.X; /-action on n.X; / is induced by a map n W S.n/! S.n/ _ S.1/ by an application of the functor Œ; X0. If we use the model Dn=S n1 for the n-sphere, then an explicit map n is x 7!.2x; / for 2kxk 1 and x 7!.; 2kxk 1/ for 2kxk 1. 6.3 Serre Fibrations The notion of a Serre ﬁbration is adapted to the investigation of homotopy groups, only the homotopy lifting property for cubes is used. (6.3.1) Theorem. Let p W E! B be a Serre ﬁbration. For B0 B set E0 D p1B0. Choose base points 2B 0 and 2E 0 with p./ D. Then p induces for n 1 a bijection p W n.E; E0; /! n.B; B0; /. 130 Chapter 6. Homotopy Groups Proof. p surjective. Let x 2 n.B; B0; / be
represented by h W.I n; @I n; J n1/!.B; B0; /: By (3.2.4), there exists a lifting H W I n! E with H.J n1/ D fg and pH D h. We then have H.@I n/ E0, and therefore H represents a pre-image of x under p. p injective. Let x0; x1 2 n.E; E0; / be represented by f0; f1 and have the same image under p. Then there exists a homotopy t W.I n; @I n; J n1/!.B; B0; / such that 0.u/ D pf0.u/, 1.u/ D pf1.u/ for u 2 I n. Consider the subspace T D I n @I [ J n1 I and deﬁne G W T! E by ( G.u; t/ D ft.u/; u 2 I n; t 2 f0; 1g; ; u 2 J n1; t 2 I: The set T @.I n I / is transformed into J n, if one interchanges the last two coordinates. By (3.2.4) again, there exists a map H W I n I! E such that H jT D G and pH D. We can view H as a homotopy from f0 to f1. We use the isomorphism n.E; F; / Š n.B; /, F D p1./ in the exact sequence of the pair.E; F; / and obtain as a corollary to (6.3.1) the exact sequence of a Serre ﬁbration: (6.3.2) Theorem. For a Serre ﬁbration p W E! B with inclusion i W F D p1.b/ E and x 2 F the sequence! n.F; x/ i n.E; x/ p n.B; b/ @ n1.F; x/! is exact. The sequence ends with 0.E; x/! 0.B; b/. The new map @ has the following description: Let f W.I n; @I n/!.B; b/ be given. View f as I n1 I! B.
Lift to W I n! E, constant on J n1. Then @Œf is represented by jI n1 1. The very end of the sequence requires a little extra argument. For additional algebraic structure at the beginning of the sequence see the discussion of the special case in (3.2.7). (6.3.3) Theorem. Let p W E! B be a continuous map and U a set of subsets such that the interiors cover B. Assume that for U 2 U the map pU W p1.U /! U induced by p is a Serre ﬁbration. Then p is a Serre ﬁbration. Q Proof. A subdivision of width ı D 1=N, N 2 N of I n consists of the cubes n I.a1; : : : ; an/ D j D1 I.aj / where I.k/ D Œk=N;.k C 1/=N for 0 k < N, k 2 Z. A k-dimensional face of I.a1; : : : ; an/ is obtained by replacing n k of the intervals I.aj / by one of its boundary points. (The aj are integers, 0 aj < n.) It sufﬁces to work with an open covering U. Choose N such that each cube I.a1; : : : ; an/ I.b/ is mapped under h into some U 2 U. This is possible by 6.3. Serre Fibrations 131 the Lebesgue lemma (2.6.4). Let V k I n denote the union of the k-dimensional faces of the subdivision of I n. We have to solve a lifting problem for the space I n with initial condition a. We begin by extending a over I n Œ0; ı to a lifting of h. We solve the lifting problems I n 0 [ V k1 Œ0; ı H.k1/ H.k/ \ I n 0 [ V k Œ0; ı h E p B for k D 0; : : : ; n with V 1 D ; and H.1/ D a by induction over k. Let W be a k-dimensional cube and @W the union of its.k 1/-dimensional faces. We can solve the lifting problems W 0 [ @W Œ0; ı H.
k1/ HW p1U pU U h \ W Œ0; ı by a map HW, since pU is a Serre ﬁbration; here U 2 U was chosen such that h.W Œ0; ı/ U. The HW combine to a continuous map H.k/ W V k Œ0; ı! E which lifts h and extends H.k 1/. We deﬁne H on the ﬁrst layer I n Œ0; ı as H.n/. We now treat I n Œı; 2ı similarly with initial condition given by H.n/jI n fıg and continue in this manner inductively. (6.3.4) Example. Since a product projection is a ﬁbration we obtain from (6.3.3): Þ A locally trivial map is a Serre ﬁbration. (6.3.5) Example. Let p W E! B be a covering with typical ﬁbre F. Since each map I n! F is constant, n.F; / is for n 1 the trivial group. The exact sequence of p then shows p W n.E/ Š n.B/ for n 2. The covering R! S 1 then yields n.S 1/ Š 0 for n 2. Moreover we have the exact sequence 1! 1.E; / p! 1.B; / @! 0.F; / i! 0.E; / p! 0.B; /! 1 with the inclusion i W F D p1./ E and 0.F; / D F. It yields for p W R! S 1 the bijection @ W 1.S 1/ Š Z. A lifting of the loop sn W I! S 1, t 7! exp.2 i nt/ with initial condition 0 is t 7! nt. Hence @Œsn D n. Thus we Þ have another method for the computation of 1.S 1/. 132 Chapter 6. Homotopy Groups (6.3.6) Example. Recall the Hopf ﬁbration p W S 2nC1! CP n (14.1.9). The exact sequence (6.3.2) and i.S 1/ D 0 for i > 1 yield the isomorphisms p W i.S 2nC
1/ Š i.CP n/; for i 3I and in particular i.S 3/ Š i.S 2/ for i 3, since CP 1 is homeomorphic to S 2 Þ (the Riemann sphere). (6.3.7) Example. From linear algebra one knows a surjective homomorphism SU.2/! SO.3/ with kernel f˙Eg ŠZ=2. The space SU.2/ is homeomorphic to S 3. Hence SO.3/ is homeomorphic to RP 3 and 1.SO.3// Š Z=2. (6.3.8) Proposition. Let p W.E1; E0/! B be a relative Serre ﬁbration, i.e., p W E1! B is a Serre ﬁbration and the restriction of p to E0 is also a Serre ﬁbration. Let.F b (1) The inclusion induces bijections n.F b (2) 0.E0/! 0.E1/ is surjective if and only if 0.F b 0 / be the pair of ﬁbres over p.e/ D b 2 B. Then: 1 ; F b 0 ; e/ Š n.E1; E0; e/. 1 ; F b 0 /! 0.F b 1 / is surjective for each b 2 B. Proof. (1) We ﬁrst prove the claim for n D 1 and begin with the surjectivity. Let f W.I; @I; 0/!.E1; E0; e/ be given. The path.pf / W I! B is lifted to g W I! E0 with initial point f.1/. Then g.1/ 2 F0, and f and f g represent the same element in 1.E1; E0; e/. The projection p.f g/ is a null homotopic loop with base point b. We lift a null homotopy to E1 with initial condition f g on I 0 and constant on @I I. The lifting is a homotopy.I; @I; 0/ I!.E1; E0; e/ from f g to a map into.F1; F0; e/. This proves the surjectivity. Suppose f0
; f1 W.I; @I; 0/!.F1; F0; e/ are given, and let K W.I; @I; 0/I!.E1; E0; e/ be a homotopy from f0 to f1. We lift pK to L W I I! E0 with initial condition L.s; 0/ D K.s; 1/ and L.0; t/ D L.1; t/ D e. The homotopy p.K 2 L/ is a homotopy of loops which is relative to @I 2 homotopic to the constant map. We lift a homotopy to E1 with initial condition K 2 L on I 2 0 and constant on @I 2 I. The end is a homotopy from f0 ke to f1 ke. This proves the injectivity. The higher dimensional case is obtained by an application to the relative Serre ﬁbration.nF1; nF0/!.nE1; nE0/! B. (2) Suppose 0.E0/! 0.E1/ is surjective. The argument above for the 1 /. The other surjectivity is used to show the surjectivity of 0.F b implication is easy. 0 /! 0.F b Problems 1. The 2-fold covering S n! RP n yields for n 2 the isomorphism 1.RP n/ Š Z=2. 2. Prove directly the exactness of the sequence (6.3.2) without using (6.1.2). 3. The map C! C, z 7! z2 has the HLP for I 0 but not for I 1. 4. Let p W.E; e/!.B; b/ be a Serre ﬁbration with ﬁbre F D p1.b/. Then n.p/ W n.E; e/! n.B; b/ 6.4. The Excision Theorem 133 is a Serre ﬁbration with ﬁbre n.F; e/. 6.4 The Excision Theorem A basic result about homotopy groups is the excision theorem of Blakers and Massey [22]. (6.4.1) Theorem (Blakers–Massey). Let Y be the union of open subspaces
Y1 and Y2 with non-empty intersection Y0 D Y1 \ Y2. Suppose that i.Y1; Y0; / D 0 for 0 < i < p; p 1 i.Y2; Y0; / D 0 for 0 < i < q; q 1 for each base point 2Y 0. Then the excision map, induced by the inclusion, W n.Y2; Y0; /! n.Y; Y1; / is surjective for 1 n p C q 2 and bijective for 1 n < p C q 2 ( for each choice of the base point 2Y 0/. In the case that p D 1, there is no condition on i.Y1; Y0; ). We defer the proof of this theorem for a while and begin with some applications and examples. We state a special case which has a somewhat simpler proof and already interesting applications. It is also a special case of (6.7.9). (6.4.2) Proposition. Let Y be the union of open subspaces Y1 and Y2 with nonempty intersection Y0. Suppose.Y2; Y0/ D 0 is q-connected. Then.Y; Y1/ is q-connected. We apply the excision theorem (6.4.1) to the homotopy group of spheres. We use the following subspaces of S n, n 0, Dn ˙ D f.x1; : : : ; xnC1/ 2 S n j ˙xnC1 0g H n ˙ D fx 2 S n j x 6D enC1g: We use Rn RnC1,.z1; : : : ; zn/ 7!.z1; : : : ; zn; 0/ and similar inclusions for subsets of Rn. We choose D e1 as a base point; ei is the standard unit vector. (6.4.3) Lemma. We have isomorphisms @ W iC1.DnC1 i 0; n 0 and i.S n; /! i.S n; Dn ˙; / for i 0; n 1. ; S n; /! i.S n; / for Proof. In the ﬁrst case we use the exact sequence of the pair.DnC1 space Dn
C1 is contractible and hence i.DnC1 ; / D 0 for i 0 and n 0. ; S n/. The In the second case we consider similarly the exact sequence of.S n; Dn ˙/. Note that D e1 2 Dn ˙ for n 1. 134 Chapter 6. Homotopy Groups For n 0 we have a diagram with the isomorphisms (6.4.3) i.S n; / Š @ E iC1.S nC1; / Š iC1.DnC1 ; S n; / iC1.S nC1; DnC1 C ; /: The morphism is induced by the inclusion and E is deﬁned so as to make the diagram commutative. Note that the inductive proof of (1) in the next theorem only uses (6.4.2). (6.4.4) Theorem..1/ i.S n/ D 0 for i < n..2/ The homomorphism is an isomorphism for i 2n 2 and an epimorphism for i D 2n 1. A similar statement holds for E. Proof. Let N.n/ be the statement (1) and E.n/ the statement (2). Obviously N.1/ holds. Assume N.n/ holds. We then deduce E.n/. We apply the excision theorem to.Y; Y1; Y2; Y0/ D.S nC1; DnC1 ; S n/. By N.n/ and (6.4.3) we have i.S n/ Š iC1.DnC1 ˙ ; S n/ D 0 for 0 i < n. We use the excision theorem for p D q D n C 1 and see that is surjective for i C 1 2n and bijective for i C 1 2n 1. Finally, E.n/ and N.n/ imply N.n C 1/. C ; DnC1 In order to have the correct hypotheses for the excision theorem, we thicken ˙ and ˙ and note that the inclusions Dn˙ H n the spaces, replace Dn S n1 H n ˙ by H n are h-equivalences. C \ H n (6.4.
5) Proposition. The homomorphism i.DnC1 =S n; / induced by the quotient map is an isomorphism for i 2n 1 and an epimorphism for i D 2n. ; S n; /! i.DnC1 Proof. Consider the commutative diagram i.DnC1 ; S n; / i.DnC1 =S n; /.1/ i.S nC1; DnC1 C ; /.2/ i.S nC1=DnC1 C ; /: The map (1) is induced by a homeomorphism and the map (2) by a homotopy equivalence, hence both are isomorphisms. Now apply (6.4.4). The homomorphism E is essentially the suspension homomorphism. In order to see this, let us work with (6.1.4). The suspension homomorphism † is the composition † W n.X; / @ Š nC1.CX; X; / q nC1.CX=X; / D nC1.†X; / 6.5. The Degree 135 with the quotient map q W D.n C 1/! D.n C 1/=S.n/ D S.n C 1/. The next result is the famous suspension theorem of Freudenthal ([66]). (6.4.6) Theorem. The suspension † W i.S.n//! iC1.S.n C 1// is an isomorphism for i 2n 2 and an epimorphism for i D 2n 1. Proof. We have to show that q W iC1.CX; X/! iC1.CX=X/ is for X D S.n/ an isomorphism (epimorphism) in the appropriate range. This follows from (6.4.5); one has to use that S n is homeomorphic to S.n/ and that DnC1 is the (pointed) cone on S n. (6.4.7) Theorem. n.S.n// Š Z and † W n.S.n//! nC1.S.n C 1// is an isomorphism (n 1). The group n.S.n// is generated by the identity of
S.n/. Proof. From the exact sequence 2.S 3/! 2.S 2/! 1.S 1/!.S 3/ of the Hopf ﬁbration S 1! S 3! S 2 and j.S 3/ D 0 for j D 1; 2 we obtain an isomorphism @ W 2.S 2/ Š 1.S 1/ Š Z. From (6.4.6) we obtain a surjection † W 1.S.1//! 2.S.2//; this is an isomorphism, since both groups are isomorphic to Z. For n 2, (6.4.6) gives directly an isomorphism †. We know that 1.S.1// Š Z is generated by the identity, and † respects the identity. (6.4.8) Example. We continue the discussion of the Hopf ﬁbrations (6.3.6). The Hopf ﬁbration S 1! S 2nC1! CP n and i.S 2nC1/ D 0 for i 2n yield 2.CP n/ Š 1.S 1/ Š Z and i.CP n/ D 0 for 0 i 2n, i 6D 2. The inclusion S 2nC1! S 2nC3, z 7!.z; 0/ induces an embedding CP n CP nC1. We compare the corresponding Hopf ﬁbrations and their exact sequences and conclude S 2.CP n/ Š 2.CP nC1/. Let CP 1 D CP n be the colimit. The canonical inclusion CP n CP 1 induces i.CP n/ Š i.CP 1/ for i 2n. A proof uses the fact that a compact subset of CP 1 is contained in some ﬁnite CP N. Therefore CP 1 is a space with a single non-trivial homotopy group 2.CP 1/ Š Z. Note also the special case 3.S 2/ Š 3.S 3/ Š Z. We have similar results for real projective spaces. The twofold coverings Z=2! S n! RP n are use to show that 1.RP 2/ Š 1.RP 3/ Š Š 1.RP 1/ Š Z=2, induced by the inclusions
, i.RP n/ Š i.RP nC1/ for i < n and i.RP n/ D 0 for 0 i < n, i 6D 1. The space RP 1 has a single non-trivial Þ homotopy group 1.RP 1/ Š Z=2. n1 6.5 The Degree Let d W n.S.n//! Z be the isomorphism which sends Œid to 1. If f W S.n/! S.n/ is a pointed map, then f W n.S.n//! n.S.n// is the multiplication by the integer d.f / D d.fŒid/ D d.Œf /. Since the map ŒS.n/; S.n/0! ŒS.n/; S.n/ which forgets about the base point is bijective (see (6.2.8)), we can transport d to a bijection d W ŒS.n/; S.n/! Z. The functoriality fg D.fg/ shows 136 Chapter 6. Homotopy Groups d.fg/ D d.f /d.g/; therefore d.h/ D ˙1 if h is a homeomorphism. The suspension sends Œf to Œf ^ id; hence d.f / D d.f ^ id/. (6.5.1) Proposition. Given pointed maps f W S.m/! S.m/ and g W S.n/! S.n/. Then d.f ^ g/ D d.f /d.g/. Proof. We use the factorization f ^ g D.f ^ id/.id ^g/. The map f ^ id is a suspension of f, and suspension does not change the degree. Let W S.m/^S.n/! S.n/ ^ S.m/ interchange the factors. From.g ^ id/ D id ^g we conclude d.id ^g/ D d.g ^ id/ D d.g/. Let kn W S.n/! S n be a homeomorphism. The bijection ŒS n; S n! ŒS.n/; S.n/; Œf 7! Œknf k1 n is independent of the choice of kn. We use
this bijection to transport d to a bijection d W ŒS n; S n! Z. If d.Œf / D k we call k the degree d.f / of f. We still have the properties d.f /d.g/ D d.fg/, d.id/ D 1, d.h/ D ˙1 for a homeomorphism h. By a similar procedure we deﬁne the degree d.f / for any self-map f of a space S which is homeomorphic to S.n/. Matrix multiplication lA W Rn! Rn, x 7! Ax induces for each A 2 GLn.R/ a pointed map LA W S.n/! S.n/. For the notation see (6.1.4). (6.5.2) Proposition. The degree of LA is the sign of the determinant det.A/. Proof. Let w W I! GLn.R/, t 7! A.t/ be a path. Then.x; t/ 7! LA.t/x is a homotopy. Hence d.LA/ only depends on the path component of A in GLn.R/. The group GLn.R/ has two path components, distinguished by the sign of the determinant. Thus it sufﬁces to show that for some A with det.A/ D 1 we have d.LA/ D 1. By the preceding discussion and (6.1.4) we see that.x1; : : : ; xn/ 7!.x1; x2; : : : ; xn/ has degree 1. The stereographic projection (6.1.4) now shows that the map S n! S n which changes the sign of the ﬁrst coordinate has degree 1. (6.5.3) Proposition. Let A 2 O.n C 1/. Then A W S n! S n, x 7! Ax has degree det.A/. Proof. Again it sufﬁces to verify this for appropriate elements in the two path components of O.n C 1/, and this we have already achieved. (6.5.4) Corollary. The map S n! S n, x 7! x has degree.1/nC1. A vector ﬁeld on S n is a continuous map F W S n!
RnC1 such that for each x 2 S n the vector F.x/ is orthogonal to x. For the maximal number of linearly independent vector ﬁelds see [3]. 6.6. The Brouwer Fixed Point Theorem 137 (6.5.5) Theorem. There exists a vector ﬁeld F on S n such that F.x/ 6D 0 for each x 2 S n if and only if n is odd. Proof. Let n D 2k 1. Then.x1; x2; : : : ; x2k1; x2k/ 7!.x2; x1; : : : ; x2k; x2k1/ is a vector ﬁeld with the desired property. Let F be a vector ﬁeld such that F.x/ 6D 0. Set V.x/ D F.x/=kF.x/k. Then.x; t/ 7! cos t x C sin t V.x/ is a homotopy from the identity to the antipodal map. Hence the antipodal map has degree 1. By (6.5.4), n is odd. (6.5.6) Proposition. Let W S.m/ ^ S.n/! S.n/ ^ S.m/ interchange the factors. Then d. / D.1/mn. Proof. By (6.5.2) we know the analogous assertion for the models S.m/. 6.6 The Brouwer Fixed Point Theorem We prove the ﬁxed point theorem of Brouwer and a number of equivalent results. As an application we discuss the problem of topological dimension. Let us ﬁrst introduce some notation. Consider the cube W D W n D f.xi / 2 Rn j 1 xi 1g with the faces Ci.˙/ D fx 2 W n j xi D ˙1g. We say, Bi W n separates Ci.C/ and Ci./, ifB i is closed in W, and if W X Bi D Bi.C/ [ Bi./; ; DB i.C/ \ Bi./; Ci.˙/ Bi.˙/; with open subsets Bi.C/ and Bi./ of W X Bi. The n-dimensional standard simplex is n.
Its boundary @n is the union of the faces @i n D f.t0; : : : ; tn/ 2 n j ti D 0g. (6.6.1) Theorem. The following statements are equivalent: (1) A continuous map b W Dn! Dn has a ﬁxed point (Brouwer Fixed Point Theorem). (2) There does not exist a continuous map r W Dn! S n1 which is the identity on S n1 (Retraction Theorem). (3) The identity of S n1 is not null homotopic (Homotopy Theorem). (4) Let f W Dn! Rn be a continuous map such that f.z/ D z for z 2 S n1. Then Dn is contained in the image of f. (5) Let g W Dn! Rn be continuous. Then there exists a ﬁxed point or there exists z 2 S n1 such that g.z/ D z with > 1. 138 Chapter 6. Homotopy Groups (6) Let vi W W n! R, 1 i n be functions such that vi.x/ < 0 for x 2 Ci./ and vi.x/ > 0 for x 2 Ci.C/. Then there exists x 2 W n such that vi.x/ D 0 for each i (Intermediate Value Theorem). (7) Suppose Bi separates Ci./ and Ci.C/ for 1 i n. Then the intersection B1 \ B2 \ \ Bn is non-empty. (8) Let B0; : : : ; Bn be a closed covering of n such that ei … Bi and @i n Bi. n iD0 Bi 6D ;. The same conclusion holds if we assume that the Bi are T Then open. (9) Let B0; : : : ; Bn be a closed covering of n such that ei 2 Bi and @i n\Bi D T ;. Then n iD0 Bi 6D ;. The ﬁxed point theorem expresses a topological property of Dn. If h W X! Dn is a homeomorphism and f W X! X a self-map, then hf h1 has a ﬁxed point z and therefore f has the ﬁxed point h.z/. We can apply (2) to the pairs.W n
; @W n/ and.n; @n/, since they are homeomorphic to.Dn; S n1/. Statement (3) is also equivalent to the inclusion S n1 Rn X f0g not being null homotopic (similarly for @W n in place of S n1). Proof..1/ ).2/. Suppose r is a retraction. Then x 7! r.x/ is a map without ﬁxed point..2/ ).3/. The map r W Dn! S n1 which corresponds by (2.3.4) to a null homotopy of the identity is a retraction..3/ ).1/. Suppose b has no ﬁxed point. Then S n1 I! S n1;.x; t/ 7! x tb.x/ kx tb.x/k D N.x tb.x// is a homotopy from the identity to the map f W x 7! N.x b.x//. Since b has no ﬁxed point, the formula for f deﬁnes a map on the whole of Dn, and then.x; t/ 7! f.tx/ is a homotopy from the constant map to f. Thus f is null homotopic, and therefore also id.S n1/..2/ ).4/. If x is contained in the interior of Dn, then there exists a retraction r W Rn X x! S n1 of S n1 Rn X x. If x is not contained in the image of f, then r ı f W Dn! S n1 contradicts the retraction theorem..4/ ).5/. Deﬁne a map f W Dn! Rn by (i) f.x/ D 2x g.2x/; (ii) f.x/ D kxk1x 2 1 kxk g kxk1x ; kxk 1=2; kxk 1=2: For kxk D 1 continuous map. 2 we obtain in both cases 2x g.2x/. Thus f is a well-deﬁned For kxk D1 we have f.x/ D x. By (4), there exists y with f.y/ D 0. If 2, then k
yk 6D 1, and 2, then (i) shows that 2y is a ﬁxed point. If kyk > 1 kyk 1 (ii) shows the second case with D.2 2kyk/1 > 1. 6.6. The Brouwer Fixed Point Theorem 139.5/ ).1/. A special case..3/ ).6/. Set v W W n! Rn, x 7!.v1.x/; : : : ; vn.x//. Suppose v.x/ 6D 0 for each x 2 W n. Then v W W n! Rn X 0. Consider h W.t; x/ 7!.1 t/x C tv.x/. If x 2 Ci./, i.e., xi < 0, then.1 t/xi C tvi.x/ < 0 for each t 2 I. Hence ht W @W n! Rn X0 is a homotopy from the inclusion to v. Since v has an extension to W n, it is null homotopic, but the inclusion is not null homotopic. A contradiction..6/ ).7/. Let d denote the Euclidean distance. Deﬁne vi W W! R by ( vi.x/ D d.x; Bi /; x 2 Bi./; d.x; Bi /; x 2 Bi.C/ [ Bi ; and apply (6)..7/ ).2/. Let r W W n! @W n be a retraction. We deﬁne Bi.˙/ D r 1.˙xi > 0/ and Bi D r 1.xi D 0/. We apply (7) and obtain a contradiction..3/ ).8/. We use the functions vi.x/ D d.x; Bi /. Our assumptions imply vi.ei / > 0, and vi.x/ D 0 provided x 2 @i n. If the Bi have empty intersection, then v.x/ D.v0.x/; : : : ; vn.x// 6D 0 for every x 2 n. This gives us a map ˛ W n! @n; x 7!. P vi.x//1v.x/; because, since the Bi cover n, for each x at least one coordinate vi.x/ is zero. If x
2 @i n, then ˛.x/ 2 @i n, hence.1 t/x C tv.x/ 2 @i n for each t 2 Œ0; 1. The identity of @n is therefore homotopic to ˇ D ˛j@n. Since ˇ has the extension ˛ it is null homotopic, and therefore also id.@n/ is null homotopic. This contradicts (3). Now suppose the Bi are open. By a general result of point-set topology there exist closed sets Ci Bi and the Ci still form a covering. In order to make sure that the Ci satisfy the hypotheses of (8) we can replace the Ci by Ci [ @i n. The ﬁrst part of the proof now shows that the Ci have non-empty intersection..8/ ).9/. Set Ui D n X Bi. Suppose the Bi have empty intersection. Then the Ui cover n. Since the Bi are a covering, the Ui have empty intersection. By construction, ei … Ui and @i n Ui. We therefore can apply (8) in the case of the open covering by the Ui and see that the Ui have non-empty intersection. Contradiction..9/ ).2/. Let Aj D f.t0; : : : ; tn/ 2 @n j tj 1=ng. Let r W n! @n be a retraction and set Bj D r 1.Aj /. Then (9) tells us that the Bj have non-empty intersection, and this is impossible. Theorem (6.6.1) has many different proofs. For a proof which uses only basic results in differential topology see [79]. Another interesting proof is based on a combinatorial result, called Sperner’s Lemma [173]. 140 Chapter 6. Homotopy Groups The retraction theorem does not hold for inﬁnite-dimensional spaces. In [70, Chapter 19] you can ﬁnd a proof that the unit disk of an inﬁnite-dimensional Banach space admits a retraction onto its unit sphere. Does there exist a sensible topological notion of dimension for suitable classes of spaces? Greatest generality is not necessary at this point. As an example we introduce the covering dimension of compact metric spaces X. (For dimension theory in
general see [94].) Let C be a ﬁnite covering of X and " > 0 a real number. We call C an "-covering, if each member of C has diameter less than ". We say C has order m, if at least one point is contained in m members but no point in mC1. The compact metric space X has covering dimension dim X D k, if there exists for each " > 0 a ﬁnite closed "-covering of X of order k C 1 and k 2 N0 is minimal with this property. Thus X is zero-dimensional in this sense, if there exists for each " > 0 a ﬁnite partition of X into closed sets of diameter at most ". We verify that this notion of dimension is a topological property. (6.6.2) Proposition. Let X and Y be homeomorphic compact metric spaces. If X is k-dimensional then also is Y. Proof. Let h W X! Y be a homeomorphism. Fix " > 0 and let U be the covering of Y by the open "-balls U".y/ D fx j d.x; y/ < "g. (We use d for the metrics.) Let ı be a Lebesgue number of the covering.h1.U / j U 2 U/. Since dim X D k, there exists a ﬁnite closed ı-covering C of X of order k C 1. The ﬁnite closed covering D D.h.C / j C 2 C/ of Y has then the order k C 1, and since each member of C is contained in a set h1.U /, the covering D is an "-covering. Thus we have shown dim Y k. We now show that dim Y k, i.e., there exists ı > 0 such that each ﬁnite closed ı-covering has order at least k C 1. Let " > 0 be a corresponding number for X. A homeomorphism g W Y! X is uniformly continuous: There exists a ı > 0 such that d.y1; y2/ < ı implies d.g.y1/; g.y2// < ". So if C is a ı-covering of Y, then D D.g.C / j C 2 C/ is an "-covering of X. Since D has order at least k
C 1, so has C. (6.6.3) Proposition. There exists " > 0 such that each ﬁnite closed "-covering.Bj j j 2 J / of n has order at least n C 1. Proof. Let " be a Lebesgue number of the covering Ui D n X @i n, i D 0; : : : ; n. Hence for each j 2 J there exist i such that Bj Ui, and the latter is equivalent to Bj \ @i n D ;. Suppose ek 2 Bj. Since ek 2 @i n for i 6D k, we cannot have Bj Ui ; thus ek 2 Bj implies Bj Uk. Since each ek is contained in at least one of the sets Bj we conclude jJ j n C 1. For each j 2 J we now choose g.j / 2 f0; : : : ; ng such that Bj \ @g.j /n D ; and set Ak D [fBj j g.j / D kg; this is a closed set because J is ﬁnite. Each Bj is contained in some Ak, hence the Ak cover n. Moreover, by construction, Ak \ @kn D ;. We can therefore apply part (9) of (6.6.1) and ﬁnd an x in the intersection of the Ak. Hence for each 6.7. Higher Connectivity 141 k there exists ik such that x 2 Bik. Since each Bj is contained in exactly one of the sets Ak, the element x is contained in the n C 1 members Bik, k D 0; : : : ; n of the covering. We can now compare the covering dimension and the algebraic dimension. (6.6.4) Theorem. n has covering dimension n. A compact subset of Rn has covering dimension at most n. Proof. By (6.6.3), n has covering dimension at least n. It remains to construct ﬁnite closed "-coverings of order n C 1 for each ". See Problem 4. Problems 1. Let U; V be an open covering of I 2. Then there exists either a path u W I! U such that u.0/ 2 I 0; u.1/ 2 I 1 or a path v W I! V such that v.0/ 2 0 I; v.1/ 2 1 I. 2. Let U
; V be an open covering of 2. Then there exists a path component QU of U such that QU \ @i 2 6D ; for each i or a path component of V with a similar property. 3. Generalize the preceding two exercises to n dimensions. 4. The following ﬁgure indicates the construction of closed "-coverings of order 3 for the square. Generalize this construction to the cube I n by a suitable induction. 5. Suppose I n is the union of a ﬁnite number of closed sets, none of which contains points of two opposite faces. Then at least n C 1 of these closed sets have a common point. 6.7 Higher Connectivity For many applications it is important to know that the homotopy groups of a space vanish in a certain range. We discuss several reformulations of this fact. In the following 0.X; x/ D 0.X/ with base point Œx. The space D0 is a singleton and S 1 D ;. (6.7.1) Proposition. Let n 0. The following are equivalent: (1) n.X; x/ D 0 for each x 2 X. (2) Each map S n! X has an extension to DnC1. (3) Each map @I nC1! X has an extension to I nC1. 142 Chapter 6. Homotopy Groups Proof. The case n D 0 is trivial. The equivalence of (2) and (3) is a consequence of the homeomorphism.DnC1; S n/ Š.I nC1; @I nC1/. Suppose f W S n! X is given. Use e1 D.1; 0; : : : / 2 S n as a base point and think of f representing an element in n.X; x/. If (1) holds, then f is pointed null homotopic. A null homotopy S n I! X factors over the quotient map S n I! DnC1,.x; t/ 7!.1 t/e1 C tx and yields an extension of f. Conversely, let an element ˛ of n.X; x/ be represented by a pointed map f W.S n; e1/!.X; x/. If this map has an extension F to DnC1, then.F; f / represents ˇ 2 n
C1.X; X; x/ D 0 with @ˇ D ˛. (6.7.2) Proposition. Let n 0. Let f W.Dn; S n1/!.X; A/ be homotopic as a map of pairs to a map k W.Dn; S n1/!.A; A/. Then f is relative to S n1 homotopic to a map g such that g.Dn/ A. Proof. The case n D 0 is trivial. Let Gt W.Dn; S n1/!.X; A/ be a homotopy from f to k according to the assumption. Deﬁne W Dn I! Dn I by.x; t/ D.2˛.x; t/1x; 2˛.x; t// with the function ˛.x; t/ D max.2kxk; 2t/. Then H D G ı is a homotopy with the desired property from f to g D H1. 1 0 a.............................. c b Dn d!........................................ b c a d (6.7.3) Proposition. Let n 1. The following assertions about.X; A/ are equivalent: (1) n.X; A; / D 0 for each choice of 2A. (2) Each map f W.I n; @I n/!.X; A/ is as a map of pairs homotopic to a constant map. (3) Each map f W.I n; @I n/!.X; A/ is homotopic rel @I n to a map into A. Proof. (1) ) (2). Let f W.I n; @I n/!.X; A/ be given. Since J n1 is contractible, there exists a homotopy of the restriction f W J n1! A to a constant map. Since J n1 @I n and @I n I n are coﬁbrations, f is as a map of pairs homotopic to g W.I n; @I n/!.X; A/ such that g.J n1/ D fa0g. Since n.X; A; a0/ D 0, the map g W.I n; @I n; J n1/!
.X; A; a0/ is null homotopic as a map of triples. (2) ) (3). (6.7.2). (3) ) (1). Let f W.I n; @I n; J n1/!.X; A; / be given. By assumption (3) Œf is contained in the image of n.A; A; /!.X; A; /. Now use n.A; A; / D 0. 6.7. Higher Connectivity 143 We call.X; A/ n-compressible if one of the assertions in (6.7.3) holds. More generally, we call a map f W X! Y n-compressible if the following holds: For each commutative diagram @ there exists ‰ W I n! X such that ‰j@I n D'and f ‰'ˆ relative to @I n. (This amounts to part (3) in (6.7.3).) This notion is homotopy invariant in the following sense: (6.7.4) Proposition. Given f W X! Y and a homotopy equivalence p W Y! Z. Then f is n-compressible if and only pf is n-compressible. (6.7.5) Proposition. Let n 0. The following assertions about.X; A/ are equivalent: (1) Each map f W.I q; @I q/!.X; A/, q 2 f0; : : : ; ng is relative to @I q homo- topic to a map into A. (2) The inclusion j W A! X induces for each base point a 2 A a bijection j W q.A; a/! q.X; a/ for q < n and a surjection for q D n. (3) 0.A/! 0.X/ is surjective, and q.X; A; a/ D 0 for q 2 f1; : : : ; ng and each a 2 A. Proof. (1), (3). The surjectivity of 0.A/! 0.X/ is equivalent to (1) for q D 0. The other cases follow from (6.7.3). (2), (3). This follows from the exact sequence (6.1
.2). A pair.X; A/ is called n-connected if (1)–(3) in (6.7.5) hold. We call.X; A/ 1-connected if the pair is n-connected for each n. A pair is 1-connected if and only if j W n.A; a/! n.X; a/ is always bijective. If X 6D ; but A D ; we say that.X; A/ is.1/-connected, and.;; ;/ is 1-connected. (6.7.6) Proposition. Let n 0. The following assertions about X are equivalent: (1) q.X; x/ D 0 for 0 q n and x 2 X. (2) The pair.CX; X/ is.n C 1/-connected. (3) Each map f W @I q! X, 0 q n C 1 has an extension to I q. Proof. The cone CX is contractible. Therefore @ W qC1.CX; X; / Š q.X; /. This and (6.7.5) shows the equivalence of (1) and (2). The equivalence of (1) and (3) uses (6.7.1). A space X is n-connected if (1)–(3) in (6.7.6) hold for X. Note that this is compatible with our previous deﬁnitions for n D 0; 1. 144 Chapter 6. Homotopy Groups Let f W X! Y be a map and X Z.f / the inclusion into the mapping cylinder. Then f is said to be n-connected if.Z.f /; X/ is n-connected. We then also say that f is an n-equivalence. Thus f is n-connected if and only if f W q.X; x/! q.Y; f.x// is for each x 2 X bijective (surjective) for q < n If f is an 1-equivalence we also say that f is a weak (homotopy) (q D n). Thus f is a weak equivalence if and only if f W n.X; x/! equivalence. n.Y; f.x// is bijective for each n 0 and each x 2 X. (6.7.
7) Proposition. Let.p1; p0/ W.E1; E0/! B be a relative Serre ﬁbration. Let F b j denote the ﬁbre of pj over b. Then the following are equivalent: (1).E1; E0/ is n-connected. (2).F b 0 / is n-connected for each b 2 B. 1 ; F b Proof. This is a direct consequence of (6.3.8). The compression properties of an n-connected map can be generalized to pairs of spaces which are regular unions of cubes of dimension at most n. We use this generalization in the proof of theorem (6.7.9). Consider a subdivision of a cube I n. Let us call B a cube-complex if B is the union of cubes of this subdivision. A subcomplex A of B is then the union of a subset of the cubes in B. We understand that B and A contain with each cube all of its faces. The k-skeleton B.k/ of B consists of the cubes in B of dimension k; thus A.k/ D B.k/ \ B. (6.7.8) Proposition. Let f W X! Y be n-connected. Suppose.C; A/ is a pair of cube-complexes of dimension at most n. Then to each commutative diagram A \ C'ˆ X f Y there exists ‰ W C! X such that ‰jA D'and f ‰'ˆ relative to A. Proof. Induction over the number of cubes. Let A B C such that C is obtained from B by adding a cube W of highest dimension. Then @W B. By induction there exists ‰0 W B! X such that ‰0jA D'and a homotopy H W f ‰0'ˆjB relative to A. Extend H to a homotopy of ˆ. The end ˆ1 of this homotopy satisﬁes ˆ1jB D f ‰0. We now use that f is n-connected and extend ‰0 over W to ‰ W C! X such that f ‰'ˆ1 relative to B. Altogether we have f ‰ 'B ˆ1 'A ˆ and ˆjA
D ‰0jA D '. (6.7.9) Theorem. Let'W.X; X0; X1/!.Y; Y0; Y1/ be a map such that the restrictions 'i W Xi! Yi are n-connected and '01 W X0 \ X1! Y0 \ Y1 is.n 1/1 and Y D Y ı connected. Suppose X D X ı 1. Then'is an n-equivalence.7. Higher Connectivity 145 Proof. We use mapping cylinders to reduce to the case of inclusions'W X Y; 'i W Xi Yi. Let.F; f / W.I n; @I n/!.Y; X/ be given. We have to show that this map is homotopic relative to @I n to a map into X. Let i / [ f 1.X X X ı Ai D F 1.Y X Y ı i /: These sets are closed and disjoint. By the Lebesgue lemma we choose a cubical subdivision of I n such that no cube W of the subdivision intersects both A0 and A1. Let Kj be the union of the cubes W which satisfy F.W / Y ı i ; Then Ki is a cubical subcomplex and f.W \ @I n/ X ı i : I n D K0 [ K1; F.Ki / Y ı i ; f.Ki \ @I n/ X ı i : We denote by K the.n 1/-skeleton of a cubical complex; then K \ @I n D K \ @I n and K0 \ K 1. We have a commutative square 1 D K 0 \ K X01 f01 @I n \ K01 g01 Y01 F01 K 01. Since.Y01; X01/ is.n 1/-connected there exists a homotopy relative to @I n \ K01 from F01 to a map g01 W K 01! X01. Deﬁne g0 W K0 \.@I n [ K/! X0 by g0jK0 \ @I n D f0; g0jK0 \ K 1 D g01: (Both maps agree on the intersection.) The homotopy F01'g01 and the constant homotopy of f0 combine to a
homotopy of F0jK0 \.@I n [ K 1/ to g0 which is constant on K0 \ @I n. Since the inclusion of a cube complex into another one is a coﬁbration, this homotopy can be extended to a homotopy W K0 I! Y0 from F0 to H0. We obtain a diagram X0 h0 g0 Y0 H0 K0 K0 \.@I n [ K 1/ where H0 is homotopic to h0 W K0! X1 relative to K0 \.@I n [ K.Y0; X0/ is n-connected. We prove the second part similarly. We obtain a map g1 W K1\.@I n[K 1/, since 0/! Y1 with g1jK1 \ @I n D f1 and g1jK1 \ K 0 D g01 and then X1 h1 g1 Y1 H1 K1. K1 \.@I n [ K 0/ 146 Chapter 6. Homotopy Groups The maps h0 and h1 coincide on K 1! X which is 1; moreover hj@I n D f. Let now W be an homotopic relative @I n to F jK n-dimensional cube, say with W K0. Then @W K 0 and h.@W / D h0.@W / X0. Since.Y0; X0/ is n-connected, we can deform the map relative to @W to a map into X0. 01 and yield a map 6.7.10) Corollary. Let f W X! Y be an n-connected map between well-pointed spaces. Then †f W †X! †Y is.n C 1/-connected. If X is n-connected, then †X is.n C 1/-connected. The sphere S kC1 is k-connected. Proof. Let †0X denote the unpointed suspension of X. This is a quotient of X I and covered by the open cones C0 D X Œ0; 1Œ=X 0 and C1 D X 0; 1=X 1 with intersection X 0; 1Œ. We can apply (6.7.7) directly; the cones are contractible and therefore the induced maps Cj.X/!
Cj.Y / 1-connected. In the case of a well-pointed space X the quotient map †0X! †X is an h-equivalence. (6.7.11) Theorem. Let f W X! Y be a continuous map. Let.Uj j j 2 J / and.Vj j j 2 J / open coverings of X and Y such that f.Uj / Vj. Suppose that for each ﬁnite E J the induced map fE W j 2E Vj is a weak equivalence. Then f is a weak equivalence j 2E Uj! T T Proof. By passage to the mapping cylinder we can assume that f is an inclusion. Let h W.I n; @I n/!.Y; X/ be given. We have to deform h relative to @I n into X. By compactness of I n it sufﬁces to work with ﬁnite J. A simple induction reduces the problem to J D f0; 1g. Then we apply (6.7.9). Problems 1. Let Y D f0g [ fn1 j n 2 Ng and X the same set with the discrete topology. Then the identity X! Y is a weak equivalence but there does not exist a weak equivalence Y! X. 2. Identify in S 1 the open sets f.x; y/ j y > 0g and f.x; y/ j y < 0g to a point. The quotient map S 1! S onto the quotient space S, consisting of four points, is a weak equivalence (but not a homotopy equivalence). In particular 1.S/ Š Z. Show that S has a universal covering. 6.8 Classical Groups We use exact sequences of Serre ﬁbrations and deduce from our knowledge of i.S n/ other results about homotopy groups of classical groups and Stiefel manifolds. We use a uniform notation for the (skew) ﬁelds F D R; C; H and the 6.8. Classical Groups 147 corresponding groups (orthogonal, unitary, symplectic) O.n/ D O.n; R/; SO.n/ D SO.n; R/; U.n/ D O.n; C/; SU.n/
D SO.n; C/; Sp.n/ D O.n; H/: Let d D dimR F. The starting point are the (Serre) ﬁbrations which arise from the action of the orthogonal groups on the unit spheres by matrix multiplication O.n; F / SO.n; F / j! O.n C 1; F /! S d.nC1/1; j! SO.n C 1; F /! S d.nC1/1: The inclusions j of the groups arise from A 7!. We also pass to the colimit and obtain O.1; F / D colimn O.n; F / and SO.1; F / D colimn SO.n; F /. From i.S n/ D 0, i < n and the exact homotopy sequences of the ﬁbrations we deduce that the inclusions j W O.n; F /! O.n C 1; F / and j W SO.n; F /! SO.n C 1; F / are d.n C 1/ 2 connected. By induction and passage to the colimit we obtain A 0 0 1 (6.8.1) Proposition. For n < m 1, the inclusions O.n; F /! O.m; F / and SO.n; F /! SO.m; F / are d.n C 1/ 2 connected; in particular, the homomorphisms i.O.n; F //! i.O.m; F // are isomorphisms in the range i n 2.R/, i 2n 1.C/, and i 4n C 1.H/. We turn our attention to Stiefel manifolds of orthonormal k-frames in F n: Vk.Rn/ Š O.n/=O.n k/ Š SO.n/=SO.n k/; Vk.Cn/ Š U.n/=U.n k/ Š SU.n/=SU.n k/; Vk.Hn/ Š Sp.n/=Sp.n k/: We have the corresponding (Serre) ﬁbrations of the type H! G! G=H for these homogeneous spaces. We use (6.8.1) in the
exact homotopy sequences of these ﬁbrations and obtain: (6.8.2) Proposition. i.Vk.F n// D 0 for i d.n k C 1/ 2. We have the ﬁbration p W VkC1.F nC1/! V1.F nC1/;.v1; : : : ; vkC1/ 7! vkC1: The ﬁbre over ekC1 is homeomorphic to Vk.F n/; with W v 7!.v; 0/ we obtain a homeomorphism j W.v1; : : : ; vk/ 7!.v1; : : : ; vk; ekC1/ onto this ﬁbre. From the homotopy sequence of this ﬁbration we obtain (6.8.3) Proposition. j W i.Vk.F n//! i.VkC1.F nC1// is an isomorphism for i d.n C 1/ 3. 148 Chapter 6. Homotopy Groups We use V1.F n/ D S d m1 and t.S t / Š Z and obtain from (6.8.3) by induction (6.8.4) Proposition. 2.nk/C1.Vk.Cn// Š Z, 4.nk/C3.Vk.Hn// Š Z. The real case is more complicated. The result is (6.8.5) Proposition. ( nk.Vk.Rn// Š k D 1; or n k even ; Z; Z=2; k 2; n k odd. Proof. By (6.8.3) and induction it sufﬁces to consider the case k D 2. Later we compute the homology groups of V2.Rn/, and the theorem of Hurewicz will then give us the desired result. Problems 1. The group O.n/ has two path components. The groups SO.n/, U.n/, SU.n/, and Sp.n/ are path connected. 2. In low dimensions we have some special situations, namely U.1/ Š SO.2/ Š S 1; Spin.3/ Š SU.2/ Š Sp.1/ Š
S 3; Z=2! SU.2/! SO.3/; a 2-fold covering, SU.n/! U.n/! S 1; a ﬁbration. Use these data in order to verify m 3; n 1; 1.SO.2// Š 1.O.2// Š Z; 1.SO.3// Š 1.SO.n// Š Z=2; 1.U.1// Š 1.U.n// Š Z; 1.SU.n// Š 1.Sp.n// Š 0; 2.SU.n// Š 2.U.n// Š 2.Sp.n// Š 0; n 1; 2.SO.n// Š 0; n 3; 3.U.2// Š 3.U.k// Š Z; 3.SU.2// Š 3.SU.k// Š Z; 3.Sp.1// Š 3.Sp.k// Š Z; 3.SO.3// Š Z: n 1; k 1; k 2; k 2; 6.9 Proof of the Excision Theorem In this section we present an elementary proof of the excision theorem (6.4.1). The proof is due to D. Puppe [46]. We derive the excision theorem from a more 6.9. Proof of the Excision Theorem 149 conceptual reformulation (6.9.3). The reformulation is more satisfactory, because it is “symmetric” in Y1; Y2. In (6.4.1) we have a second conclusion with the roles of Y1 and Y2 interchanged. We begin with a technical lemma used in the proof. A cube in Rn, n 1 will be a subset of the form W D W.a; ı; L/ D fx 2 Rn j ai xi ai C ı for i 2 L; ai D xi for i … Lg for a D.a1; : : : ; an/ 2 Rn, ı > 0, L f1; : : : ; ng. (L can be empty.) We set dim W D jLj. A face of W is a subset of the form W 0 D fx 2 W j xi D ai for i
2 L0; xj D aj C ı for j 2 L1g (W 0 can be empty.) Let @W denote the union for some L0 L; L1 L. of all faces of W which are different from W. We use the following subsets of W D W.a; ı; L/: ˚ ˚ Kp.W / D Gp.W / D x 2 W j xi < ai C ı x 2 W j xi > ai C ı 2 for at least p values i 2 L 2 for at least p values i 2 L ; : Here 1 p n. For p > dim W we let Kp.W / and Gp.W / be the empty set. (6.9.1) Lemma. Let f W W! Y and A Y be given. Suppose that for p dim W the inclusions f 1.A/ \ W 0 Kp.W 0/ for all W 0 @W hold. Then there exists a map g which is homotopic to f relative to @W such that g1.A/ Kp.W /. (Similarly for Gp in place of Kp.) n 4 ; : : : ; 1 0; 1 2 Proof. We can assume that W D I n, n 1. We deﬁne h W I n! I n in the following manner: Let x D. 1 4 /. For a ray y which begins in x we consider and its intersection Q.y/ with @I n. Let h map its intersection P.y/ with @ the segment from P.y/ to Q.y/ onto the single point Q.y/ and the segment from x to P.y/ afﬁnely to the segment from x to Q.y/. Then h is homotopic relative to @I n to the identity. We set g D f h. Let z 2 I n and g.z/ 2 A. If zi < 1 2 for all i, then z 2 Kn.I n/ Kp.I n/. Suppose now that for at least one i we have zi 1 2, then h.z/ 2 @I n and hence h.z/ 2 W 0 for some face W 0 with dim W 0 D n 1. Since also h.z/ 2 f 1.A/, by assumption h.z/ 2 K
p.W 0/. Hence we have for at 4 C t.zi 1 least p coordinates 1 4 / with t 1. We conclude that for at least p coordinates 1 2 > h.z/i. By deﬁnition of h, we haveh.z/ i D 1 2 > zi. The next theorem is the basic technical result. In it we deform a map I n! Y into a kind of normal form. We call it the preparation theorem. Let Y be the union 150 Chapter 6. Homotopy Groups of open subspaces Y1; Y2 with non-empty intersection Y0. Let f W I n! Y be given. By the Lebesgue lemma (2.6.4) there exists a subdivision of I n into cubes W such that either f.W / Y1 or f.W / Y2 for each cube. In this situation we claim: (6.9.2) Theorem. Suppose.Y1; Y0/ is p-connected and.Y2; Y0/ is q-connected (p; q 0). Then there exists a homotopy ft of f with the following properties: (1) If f.W / Yj, then ft.W / Yj. (2) If f.W / Y0, then ft is constant on W. (3) If f.W / Y1, then f 1 (4) If f.W / Y2, then f 1.Y1 X Y0/ \ W KpC1.W /..Y2 X Y0/ \ W GqC1.W /. 1 1 Here W is any cube of the subdivision. Proof. Let C k be the union of the cubes W with dim W k. We construct the homotopy inductively over C k I. Let dim W D 0. If f.W / Y0 we use condition (2). If f.W / Y1; f.W / 6 Y2, there exists a path in Y1 from f.W / to a point in Y0, since.Y1; Y0/ is 0connected. We use this path as our homotopy on W. Then (1) and (3) hold. Similarly if f.W / Y2; f.W / 6 Y1. Thus we have found a suitable homotopy on C 0. We extend this homotopy to
the higher dimensional cubes by induction over the dimension; we use that @W W is a coﬁbration, and we take care of (1) and (2). Suppose we have changed f by a homotopy such that (1) and (2) hold and (3), (4) for cubes of dimension less than k. Call this map again f. Let dim W D k. If f.W / Y0, we can use (2) for our homotopy. Let f.W / Y1; f.W / 6 Y2. If dim W p, there exists a homotopy f W W W! Y1 relative to @W of f jW with f W 1.W / Y0, since.Y1; Y0/ is p-connected. If dim W > p we use (6.9.1) in order to ﬁnd a suitable homotopy of f jW. We treat the case f.W / Y2; f.W / 6 Y1 in a similar manner. Again we extend the homotopy to the higher dimensional cubes. This ﬁnishes the induction step. t Let us denote by F.Y1; Y; Y2/ the path space fw 2 Y I j w.0/ 2 Y1; w.1/ 2 Y2g. We have the subspace F.Y1; Y1; Y0/. (6.9.3) Theorem. Under the hypothesis of the previous theorem the inclusion F.Y1; Y1; Y0/ F.Y1; Y; Y2/ is.p C q 1/-connected. Proof. Let a map'W.In; @In/!.F.Y1; Y; Y2/; F.Y1; Y1; Y0// be given where n p C q 1. We have to deform this map of pairs into the subspace. By adjunction, a map of this type corresponds to a map ˆ W I n I! Y with the following properties: (1) ˆ.x; 0/ 2 Y1 for x 2 I n, (2) ˆ.X; 1/ 2 Y2 for x 2 I n, 6.9. Proof of the Excision Theorem 151 (3) ˆ.y; t/ 2 Y1 for y 2 @I n and
t 2 I. Let us call maps of this type admissible. The claim of the theorem is equivalent to the statement, that ˆ can be deformed as an admissible map into a map with image in Y1. We apply the preparation theorem to ˆ and obtain a certain map ‰. The deformation in (6.9.2) stays inside admissible maps. Consider the projection W I n I! I n. We claim that the images of ‰1.Y X Y1/ and ˆ1.Y X Y2/ under are disjoint. Let y 2 ‰1.Y X Y2/, y D.z/ and z 2 ‰1.Y X Y2/ \ W for a cube W. Then z 2 KpC1.W / and hence y has at least p small coordinates. In a similar manner we conclude from y 2 ‰1.Y X Y1/ that y has at least q large coordinates. In the case that n < p C q the point y cannot have p small and q large coordinates. The set ‰1.Y X Y1/ is disjoint to @I n, since ‰.@I n/ I / A. There exists a continuous function W I n! I which assumes the value 0 on ‰1.Y X Y1/ and the value 1 on @I n [ ‰.Y X Y2/. The homotopy..x; t/; s/ 7! ‰.x;.1 s/t C st.x// is a homotopy of admissible maps from ‰ to a map with image in Y1. (6.9.4) Theorem. Under the hypothesis of (6.9.2) the inclusion induces an isomorphism j.Y1; Y0/! j.Y; Y2/ for j < p C q and an epimorphism for j D p C q. Proof. We have the path ﬁbration F.Y; Y; Y2/! Y, w 7! w.0/. The pullback along Y1 Y yields the ﬁbration F.Y1; Y; Y2/! Y1, w 7! w.0/. The ﬁbre over is F.; Y; Y2/. We obtain a commutative diagram of ﬁbrations:
F.; Y1; Y0/ F.Y1; Y1; Y0/ A ˇ ˛ D F.; Y; Y2/ F.Y1; Y; Y2/ A. The inclusion ˛ is.p C q 1/-connected (see (6.9.3)). Hence ˇ has the same connectivity (see (6.7.8)), i.e., the inclusion.Y1; Y0/.Y; Y2/, n.F.; Y1; Y0// n.F.; Y; Y2// Š nC1.Y1; Y0; / Š nC1.Y; Y2; / induces an isomorphism for n < p C q 1 and an epimorphism for n D p C q 1. 152 Chapter 6. Homotopy Groups Problems 1. The hypothesis of (6.4.1) is a little different from the hypothesis of (6.9.4), since we did not assume in (6.4.1) that.Y1; Y0/ and.Y2; Y0/ are 0-connected. Let Y 0 be the subset of points that can be connected by a path to Y0. Show that Y 0 has the open cover Y 0 2 and the inclusion induces isomorphisms.Y 0 1; Y0/ Š.Y1; Y0/ and.Y; Y2/ Š.Y 0; Y 0 2/. This reduces (6.4.1) to (6.9.4). 2. The map Y0! F.Y1; Y1; Y0/ which sends y 2 Y0 to the constant path with value y is an h-equivalence. 3. The map a1 W F.Y; Y; Y1/! Y, w 7! w.0/ replaces the inclusion Y1! Y by a ﬁbration. There is a pullback diagram 1; Y 0 F.Y1; Y; Y2/ F.Y; Y; Y1/ F.Y; Y; Y2/ a2 a1 Y. Thus (6.9.3) compares the pushout Y of Y1 Y0! Y2 and the pullback of a1; a2 with Y0. For generalizations see [73]. 4. Show that the proof of (6
.4.2) along the lines of this section does not need (6.9.1). 6.10 Further Applications of Excision The excision theorem is a fundamental result in homotopy theory. For its applications it is useful to verify that it holds under different hypotheses. In the next proposition we show and use that Y is the homotopy pushout. (6.10.1) Proposition. Let a pushout diagram be given with a coﬁbration. Suppose i.X; A; a/ D 0 for 0 < i < p and each a 2 A, and i.f; a/ D 0 for 0 < i < q and each a 2 A. Then the map.F; f / W n.X; A; a/! n.Y; B; f.a// is surjective for 1 n p C q 2 and bijective for 1 n < p C q 2. Proof. We modify the spaces up to h-equivalence such that (6.4.1) can be applied. Let Z.f / D B[f AŒ0; 1 D BCAŒ0; 1=f.a/.a; 0/ be the mapping cylinder of f with inclusion k W A! Z.f /, a 7!.a; 1/ and projection p W Z.f /! B a homotopy equivalence. We form the pushout diagrams A j X k Z..10. Further Applications of Excision 153 with pk D f and PK D F. Then P is a homotopy equivalence by (5.1.10) and.P; p/ induces an isomorphism of homotopy groups. Therefore it sufﬁces to analyze.K; k/. The space Z can be constructed as Z D B [f A Œ0; 1 [ X D Z.f / C X=.a; 1/ a: The map.K; k/ is the composition of.X; A; a/!.A0; 1 [ X; A0; 1;.a; 1//; x 7!.x; 1/ with the inclusion into.Z; Z.f /;.a; 1//. The ﬁrst map induces an isomorphism of homotopy groups, by homotopy equivalence. In order to exhibit n./ as an isomorph
ism, we can pass to the base point.a; 1=2/, by naturality of transport. With this base point we have a commutative diagram n.A0; 1 [ X; A0; 1/ n.Z; Z.f // n.A0; 1 [ X; A0; 1Œ/ n.Z; B [ A Œ0; 1Œ/. The vertical maps are isomorphism, by homotopy invariance. We apply (6.4.1) to the bottom map. Note that i.A0; 1 [ X; A0; 1/ Š i.X; A/ and i.B [ A Œ0; 1Œ; A0; 1Œ/ Š i.Z.f /; A/; again by homotopy invariance. (6.10.2) Theorem (Quotient Theorem). Let A X be a coﬁbration. Let further p W.X; A/!.X=A; / be the map which collapses A to a point. Suppose that for each base point a 2 A, i.CA; A; a/ D 0 for 0 < i < m; i.X; A; a/ D 0 for 0 < i < n: Then p W i.X; A; a/! i.X=A; / is bijective for 0 < i < m C n 2 and surjective for i D n C m 2. Proof. By pushout excision, i.X; A/! i.X [ CA; CA/ is bijective (surjective) in the indicated range. Note that @ W i.CA; A; a/ Š i1.A; a/, so that the ﬁrst hypothesis is a property of A. The inclusion CA X [CA is an induced coﬁbration. Since CA is contractible, the projection p W X [ CA! X [ CA=CA Š X=A is a homotopy equivalence. (6.10.3) Corollary. Let A X be a coﬁbration. Assume that i.A/ D 0 for 0 i m 1 and i.X/ D 0 for 0 i m 2. Then i.X; A/! i.X=A/ 154 Chapter 6
. Homotopy Groups is an isomorphism for 0 < i 2m 1. We use this isomorphism in the exact sequence of the pair.X; A/ and obtain an exact sequence 2m1.A/! 2m1.X/! 2m1.X=A/! 2m2.A/!! mC1.X/! mC1.X=A/! m.A/! 0: A similar exact sequence exists for an arbitrary pointed map f W AX where a typical portion comes from the coﬁbre sequence i.A/ f1! i.C.f //. f! i.X/ We now generalize the suspension theorem. Let.X; / be a pointed space. Recall the suspension †X and the homomorphism † W n.X/! nC1.†X/. (6.10.4) Theorem. Let X be a well-pointed space. Suppose i.X/ D 0 for 0 i n. Then † W j.X/! j C1.†X/ is bijective for 0 j 2n and surjective for j D 2n C 1. Proof. Let CX D X I =.X 1[fgI / be the cone on X. We have an embedding i W X! CX, x 7! Œx; 0 which we consider as an inclusion. The quotient CX=X can be identiﬁed with †X. From the assumption that fg X is a coﬁbration one concludes that i is a coﬁbration (Problem 1). Since CX is contractible, the exact sequence of the pair.CX; X/ yields an isomorphism @ W j C1.CX; X/'j.X/. The inverse isomorphism sends an element represented by f W I n! X to the element represented by f id.I /. From this fact we see † D p ı @1 W j.X/ @ j C1.CX; X/ p j C1.†X/; with the quotient map p W CX! CX=X D †X. We can therefore prove the theorem by showing that p is bijective or surjective in the same range. This follows from the quotient theorem (6.10
.2). (6.10.5) Theorem. Let X and Y be well-pointed spaces. Assume i.X/ D 0 for i < p. 2/ and i.Y / D 0 for i < q. 2/. Then the inclusion X _ Y! X Y induces an isomorphism of the i -groups for i p C q 2. The groups i.X Y; X _ Y / and i.X ^ Y / are zero for i p C q 1. Proof. We ﬁrst observe that j W i.X _ Y /! i.X Y /, induced by the inclusion, is always surjective. The projections onto the factors induce isomorphisms k W i.X Y / Š i.X/ i.Y /. Let j X W X! X _ Y and j Y W Y! X _ Y denote the inclusions. Let s W i.X/ i.Y /! i.X _ Y /;.x; y/ 7! j X.x/ C j Y.y/: Then sk is right inverse to j. Hence the exact sequence of the pair.X Y; X _ Y / yields an exact sequence./ 0! iC1.X Y; X _ Y /! i.X _ Y /! i.X Y /! 0: 6.10. Further Applications of Excision 155 In the case that i 2, the sequence splits, since we are then working with abelian groups; hence i.X _ Y / Š i.X/ ˚ i.Y / ˚ iC1.X Y; X _ Y /; i 2: Since the spaces are well-pointed, we can apply the theorem of Seifert–van Kampen to.X _ Y; X; Y / and see that 1.X _ Y / D 0. We now consider the diagram i.X _ Y; Y /.1/ i.X/ i.X _ Y / i.X _ Y; X/.2/ i.Y / with exact row and column. The diagonal arrows are always injective and split; this is seen by composing with the projections. Since the spaces are well-pointed, we can apply the pushout excision to the triad.X _ Y; X; Y; /. It says that (1) and (2) are sur
jective for i p C q 2, and hence bijective (since we already know the injectivity). We now apply the Sum Lemma (11.1.2) to the diagram and conclude that hj X ; j Y i is an isomorphism, and therefore also the map of the theorem is an isomorphism. The exact sequence now yields i.X Y; X _Y / D 0 for i p Cq 1. We apply (6.10.2) to i.X Y; X _ Y /! i.X ^ Y /. By what we have already proved, we can apply this theorem with the data n D p C q 1 and m D min.p 1; q 1/. We also need that X _ Y! X Y is a coﬁbration. This is a consequence of the product theorem for coﬁbrations. (6.10.6) Proposition. Let.Yj j j 2 J / be the family of path components of Y and cj W Yj! Y the inclusion. Then hcj i W L j 2J k.Y C j ^ S n/! k.Y C ^ S n/ is an isomorphism for k n. W Proof. Suppose Y is the topological sum of its path components. Then we have j 2J Y C a homeomorphism, and the assertion follows for ﬁnite J by induction on the cardinality of J from (6.10.5) and for general J then by a compactness argument. For general Y it sufﬁces to ﬁnd a 1-connected map X! Y such that X is the topological sum of its path components, because then is.n C 1/-connected by (6.7.10) (and similarly for the path components). 156 Chapter 6. Homotopy Groups (6.10.7) Proposition. Let Y be k-connected.k 0/ and Z be l-connected.l 1/ and well-pointed. Then the natural maps j.Z/! j.Y Z; Y /! j.Y Z=Y /! j.Z/ are isomorphisms for 0 < j k C l C 1. Proof. The ﬁrst map is always bijective for j 1; this is a consequence of the exact sequence of the pair.Y Z; Y / and the
isomorphism j.Y / j.Z/ Š j.Y Z/. Since the composition of the maps is the identity, we see that the second map is always injective and the third one surjective. Thus if p W j.Y Z; Y /! j.Y Z=Y is surjective, then all maps are bijective. From our assumption about Z we conclude that j.Y Z; Y / D 0 for 0 < j l (thus there is no condition for l D 0; 1). The quotient theorem now tells us that p is surjective for 0 < j k C l C 1. (6.10.8) Corollary. Let Y be path connected. The natural maps k.S n/! k.Y S n; S n/! k.Y S n= S n/! k.S n/ are isomorphisms for 1 k n. (6.10.9) Proposition. Suppose i.X/ D 0 for i < p. 0/ and i.Y / D 0 for i < q. 0/. Then i.X? Y / D 0 for i < p C q C 1. Proof. In the case that p D 0 there is no condition on X. From the deﬁnition of the join we see that X? Y is always path connected. For p D 0 we claim that i.X? Y / D 0 for i < q C 1. Consider the diagram pr X X X Y pr X pr Y fg and apply (6.7.9). In the general case the excision theorem says that the map i.CX Y; X Y /! i.X? Y; X C Y / is an epimorphism for i < p C q C 1. Now use diagram chasing in the diagram i.X C Y / i.X? Y / i.X? Y; X C Y / i1.X C Y / i.X Y / i.CX Y / i.CX Y; X Y / i1.X Y / (a morphism between exact homotopy sequences). 6.10. Further Applications of Excision 157 The excision theorem in the formulation of (6.9.4) has a dual. Suppose given a pullback diagram E G Y F g X f B with ﬁbrations f and g. The double mapping cylinder
Z.F; G/ can be considered as the ﬁbrewise join of f and g. It has a canonical map W Z.F; G/! B. (6.10.10) Proposition. Suppose f is p-connected and g is q-connected. Then is p C q C 1-connected. Proof. Use ﬁbre sequences and (6.10.9). Problems 1. Let 0.X/ D 0 and i.Y / D 0 for i < q. 2/. Then i.X/! i.X _ Y / is an isomorphism for i < q. Show also 2.X Y; X _ Y / D 0. 2. Let X and Y be 0-connected and well-pointed. Show 1.X ^ Y / D 0. 3. Show that 3.D2; S 1/! 3.D2=S 1/ is not surjective. 4. Show 1.S 2_S 1; S 1/ D 0. Show that 2.S 2_S 1; S 1/! 2.S 2_S 1=S 1/ Š 2.S 2/ is surjective but not injective. 5. For X D Y D S 1 and i D 1 the sequence./ does not split. The fundamental group 1.S 1 _ S 1/ D Z Z has no subgroup isomorphic to Z ˚ Z. 6. Show that the diagram X @I [ fg I p \ X I X i CX with p.x; 0/ D x; p.x; 1/ D ; p.; t/ D is a pushout. 7. If X is well-pointed, then †X is well-pointed. 8. Some hypothesis like e.g. well-pointed is necessary in both (6.10.1) and (6.10.4). Let A D f0g [ fn1 j n 2 Zg and A D A 0 X D A I =A 1 with base point.0; 0/. Then 1.†X/ and 1.A/ are uncountable; † W 0.X/! 1.†X/ is not surjective. Note: A X is a coﬁbration and X=A is well-pointed. 9. Let e1; : : : ; enC1 be the standard
basis of unit vectors in RnC1, and let e1 be the base point of S n. A pointed homeomorphism hn W †S n Š S nC1 is hn W †S n! S nC1;.x; t/ 7! 1 2.e1 C x/ C 1 2 cos 2 t.e1 x/ C 1 2 je1 xj sin 2 t enC2 where RnC1 D RnC1 0 RnC2. 10. Let K RnC1 be compact. Show that each map f W K! S n has an extension to the complement RnC1 X E of a ﬁnite set E. One can choose E such that each component of RnC1 X K contains at most one point of E. 158 Chapter 6. Homotopy Groups 11. Determine 2n1.S n _ S n/ for n 2. 12. Let fj be a self-map of S n.j /. Show d.f1? f2/ D d.f1/d.f2/. 13. Let H W 3.S 2/! Z be the isomorphism which sends (the class of the) Hopf map W S 3! S 2 to 1 (the Hopf invariant). Show that for f W S 3! S 3 and g W S 2! S 2 the relations H.˛ ı f / D d.f /H.˛/ and H.g ı ˛/ D d.g/2H.˛/ hold. Chapter 7 Stable Homotopy. Duality The suspension theorem of Freudenthal indicates that homotopy theory simpliﬁes by use of iterated suspensions. We use this idea to construct the simplest stable homotopy category. Its construction does not need extensive technical considerations, yet it has interesting applications. The term “stable” refers to the fact that iteration of suspension induces after a while a bijection of homotopy classes. We use the stable category to give an introduction to homotopical duality theory. In this theory the stable homotopy type of a closed subspace X Rn and its complement Rn X X are compared. This elementary treatment of duality theory is based on ideas of Albrecht Dold and Dieter Puppe; see in particular [54]. It is related
to the classical Alexander duality of homology theory and to Spanier–Whitehead duality. We introduced a naive form of spectra and us them to deﬁne spectral homology and cohomology theories. The homotopical Euclidean complement duality is then used to give a simple proof for the Alexander duality isomorphism. In a later chapter we reconsider duality theory in the context of product structures. 7.1 A Stable Category Pointed spaces X and Y are called stably homotopy equivalent, in symbols X's Y, if there exists an integer k 0 such that the suspensions †kX and †kY are homotopy equivalent. Pointed maps f; g W X! Y are called stably homotopic, in symbols f's g, if for some integer k the suspensions †kf and †kg are homotopic. We state some of the results to be proved in this chapter which use these notions. (7.1.1) Theorem (Stable Complement Theorem). Let X and Y be homeomorphic closed subsets of the Euclidean space Rn. Then the complements Rn XX and Rn XY are either both empty or they have the same stable homotopy type with respect to arbitrary base points. In general the complements themselves can have quite different homotopy type. A typical example occurs in knot theory, the case that X Š Y Š S 1 are subsets of R3. On the other hand the stable homotopy type still carries some interesting geometric information: see (7.1.10). (7.1.2) Theorem (Component Theorem). Let X and Y be closed homeomorphic subsets of Rn. Then 0.Rn X X/ and 0.Rn X Y / have the same cardinality. 160 Chapter 7. Stable Homotopy. Duality Later we give another proof of Theorem (7.1.2) based on homology theory, see (10.3.3). From the component theorem one can deduce classical results: The Jordan separation theorem (10.3.4) and the invariance of domain (10.3.7). Theorem (7.1.1) is a direct consequence of (7.1.3). One can also compare complements in different Euclidean spaces. The next result gives some information about how many suspensions
sufﬁce. (7.1.3) Theorem. Let X Rn and Y Rm be closed subsets and h W X! Y a homeomorphism. Suppose n m. Then the following holds: (1) If Rn 6D X, then Rm 6D Y, and h induces a canonical homotopy equivalence †mC1.Rn X X/'†nC1.Rm X Y / with respect to arbitrary base points. (2) If Rn D X and Rm 6D Y, then n < m and †nC1.Rm XY /'S m, i.e., Rm XY has the stable homotopy type of S mn1. (3) If Rn D X and Rm D Y, then n D m. In many cases the number of suspensions is not important. Since it also depends on the situation, it is convenient to pass from homotopy classes to stable homotopy classes. This idea leads to the simplest stable category. The objects of our new category ST are pairs.X; n/ of pointed spaces X and integers n 2 Z. The consideration of pairs is a technical device which allows for a better formulation of some results. Thus we should comment on it right now. The pair.X; 0/ will be identiﬁed with X. The subcategory of the objects.X; 0/ D X with morphisms the so-called stable homotopy classes is the geometric input. For positive n the pair.X; n/ replaces the n-fold suspension †nX. But it will be convenient to have the object.X; n/ also for negative n (“desuspension”). Here is an interesting example. In the situation of (7.1.3) the homotopy equivalence †mC1.Rn X X/! †nC1.Rm X Y / induced by h represents in the category ST an isomorphism h W.Rn X X; n/!.Rm X Y; m/. In this formulation it then makes sense to say that the assignment h 7! h is functor. (Otherwise we would have to use a mess of different suspensions.) Thus if X is a space which admits an embedding i W X! Rn as a proper closed subset for some n, then the isomorph
ism type of.Rn X i.X/; n/ in ST is independent of the choice of the embedding. Hence we have associated to X a “dual object” in ST (up to canonical isomorphism). Let †t X D X ^ S t be the t-fold suspension of X. As a model for the sphere S t we use either the one-point compactiﬁcation Rt [ f1g or the quotient space S.t/ D I t =@I t. In these cases we have a canonical associative homeomorphism S a ^ S b Š S aCb which we usually treat as identity. Suppose n; m; k 2 Z are integers such that n C k 0; m C k 0. Then we have the suspension morphism † W ŒX ^S nCk; Y ^S mCk0! ŒX ^S nCkC1; Y ^S mCkC10; f 7! f ^id.S 1/: We form the colimit over these morphisms, colimkŒX ^ S nCk; Y ^ S mCk0. For n C k 2 the set ŒX ^ S nCk; Y ^ S mCk0 carries the structure of an abelian group 7.1. A Stable Category 161 and † is a homomorphism. The colimit inherits the structure of an abelian group. We deﬁne as morphism group in our category ST ST..X; n/;.Y; m// D colimkŒX ^ S nCk; Y ^ S mCk0: Formation of the colimit means the following: An element of ST..X; n/;.Y; m// is represented by pointed maps fk W X ^ S nCk! Y ^ S mCk, and fk, fl ; l k represent the same element of the colimit if †lkfk'fl. Composition of morphisms is deﬁned by composition of representatives. Let fk W X ^ S nCk! Y ^ S mCk and gl W Y ^ S mCl! Z ^ S pCl be representatives of morphisms and let r k; l. Then the following composition of maps represents the composition of the morph
isms (dotted arrow): X ^ S nCr D X ^ S nCk ^ S rk Z ^ S pCr D Z ^ S pCl ^ S rl †rk fk †rl gl Y ^ S mCk ^ S rk D Y ^ S mCl ^ S rl : One veriﬁes that this deﬁnition does not depend on the choice of representatives. The group structure is compatible with the composition ˇ ı.˛1 C ˛2/ D ˇ ı ˛1 C ˇ ı ˛2;.ˇ1 C ˇ2/ ı ˛ D ˇ1 ı ˛ C ˇ2 ı ˛: The category ST has formal suspension automorphisms †p W ST! ST, p 2 Z.X; n/ 7!.X; n C p/; f 7! †pf: If f W.X; n/!.Y; m/ is represented by fk W †nCkX! †mCkY (with n C k 0, m C k 0, k jpj), then †pf is represented by.†pf /k D †p.fk/ W †nCkCpX! †mCkCpY; p 0;.†pf /kCjpj D fk W †nCkCpCjpjX! †mCkCpCjpjY; p 0: The rules †0 D id.ST/ and †p ı †q D †pCq show that †p is an automorphism. For p > 0 we call †p the p-fold suspension and for p < 0 the p-fold desuspension. We have a canonical isomorphism p W.X; n/!.†pX; n p/; it is represented by the identity X ^ S nCk!.X ^ S p/ ^ S nCkp for n C k p 0. We write X for the object.X; 0/. Thus for positive n the object.X; n/ can be replaced by †nX. (7.1.4) Example. Pointed spaces
X; Y are stably homotopy equivalent if and only if they are isomorphic in ST. The image ST.f / of f W X! Y in ST.X; Y / is called the stable homotopy class of f. Maps f; g W X! Y are stably homotopic if and only if they represent the same element in ST.X; Y /. The groups ST.S k; S 0/ D Þ colimn nCk.S n/ are the stable homotopy groups of the spheres. 162 Chapter 7. Stable Homotopy. Duality (7.1.5) Example. It is in general difﬁcult to determine morphism groups in ST. But we know that the category in non-trivial. The suspension theorem and the degree theorem yield ST.S n; S n/ D colimkŒS nCk; S nCk0 Š Z: The composition of morphisms corresponds to multiplication of integers. Þ (7.1.6) Proposition. Let Y be pathwise connected. We have the embedding, x 7!.; x/ and the projection,.y; x/ 7! x with pi D id. They induce isomorphisms of the k-groups for k n 1. Proof. Let n D 1. Then Y C ^ S 1 Š Y S 1=Y fg is path connected. The base point of Y C is non-degenerate. Hence the quotient †0.Y C/! †.Y C/ from the unreduced suspension to the reduced suspension is an h-equivalence. The projection Y! P onto a point induces a 2-connected map between double mapping cylinders †0.Y C/ D Z. Y C fg! /! Z0. P C fg! / D †0.P C/ Š S 1: From this fact one deduces the assertion for n D 1. We now consider suspensions k.S n/ † kC1.S nC1/ i i k.Y C ^ S n/ p †Y kC1.Y C ^ S nC1/ p k.S n/ † kC1.S nC1/: The vertical morphisms are bijective (surjective) for k 2n 2 (k D 2n
1). For n D 1 1.Y C ^ S 1/ Š Z. Since 2.Y C ^ S 2/ contains 2.S 2/ Š Z as a direct summand, we conclude that †Y is an isomorphism. For n 2 we can use directly the suspension theorem (6.10.4). (7.1.7) Proposition. Let.Yj j j 2 J / be the family of path components of Y and cj W Yj! Y the inclusion. Let n 2. Then hcj i W L j 2J k.Y C j ^ S n/! k.Y C ^ S n/ is an isomorphism for 0 k n. In particular n.Y C ^ S n/ is a free abelian group of rank j0.Y /j. Proof. (7.1.6) and (6.10.6). (7.1.8) Proposition. Let Y be well-pointed and n 2. Then n.Y ^ S n/ is a free abelian group of rank j0.Y /j 1 and the suspension n.Y ^ S n/! nC1.Y ^ S nC1/ is an isomorphism. 7.1. A Stable Category 163 From the exact homotopy sequence of the pair.Y C ^ S n; S n/ we conclude that k.Y C ^ S n; S n/ D 0 for 0 < k < n. The quotient theorem (6.10.2) shows that k.Y C ^ S n; S n/! k.Y C ^ S n=S n/ Š k.Y ^ S n/ is bijective (surjective) for 0 < k 2n 2 (k D 2n 1). From the exact sequence 0! n.S n/! n.Y C ^S n/! n.Y C ^S n; S n/! 0 we deduce a similar exact sequence where the relative group is replaced by n.Y ^ S n/. The inclusion of n.S n/ splits. Now we can use (7.1.7). (7.1.9) Corollary. Let X be a well-pointed space. Then ST.S 0; X/ is a free abelian Þ group of rank j0.X/j 1. The group
ST.S 0; X/ only depends on the stable homotopy type of X. Therefore we can state: (7.1.10) Corollary. Let X and Y be well-pointed spaces of the same stable homotopy type. Then j0.X/j D j0.Y /j. Therefore (7.1.2) is a consequence of (7.1.3). Þ The category ST has a “product structure” induced by the smash product. The category ST together with this additional structure is called in category theory a symmetric tensor category (also called a symmetric monoidal category). The tensor product of objects is deﬁned by.X; m/ ˝.Y; n/ D.X ^ Y; m C n/: Let fk W X ^ S mCk! X 0 ^ S m0Ck; gl W Y ^ S nCl! Y 0 ^ S n0Cl be representing maps for morphisms f W.X; m/!.X 0; m0/ and g W.Y; n/!.Y 0; n0/. A representing morphism.f ˝ g/kCl is deﬁned to be.1/k.nCn0/ times the composition 0 ı.fk ^ gl / ı (dotted arrow) X ^ Y ^ S mCkCnCl X 0 ^ Y 0 ^ S m0CkCn0Cl 0 X ^ S mCk ^ Y ^ S nCl fk ^gl X 0 ^ S m0Ck ^ Y 0 ^ S n0Cl where and 0 interchange two factors in the middle. Now one has to verify: (1) The deﬁnition does not depend on the representatives; (2) the functor property.f 0 ˝ g0/.f ˝ g/ D f 0f ˝ g0g holds; (3) the tensor product is associative. These requirements make it necessary to introduce signs in the deﬁnition. The neutral object is.S 0; 0/. The symmetry c W.X; m/ ˝.Y; n/!.Y; n/ ˝.X; m/ is.1/mn times the morphism represented by the interchange map X ^ Y! Y ^ X
. 164 Chapter 7. Stable Homotopy. Duality Problems 1. The spaces S 1 S 1 and S 1 _ S 1 _ S 2 are not homotopy equivalent. They have different fundamental group. Their suspensions are homotopy equivalent. 2. The inclusion X Y! X C Y induces for each pointed space Y a homeomorphism.X Y /=.X fg/. 3. Let X and Y be well-pointed spaces. Then Y!.X Y /=.X fg/, y 7!.; y/ is a coﬁbration. 4. Let P be a point. We have an embedding P C ^ Y! X C ^ Y and a canonical homeomorphism X ^ Y! X C ^ Y =P C ^ Y. 7.2 Mapping Cones We need a few technical results about mapping cones. Let f W X! Y be a pointed map. We use as a model for the (unpointed) mapping cone C.f / the double mapping cylinder Z.Y X! /; it is the quotient of Y C X I C fg under the relations f.x/.x; 0/;.x; 1/. The image of is the basepoint. For an inclusion W A X we write C.X; A/ D C./. For empty A we have C.X; ;/ D X C. Since we will meet situations where products of quotient maps occur, we work in the category of compactly generated spaces where such products are again quotient maps. The mapping cone is a functor C W TOP.2/! TOP0; a map of pairs.F; f / W.X; A/!.Y; B/ induces a pointed map C.F; f / W C.X; A/! C.Y; B/, and a homotopy.Ft ; ft / induces a pointed homotopy C.Ft ; ft /. We note for further use a consequence of (4.2.1): (7.2.1) Proposition. If F and f are h-equivalences, then C.F; f / is a pointed h-equivalence. (7.2.2) Example. We write C n D C.Rn; Rn X 0/. This space will be our model for the homotopy type of S n. In order to get
a homotopy equivalence C n! S n, we observe that S n is homeomorphic to the double mapping cylinder Z. S n1! /. We have the canonical projection from C n D Z.Rn Rn X 0! /. An explicit Þ homotopy equivalence is.x; t/ 7!.sin t x kxk ; cos t /, x 7!.0; : : : ; 0; 1/. (7.2.3) Example. Let X Rn be a closed subspace. Then C.Rn; RnXX/ D Z.Rn RnXX! /'Z. R nXX! / D †0.RnXX/; the unpointed suspension. If X D Rn, then this space is h-equivalent to S 0. If X 6D Rn, then Rn X X is well-pointed with respect to any point and †0.Rn X X/ is Þ h-equivalent to the pointed suspension †.Rn X X/. We are mainly interested in the homotopy type of C.X; A/ (under fg C X). It is sometimes convenient to provide the set C.X; A/ with a possibly different 7.2. Mapping Cones 165 topology which does not change the homotopy type. Set theoretically we can view C.X; A/ as the quotients C1.X; A/ D.X 0 [ A I /=A 1 or C2.X; A/ D.X 0 [ A I [ X 1/=X 1. We can provide C1 and C2 with the quotient topology. Then we have canonical continuous maps p W C.X; A/! C1.X; A/ and q W C1.X; A/! C2.X; A/ which are the identity on representative elements. (7.2.4) Lemma. The maps p and q are homotopy equivalences under fg C X. Proof. Deﬁne Np W C1.X; A/! C.X; A/ by Np.x; t/ D x; t 1=2; Np.a; t/ D.a; max.2t 1; 0//; Np.a; 1/ D : One veri�
�es that this assignment is well-deﬁned and continuous. A homotopy p Np'id is given by..x; t/; s/ 7!.x; st C.1 s/ max.2t 1; 0//. A similar formula works for Npp'id. Deﬁne Nq W C2.X; A/! C1.X; A/ by Nq.x; t / D.x; min.2t; 1//; t < 1; Nq.x; t/ D D fA 1g; t 1=2: Again linear homotopies in the t-coordinate yield homotopies from q Nq and Nqq to the identity. (7.2.5) Proposition (Excision). Let U A X and suppose there exists a function W X! I such that U 1.0/ and 1Œ0; 1Œ A. Then the inclusion of pairs induces a pointed h-equivalence g W C.X X U; A X U /! C.X; A/. Proof. Set.x/ D max.2.x/ 1; 0/. A homotopy inverse of g is the map f W.x; t/ 7!.x;.x/t/. The deﬁnition of f uses the notation C2 for the mapping cone. The homotopies from fg and gf to the identity are obtained by a linear homotopy in the t-coordinate. (7.2.6) Remark. Mapping cones of inclusions are used at various occasions to relate the category TOP.2/ of pairs with the category TOP0 of pointed spaces. We make some general remarks which concern the relations. They will be relevant for the investigation of homology and cohomology theories. Let Qh W TOP0! C be a homotopy invariant functor. We deﬁne an associated functor h D P Qh W TOP.2/! C by composition with the mapping cone functor.X; A/ 7! C.X; A/. The functor P Qh is homotopy invariant in a stronger sense: If f W.X; A/!.Y; B/ is a map of pairs such that the components f W X! Y and f W A! B are h-equival
ences, then the induced map h.X; A/! h.Y; B/ is an isomorphism (see (7.2.1)). Moreover h satisﬁes excision: Under the hypothesis of (7.2.5) the inclusion induces an isomorphism h.X X U; A X U / Š h.X; A/. Conversely, let h W TOP.2/! C be a functor. We deﬁne an associated functor Rh D Qh on objects by Rh.X/ D h.X; / and with the obvious induced morphisms. If h is homotopy invariant, then also Rh. 166 Chapter 7. Stable Homotopy. Duality The composition PR is given by PRh.X; A/ D h.C.X; A/; /. We have natural morphisms h.C.X; A/; /! h.C.X; A/; CA/ h.C.X; A/ X U; CA X U / h.X; A/: Here CA is the cone on A and U CA is the subspace with t-coordinates in Œ1=2; 1. If h is strongly homotopy invariant and satisﬁes excision, then these morphisms are isomorphisms, i.e., PR is naturally isomorphic to the identity. The composition RP is given by RP Qh.X/ D Qh.C.X; //. There is a canonical projection C.X; /! X. It is a pointed h-equivalence, if the inclusion fg! X is a coﬁbration. Thus if Qh is homotopy invariant, the composition RP is naturally Þ isomorphic to the identity on the subcategory of well-pointed spaces. Let.X; A/ and.Y; B/ be two pairs. We call A Y; X B excisive in X Y if the canonical map p W Z.A Y A B! X B/! A Y [ X B is a homotopy equivalence. (7.2.7) Proposition (Products). Let.A Y; X B/ be excisive. Then there exists a natural pointed homotopy equivalence ˛ W C.X; A/ ^ C.Y; B/! C..X; A/.
Y; B//: It is deﬁned by the assignments.x; y/ 7!.x; y/;.a; s; y/ 7!.a; y; s/;.x; b; s/ 7!.x; b; s/;.a; s; b; t/ 7!.a; b; max.s; t//: (See the proof for an explanation of notation). Proof. In the category of compactly generated spaces C.X; A/ ^ C.Y; B/ is a quotient of under the following relations:.a; 0; y/.a; y/,.x; b; 0/.x; b/,.a; 0; b; t/.a; b; t/,.a; s; b; 0/.a; s; b/, and A 1 B I [ A I B 1 is identiﬁed to a base point. In a ﬁrst step we show that the smash product is homeomorphic to the double mapping cylinder Z.X Y Z! fg/ where p Z D Z.A Y A B! X B/: This space is the quotient of X Y C.A Y C A B I C X B/ I C A B.I I =I 0/ 7.2. Mapping Cones 167 under the following relations:.x; b; 0/.x; b/,.a; y; 0/.a; y/,.a; b; t; 0/.a; b/,.a; b; 1; s/.a; b; s/, and.A Y C A B I C X B/ 1 is identiﬁed to a base point. The assignment I I! I I;.u; v/ 7! (.2uv; v/; u 1=2;.v; 2v.1 u//; u 1=2: induces a homeomorphism 0 W I I =.I 0/! I I. Its inverse ˇ0 has the form I I X f.0; 0/g!I.I X f0g/;.s; t / 7! (.1 t=2s; s/;.s=2t; t/; s t; s t: A homeomorphism ˇ W C.
X; A/ ^ C.Y; B/! C.p/ is now deﬁned by ˇ.x; y/ D.x; y/, ˇ.a; s; y/ D.a; y; s/, ˇ.x; b; t/ D.x; b; t/, ˇ.a; s; b; t/ D.a; b; ˇ0.s; t//. The diagram fg D fg induces W C.p/! C./. It is a pointed h-equivalence if p is an h-equivalence. One veriﬁes that ˛ D ˇ. (7.2.8) Remark. The maps ˛ are associative: For three pairs.X; A/;.Y; B/;.Z; C / the relation ˛.˛ ^ id/ D ˛.id ^˛/ holds. They are also compatible with the Þ interchange map. Finally, they yield a natural transformation. Problems 1. Verify that the map f in the proof of 7.2.5 is continuous. Similar problem for the homotopies. 2. Let.F; f / W.X; A/!.Y; B/ be a map of pairs. If F is n-connected and f.n 1/connected, then C.F; f / is n-connected. 3. Let X D A [ B and suppose that the interiors Aı; B ı still cover X. Then the inclusion induces a weak homotopy equivalence C.B; A \ B/! C.X; A/. 4. Construct explicit h-equivalences C n! Rn [ f1g D S.n/ such that C m ^ C n ˛ C mCn S.m/ ^ S.n/ Š S.mCn/ is homotopy commutative. 168 Chapter 7. Stable Homotopy. Duality 7.3 Euclidean Complements This section is devoted to the proof of (7.1.3). We need an interesting result from general topology. (7.3.1) Proposition. Let A Rm and B Rn be closed subsets and let f W A! B be a homeomorphism. Then there exists
a homeomorphism of pairs F W.Rm Rn; A 0/!.Rn Rm; B 0/ such that F.a; 0/ D.f.a/; 0/ for a 2 A. Proof. By the extension theorem of Tietze (1.1.2) there exists a continuous extension'W Rm! Rn of f W A! B Rn. The maps ˆ˙ W Rm Rn! Rm Rn;.x; y/ 7!.x; y ˙ '.x// are inverse homeomorphisms. Let G.f / D f.a; f.a// j a 2 Ag denote the graph of f. Then ˆC sends A 0 homeomorphically to G.f / by.a; 0/ 7!.a; f.a//. Let W Rn! Rm be a Tietze extension of the inverse g of f. Then we have similar homeomorphisms ‰˙ W Rn Rm! Rn Rm;.y; x/ 7!.y; x ˙.y//: The desired homeomorphism F is the composition ‰ı ıˆC where interchanges Rm and Rn (and sends G.f / to G.g/). Let X Rn and Y Rm be closed subsets and f W X! Y a homeomorphism. The induced homeomorphism F from (7.3.1) can be written as a homeomorphism F W.Rn; Rn X X/.Rm; Rm X 0/!.Rm; Rm X Y /.Rn; Rn X 0/: We apply the mapping cone functor to F and use (7.2.2) and (7.2.7). The result is a homotopy equivalence C.Rn; Rn X X/ ^ S m'C.Rm; Rm X Y / ^ S n: If X 6D Rm and Y 6D Rm we obtain together with (7.2.3) †mC1.Rn X X/'†nC1.Rm X Y /: If X 6D Rn then we have C.Rn; Rn X X/'†.Rn X X/, and if X D Rn
then we have C.Rn; Rn X X/'S 0. Suppose X 6D Rn but Y D Rm. Then †mC1.Rn X X/'S n. Since n m the homotopy group n.†mC1.Rn X X// D 0 and n.S n/ Š Z. This contradiction shows that Y 6D Rm. 7.4. The Complement Duality Functor 169 Suppose X D Rn and Y 6D Rm. Then n D m is excluded by the previous proof. Thus S m'C.Rn; Rn X X/ ^ S m'C.Rm; Rm X Y / ^ S n'†mC1.Rm X Y /: If X D Rn and Y D Rm, then S n'†nC.Rn; Rn X X/'†mC.Rm; Rm X Y /'S m and therefore m D n. This ﬁnishes the proof of (7.1.3). 7.4 The Complement Duality Functor The complement duality functor is concerned with the stable homotopy type of Euclidean complements Rn X X for closed subsets X Rn. We consider an associated category E. The objects are pairs.Rn; X/ where X is closed in Rn. A morphism.Rn; X/!.Rm; Y / is a proper map f W X! Y. The duality functor is a contravariant functor D W E! ST which assigns to.Rn; X/ the object †nC.Rn; Rn X X/ D.C.Rn; Rn X X/; n/. The associated morphism D.f / W †mC.Rm; Rm X Y /! †nC.Rn; Rn X X/ will be constructed via a representing morphism D.f /mCn. Its construction needs some preparation. Given the data X Rn; Y Rm and a proper map f W X! Y. Henceforth we use the notation AjB D.A; A X B/ for pairs B A. Note that in this notation AjB C jD D A BjC D. The basic step in the construction of the functor will be
an associated homotopy class D#f W RnjDn RmjY! RnjX Rmj0: Here Dn again denotes the n-dimensional standard disk. A scaling function for a proper map f W X! Y is a continuous function'W Y! 0; 1Œ with the property '.f.x// kxk; x 2 X: The next lemma shows the existence of scaling functions with an additional property. (7.4.1) Lemma. There exists a positive continuous function W Œ0; 1Œ! 0; 1Œ such that the inequality.kf xk/ kxk holds for x 2 X. A scaling function in the sense of the deﬁnition is then y 7!.kyk/. Proof. The set f 1D.t/ D maxfx 2 X j kf xk t g is compact, since f is proper. Let Q.t/ be its norm maximum maxfkxk jx 2 X; kf xk t g. Then Q.kf xk/ D maxfkak ja 2 X; kf ak kf xkg kxk. The function Q W Œ0; 1Œ! Œ0; 1Œ is increasing. There exists a continuous increasing function W Œ0; 1Œ! 0; 1Œ such that.t/ Q.t/ for each t 0. 170 Chapter 7. Stable Homotopy. Duality The set of scaling functions is a positive convex cone. Let '1; '2 be scaling functions and 0 1; then '1 C.1 /'2 is a scaling function. Let Q' '; if'is a scaling function then also Q'. Let'be a scaling function and set M.'/ D f.x; y/ j '.y/ kxkg. Then we have a homeomorphism RnCmjDn Y! RnCmjM.'/;.x; y/ 7!.'.y/ x; y/: The graph G.f / D f.x; f x/ j x 2 Xg of f is contained in M.'/. We thus can continue with the inclusion and obtain a map D1.f; '/ of pairs RnCmjDn Y! RnCmjG.f /;.x; y
/ 7!.'.y/ x; y/: The homotopy class of D1.f; '/ does not depend on the choice of the scaling function: If '1; '2 are scaling functions, then.x; y; t/ 7!..t'1.y/ C.1 t/'2.y// x; y/ is a homotopy from D1.f; '2/ to D1.f; '1/. A continuous map f W X! Y has a Tietze extension Qf W Rn! Rm. The homeomorphism.x; y/ 7!.x; y Qf.x// of RnCm sends.x; f.x// to.x; 0/. We obtain a homeomorphism of pairs D2.f; Qf / W RnCmjG.f /! RnCmjX 0: The homotopy class is independent of the choice of the Tietze extension: The homotopy.x; y; t/ 7!.x; y.1 t/ Qf1.x/ t Qf2.x// proves this assertion. The duality functor will be based on the composition D#.f / D D2.f; Qf / ı D1.f; '/ W RnjDn RmjY! RnjX Rmj0: We have written D#.f /, since the homotopy class is independent of the choice of the scaling function and the Tietze extension. The morphism Df W †mC.RmjY /! †nC.RnjX/ is deﬁned by a representative of the colimit:.Df /nCm W †nC.RmjY /! †mC.RnjX/: Consider the composition C.RmjY / ^ C n C n ^ C.RmjY /'C.RnjDn/ ^ C.RmjY /.1/nm.Df /nCm ˛ C.RnjX/ ^ C m ˛1 C.RnjX Rmj0/ CD#f C.RnjDn RmjY /: Explanation. interchanges the factors; the

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