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it of a covering to the colimit of the covering; see (13.2.4) for a result of this type. There are many other results of this type in the literature. This chapter only can serve as an introduction to this topic. 13.1 Partitions of Unity Let t W X! R be continuous. The closure of t 1.R X 0/ is the support supp.t / of t. A family T D.tj W X! R j j 2 J / of continuous functions is said to be locally ﬁnite if the family of supports.supp.tj / j j 2 J / is locally ﬁnite. We call T point ﬁnite if fj 2 J j tj.x/ ¤ 0g is a ﬁnite set for each x 2 X. We call a locally ﬁnite T a partition of unity if the tj assume only non-negative values and j 2J tj.x/ D 1. A covering U D.Uj j j 2 J / is if for each x 2 X we have numerable if there exists a partition of unity T such that supp.tj / Uj holds for each j 2 J ; the family T is then called a numeration of U or a partition of unity subordinate to U. P (13.1.1) Theorem. A locally ﬁnite open covering of a normal space is numerable. Proof. Let U D.Uj j j 2 J / be a locally ﬁnite open covering of the normal space X and V D.Vj j j 2 J / a shrinking of U and W D.Wj j j 2 J / a shrinking of V. By the theorem of Urysohn, there exist continuous functions j W X! Œ0; 1 which assume the value 1 on Wj and the value 0 on the complement of Vj. The function D j 2J j W X! Œ0; 1 is well-deﬁned and continuous, since by local ﬁniteness of V, in a suitable neighbourhood of a point only a ﬁnite number of j are non-zero. We set fj.x/ D j.x/.x/1. The functions.fj j j 2 J / are a numeration of U. P (13.1.2)
Lemma. Let the covering V D.Vk j k 2 K/ be a reﬁnement of the covering U D.Uj j j 2 J /. If V is numerable, then also U is numerable. 13.1. Partitions of Unity 319 Proof. Let.fk j k 2 K/ be a numeration of V. For each k 2 K choose a.k/ 2 J with Vk Ua.k/. This deﬁnes a map a W K! J. We set gj.x/ D P k;a.k/Dj fk.x/; this is the zero function if the sum is empty. Then gj is continuous; the support of gj is contained in the union of the supports of the fk with a.k/ D j and is therefore contained in Uj. Moreover, the sum of the gj is 1. The family.gj j j 2 J / is locally ﬁnite: If W is an open neighbourhood of x which meets only a ﬁnite number of supports supp.fk/, k 2 E K, E ﬁnite, then W meets only the supports of the gj with j 2 a.E/. (13.1.3) Theorem. Each open covering of a paracompact space is numerable. Proof. Let U D.Uj j j 2 J / be an open covering of the paracompact space X and let V D.Vk j k 2 K/ be a locally ﬁnite reﬁnement. Since X is normal, there exists a numeration.fk j k 2 K/ of V. Now apply the previous lemma. (13.1.4) Lemma. Let.fj W X! Œ0; 1Œ j j 2 J / be a family of continuous functions such that U D.f 10; 1Œ j j 2 J / is a locally ﬁnite covering of X. Then U is numerable and has, in particular, a shrinking. Proof. Since U is locally ﬁnite, f W x 7! max.fj.x/ j j 2 J / is continuous and nowhere zero. We set gj.x/ D fj.x/f.x/1. Then tj W X! Œ0
; 1; x 7! max.2gj.x/ 1; 0/ is continuous. Since tj.x/ > 0 if and only if gj.x/ > 1=2, we have the inclusions supp.tj / g1 Œ1=4; 1Œ f 10; 1Œ. For x 2 X and i 2 J with fi.x/ D max.fj.x// we have ti.x/ D 1. Hence the supports of the tj form a locally ﬁnite j 2J tj.x/ are a covering of X, and the functions x 7! ti.x/=t.x/, t.x/ D numeration of U. P j (13.1.5) Theorem. Let U D.Uj j j 2 J / be a covering of the space X. The following assertions are equivalent: (1) U is numerable. (2) There exists a family.sa;n W X! Œ0; 1Œ j a 2 A; n 2 N/ D S of continuous functions sa;n with the properties: (a) S, i.e.,.s1 (b) For each n the family.s1 a;n0; 1Œ j a 2 A/ is locally ﬁnite. (c) For each x 2 X there exists.a; n/ such that sa;n.x/ > 0. a;n 0;1 Œ/, reﬁnes U. Proof. (1) ) (2) is clear. (2) ) (1)..sa;n/ is, by assumption, a countable union of locally ﬁnite families. From these data we construct a locally ﬁnite family. By replacing sa;n with sa;n=.1 C sa;n/ we can assume that sa;n has an image contained in Œ0; 1. Let X qr.x/ D sa;i.x/; r 1 a2A; i<r 320 Chapter 13. Partitions of Unity in Homotopy Theory and qr.x/ D 0 for r D 0. (The sum is ﬁnite for each x 2 X.) Then qr and pa;r.x/ D max.0; sa;r.x/ rqr.x//
are continuous. Let x 2 X; then there exists sa;k with sa;k.x/ ¤ 0; we choose such a function with minimal k; then qk.x/ D 0, pa;k.x/ D sa;k.x/. Therefore the sets a;k 0; 1 also cover X. Choose N 2 N such that N > k and sa;k.x/ > 1 p1 N. Then qN.x/ > 1 N, and therefore N qN.y/ > 1 for all y in a suitable neighbourhood of x. In this neighbourhood, all pa;r with r N vanish. Hence.p1 a;n 0; 1 j a 2 A; n 2 N/ is a locally ﬁnite covering of X which reﬁnes U. We ﬁnish the proof by an appli- cation of the previous lemma. (13.1.6) Theorem. Let.Uj j j 2 J / be a numerable covering of B Œ0; 1. Then there exists a numerable covering.Vk j k 2 K/ of B and a family..k/ j k 2 K/ of positive real numbers such that, for t1; t2 2 Œ0; 1, t1 < t2 and jt1 t2j <.k/, there exist a j 2 J with Vk Œt1; t2 Uj. Proof. Let.tj j j 2 J / be a numeration of.Uj /. For each r-tuple k D.j1; : : : ; jr / 2 J r, deﬁne a continuous map vk W B! I; x 7! min tji.x; s/ j s 2 rY iD1 i1 rC1 ; iC1 rC1 : S 1 rD1 J r. We show that the Vk D v1 2r for k D Let K D.j1; : : : ; jr / satisfy the requirements of the theorem. Namely if jt1 t2j < 1 2r, there exists i with Œt1; t2 Œ i1 k 0; 1 and.k/ D 1 rC1 and hence Vk Œt1; t2 Uji. rC1 ; iC1 We show that.Vk
/ is a covering. Let x 2 B be given. Each point.x; t/ has an open neighbourhood of the form U.x; t / V.x; t/ which is contained in a suitable set W.i/ D t 1 0; 1 and meets only a ﬁnite number of the W.j /. Suppose 2 V.x; t1/; : : : ; V.x; tn/ cover the interval I D Œ0; 1; let rC1 be a Lebesgue number of this covering. We set U D U.x; t1/ \ \ U.x; tn/. Each set U Œ i1 rC1 is then contained in a suitable W.ji /. Hence x is contained in Vk, k D.j1; : : : ; jr /. There are only a ﬁnite number of j 2 J for which W.j / \.U I / ¤ ;. Since vk.x/ ¤ 0 implies the relation W.ji / \ fxg I ¤ ;, the family.Vk j k 2 J r / is locally ﬁnite for r ﬁxed. The existence of a numeration for.Vk j k 2 K/ follows now from theorem (13.1.5). rC1 ; iC1 i A family of continuous maps.tj W X! Œ0; 1 j j 2 J / is called a generalized partition of unity if for each x 2 X the family.tj.x/ j j 2 J / is summable with sum 1. 13.2. The Homotopy Colimit of a Covering 321 (13.1.7) Lemma. Let.tj j j 2 J / be a generalized partition of unity. Then.t 1 j 0; 1 j j 2 J / is a numerable covering. P Proof. Summability of.tj.a// means: For each " > 0 there exists a ﬁnite set E J such that for all ﬁnite sets F E the inequality j1 j 2F tj.a/j > 1 " j 2E tj.x/ > 1 "g is an open neighbourhood holds. In that case V D fx j P of a. If k … E, x 2 V and tk.
x/ > ", then tk.x/ C j 2E tj.x/ > 1. This is impossible. Hence for each a 2 X there exists an open neighbourhood V.a/ such that only a ﬁnite number of functions tj have a value greater than " on V.a/. Let sj;n.x/ D max.tj.x/ n1; 0/ for j 2 J and n 2 N. By what we have just shown, the sj;n are locally ﬁnite for ﬁxed n. The claim now follows from (13.1.5). P It is a useful fact that arbitrary partitions of unity can be reduced to countable ones. The method of proof is inspired by the barycentric subdivision of a simplicial complex. Let U D.Uj j j 2 J / be a covering of the space Z with subordinate partition of unity T D.tj j j 2 J /. For each ﬁnite set E J we set qE.z/ D max.0; mini2E ti.z/ maxj …E XS tj.z//. The function qe is continuous, since T is locally ﬁnite. From this deﬁnition one veriﬁes: (13.1.8) Lemma. If qE.x/ 6D 0 6D qF.x/, then either E F or F E. The family.jEjqE j E J ﬁnite/ is a locally ﬁnite partition of unity. (13.1.9) Corollary. Let U.E/ D q1 E 0; 1. Then U.E/\U.F / 6D ; and jEj D jF j ` implies E D F. We set Un D jE jDn U.E/ and deﬁne n W Un! Œ0; 1 by njU.E/ D jEjqE. Then.n j n 2 N/ is a numeration of.Un j n 2 N/. Suppose the functions tj are non-zero. Let N be the nerve of the covering 0; 1 j j 2 J /. Then the nerve of the covering.q1 E 0; 1 j E J ﬁnite/ is.t 1 j the
barycentric subdivision of N. 13.2 The Homotopy Colimit of a Covering Let K D.V; S/ be a simplicial complex. We consider it as a category: The objects are the simplices, the morphisms are the inclusions of simplices. A contravariant functor X W K! TOP is called a simplicial K -diagram (in TOP). It associates to each simplex s a space Xs and to each inclusion t s a continuous map r s t W Xs! ˚ P Xt. We also have a covariant functor W K! TOP which.s/ D v2s tvv j tv 2 I;, and for an inclusion t s we have the canonical inclusion i s t W.t/!.s/. The geometric realization jXj of a K-diagram X is the quotient ` of tv D 1 P s Xs.s/ with respect to the relation t.a/.r s Xs.s/ 3.x; i s t.x/; a// 2 Xt.t/: Restriction to the n-skeleton Kn yields a functor X n W Kn! TOP. In jXj we have the subspace jXjn which is the image of the Xs.s/ with dim s n. Since jX nj is a quotient of the sum of these products we obtain a continuous map jX nj! jXjn. 322 Chapter 13. Partitions of Unity in Homotopy Theory (13.2.1) Proposition. The space jXj is the colimit of the subspaces jXjn. The canonical map jX nj! jXjn is a homeomorphism. There exists a canonical pushout diagram ` s;dim sDn Xs @.s/ 'n jX n1j ` s;dim sDn Xs.s/ ˆn jX nj. The attaching map 'n is deﬁned as follows: @.s/ is the union of the i s the t s have one element less than s. The map 'n is deﬁned on Xs i s t.i s r s t /1 composed with the canonical map into jX n1j. t.t/, where t.s/ by Let U D fUj j j
2 J g be a covering of a space X. For each ﬁnite non-empty j 2J Ij, j 2 Uj. We deﬁne a subspace C.U/ of X Q T E J we write UE D Ij D I, as the set of families y D.xI tj / such that: (1) Only a ﬁnite number of the tj are non-zero. (2) (3) If J.y/ D fj 2 J j tj 6D 0g then x 2 UJ.y/. j tj D 1. P We have coordinate maps pr D prC W C.U/! X,.xI tj / 7! x and ti W C.U/! I,.xI tj / 7! ti. They are restrictions of the product projections and therefore continuous. The tj form a point-ﬁnite partition of unity on C.U/. We view C.U/ via prC as a space over X. We deﬁne a second space B.U/ with the same underlying set but with a new topology. Recall the nerve N.U/ of the covering U. We have the simplicial N.U/diagram which associated to a simplex E of the nerve the space UE and to an F W UE! UF. The space B.U/ is inclusion F E of simplices the inclusion r E the geometric realization of this N.U/-diagram. Thus B.U/ is the quotient space of E UE.E/ by the relation ` UE.E/ 3.x; d E F.a//.i E F.x/; a/ 2 UF.F /: The sum is taken over the ﬁnite non-empty subsets E of J. Let.E/ı be the interior of.E/ and @.E/ its boundary. Then each element of B.U/ has a unique representative of the form.xI t/ 2 UE.E/ı for a unique E. We can interpret this element as an element of C.U/, and in this manner we obtain a bijection of sets W B.U/! C.U/. This map is continuous, since the canonical maps UE.E/! C.U/ are continuous. The space B.U/ has a projection
pr D prB onto X and is a map over X. (13.2.2) Proposition. The map is a homotopy equivalence over X. Proof. We construct a map W C.U/! B.U/. For this purpose we choose a lo0; 1 of C.U/. cally ﬁnite partition of unity.j / subordinate to the open covering t 1 j 13.2. The Homotopy Colimit of a Covering 323 Then we deﬁne W y D.xI tj / 7!.xI j.y// D z: The map is well-deﬁned and continuous: Let j 2 J.z/, i.e., 0 6D tj.z/ D j.y/ hence J.z/ J.y/ and x 2 J.z/; this shows that z 2 B.U/. Let W C.U/ be an open set such that J.W / D fj j j jW 6D 0g is ﬁnite. Let y D.xI tj / 2 W. Then J.W / J.y/, therefore factors on W through a map W! UJ.W /.J.W //, and this shows the continuity. A homotopy'idC.U/ is deﬁned by.y; t/ D..xI tj /; t/ 7!.xI t tj C.1 t/j /: This assignment is clearly well-deﬁned and continuous. A homotopy'id is deﬁned by the same formula.yI t/ 7!.xI t tj.y/ C.1 t/j.y//. In order to verify the continuity, we let again W be as above, but now considered as a subset of B.U/. We consider the composition with XE.E/! B.U/. The formula for the homotopy on the pre-image of W has an image in XE.E/. Let B.U/n be the subspace of B.U/ which is the image of the UE.E/ with jEj n C 1. We state (13.2.1) for the special case at hand. (13.2.3) Proposition. B.U/ is the colimit of the
sequence of subspaces B.U/n. The dim E n UE.E/! B.U/n is a quotient map. The inclusion canonical map B n1 B n is obtained via a pushout diagram ` ` dim E Dn UE @.E/ kn B.U/n1 ` dim E Dn UE.E/ Kn B.U/n. The map kn is deﬁned on XE @.E/ as follows: Let F E be a proper subset. Then kn is deﬁned on XE.F / by XE.F / XF.F /! B n1. (13.2.4) Proposition. Let U be numerable. Then the projections C.U/! X and B.U/! X are shrinkable. Proof. Let.j j j 2 J / a numeration of U. Then x 7!.xI j.x// is a section s of prC and..xI tj /; t/ 7!.xI t tj C.1 t/j.x// is a homotopy from s prC to the identity over X. Thus prC is shrinkable, and (13.2.2) shows that also prB is shrinkable. For some applications we need a barycentric subdivision of B.U/. Recall that we have the barycentric subdivision N 0.U/ of the nerve of U. An n-simplex of N 0.U/ is an ordered set D.E0 ¤ E1 ¤ ¤ En/ such that UEn 6D ;. We write q. / D En. We have theN 0.U/-diagram which associates to the space 324 Chapter 13. Partitions of Unity in Homotopy Theory Xq./ and to the inclusion Xq./ Xq./. Let B 0.U/ denote the geometric realization of this N 0.U/-diagram. Since the simplices are ordered, we can replace./ by the standard simplex Œn spanned by Œn D f0; 1; : : : ; ng. (13.2.5) Remark. In the case of the barycentric subdivision the pushout diagram in (13.2.3) reads as follows: ` 2An Xq./ @n kn B 0.U/n1 jn ` 2An Xq./
n Kn Jn B 0.U/n: The sum is over the set An D f.0; : : : ; n/ j 0 ¤ ¤ n; n J ﬁniteg, and Þ q.0; : : : ; n/ D n. 13.3 Homotopy Equivalences The main result (13.3.1) of this section asserts that being a homotopy equivalence is in a certain sense a local property. (13.3.1) Theorem. Let p W X! B and q W Y! B be spaces over B and f W X! Y a map over B. Let X D.Xj j j 2 J / be a covering of X and Y D.Yj j j 2 J / a covering of Y. Let f.Xj / Yj and assume that for each ﬁnite E J the map E W XE! YE induced by f is a homotopy equivalence over B. Then the induced map B.f / W B.X/! B.Y/ is a homotopy equivalence over B. Thus if the coverings X and Y are numerable, then f is a homotopy equivalence over B. Proof. From (5.3.4) and (13.2.3) we prove inductively that the induced maps B.X/n! B.Y/n are h-equivalences. Now we use (5.2.9), in order to show that Bf is an h-equivalence. In the case of numerable coverings we also use (13.2.4). (13.3.2) Remark. In the situation of (13.3.1) we can conclude that f is a homotopy equivalence, if the projections pX W B.X/! X and pY are homotopy equivalences. Þ (13.3.3) Theorem. Let p W X! B and q W Y! B be spaces over B and f W X! Y a map over B. Let.Uj j j 2 J / be a numerable covering of B. Let fj W p1.Uj /! q1.Uj / be the map induced by f over Uj. Suppose each fj is a ﬁbrewise homotopy equivalence. Then
f is a ﬁbrewise homotopy equivalence. Proof. The hypothesis implies that f is a ﬁbrewise homotopy equivalence over each set V Vj. We can therefore apply (13.3.1). 13.4. Fibrations 325 We say that a covering.Uj j j 2 J / of B is null homotopic if every inclusion Uj B is null homotopic. (13.3.4) Theorem. Let f W X! Y be a map over B from p W X! B to q W Y! B. Assume that p and q are ﬁbrations. Suppose B has a numerable null homotopic covering.Vj j j 2 J / and that each path component of B contains a point b such that f is a homotopy equivalence over b. The f is a ﬁbrewise homotopy equivalence. Proof. Let Vj B be homotopic to the constant map Vj! fb.j /g. We can assume that fb.j / W Xb.j /! Yb.j / is an h-equivalence. By the homotopy theorem for ﬁbrations we obtain a homotopy commutative diagram of maps over Vj fj p1.Vj /.1/ q1.Vj /.2/ Vj Xb.j / id fb.j / Vj Yb.j / with ﬁbrewise homotopy equivalences (1) and (2). Hence fj is a ﬁbrewise equiva- lence and we can apply (13.3.3) Problems 1. For each j 2 J we let C.U/j D pr1.Uj / and similarly for B. Then the partial projection maps prC 2. If the coverings in (13.3.1) are open, then f W ŒZ; X! ŒZ; Y is for each paracompact Z a bijection. The canonical projection p W B.X/! X induces for a paracompact space Z a bijection p W ŒZ; B.X/! ŒZ; X. j W B.U/j! Uj are shrinkable. j W C.U/j! Uj and prB 13.4 Fibrations (
13.4.1) Theorem. Let V D.Vj j j 2 J / be a covering of B and p W E! B a continuous map. Assume that the map pj W p1.Bj /! Bj induced by p is for each j 2 J a ﬁbration. If the covering V is numerable, then p is a ﬁbration. If the covering V is open, then p has the HLP for paracompact spaces. Proof. We have to solve a homotopy lifting problem (left diagram: 326 Chapter 13. Partitions of Unity in Homotopy Theory We form the pullback of p along h (right diagram). The initial condition a yields a section s0 of q over X 0. A lifting of h with this initial condition amounts to a section s of q which extends s0. We pull back the numerable covering of B to a numerable covering of X I. There exists a numerable covering U D.Uk j k 2 K/ of X such that q is a ﬁbration over the sets Uk I (Problem 1). We begin by constructing a lifting t W B.U/ I! E of prB which extends the lifting t0 over B.U/ 0 determined by s0. The lifting is constructed inductively from partial liftings tn over B.U/n I. The induction step is again based on the pushout diagram (13.2.3), now multiplied by I. The extension of the lifting tn amounts to solving a lifting problem of the type ` UE.@.E/ I [.E/ 0/ Y q ` UE.E/ I X I and this is possible (by (5.5.3)), because UE.E/ I is mapped into a subset over which q is a ﬁbration. If the covering U is numerable we compose it with a section of prB and obtain the desired extension of s0. (13.4.2) Theorem. Let p W Y! X be a continuous map. Let Y D.Yj j j 2 J / be a family of subsets of Y and X D.Xj j j 2 J / a numerable covering of X. Assume that p.Yj / Xj and that for ﬁnite E J the map pE W YE! XE induced by p is shrinkable. Then p
has a section. (Note that Y is not assumed to be a covering of Y.) Proof. We work with the barycentric subdivision B 0.X/. We show the existence of a map s W B 0.X/! Y such that ps D prB. The proof does not use the numerability of the covering. We construct inductively maps sn W B 0.X/n! Y with the appropriate properties and an additional property which makes the induction work. The map B 0 D E XE! Y is given as follows: We choose sections XE! ` YE of pE and compose them with the inclusion YE Y. Suppose sn1 is given. We want to extend ` sn1kn W.Xq./ @Œn/! E `.Xq./ Œn/. If D.E0; : : : ; En/ we impose the additional hypothesis over that the image of Xq./ @Œn under sn1kn is contained in YE0. The construction of s0 agrees with this requirement. Under this additional hypothesis we have a commutative diagram Xq./ @Œn sn1kn YE0 Xq./ XE0. From (5.5.3) we see that sn1kn can be extended over Xq./ Œn. With an extension we construct sn via the pushout (13.2.5). We show that sn satisﬁes the additional hypothesis. Given D.E0; : : : ; EnC1/ we describe ` 13.4. Fibrations 327 knC1 W Xq./ @Œn C 1! B.U/n: Let di W Œn! Œni be the standard map onto the i-th face of Œn C 1 with inverse homeomorphism ei. Let @i W Xq./! Xq."i / be the inclusion where "i D.E0; : : : ; Ei1; EiC1; : : : ; EnC1/. The restriction of knC1 to the subset Xq./ Œn C 1i is Kn.@i ei /. By construction of sn the image of snKn.@i ei / is contained in XE0 (for i > 0) orX E1 (for i D 0). But XE1 XE0, hence sn has the desired property. If X
is numerable, then pr has a section t and st is a section of p. (13.4.3) Theorem. Let p W X! B be a continuous map and X D.Xj j j 2 J / a numerable covering of X. Assume that for each ﬁnite E J the restriction pE W XE! B is a ﬁbration (an h-ﬁbration, shrinkable). Then p is a ﬁbration (an h-ﬁbration, shrinkable). Proof. Recall that for a ﬁbration p W X! B the canonical map r W X I! W.p/ is shrinkable (see (5.6.5)), and that p is a ﬁbration if this map has a section (see (5.5.1)). If the pE are ﬁbrations, then the rE W X I E! W.pE / are shrinkable. The W.pj / form a numerable covering of W.p/. Theorem (13.4.2) shows that r has a section, hence p is a ﬁbration. Assume that the pE are shrinkable, i.e., homotopy equivalences over B. We apply (13.3.1) and see that p is shrinkable. Assume that the pE are h-ﬁbrations. The map p is an h-ﬁbration if and only if the canonical map b W W.p/! X is a homotopy equivalence over B (this can be taken as a deﬁnition of an h-ﬁbration). The W.pj / form a numerable covering of W.p/ and the bE W W.pE /! XE are homotopy equivalences over B, since pE are h-ﬁbrations. Thus we are in a position where (13.3.1) can be applied. The hypothesis of (13.4.3) is satisﬁed if the XE are either empty or contractible. Problems 1. If q W M! N Œa; b is a ﬁbration over N Œa; c and N Œc; b, then q is a ﬁbration. 2
. Let X D.Xj j j 2 J / be a numerable covering of X. If the spaces XE have the homotopy type of a CW-complex, then X has the homotopy type of a CW-complex. 3. Let p W E! B be an h-ﬁbration. Suppose B and each ﬁbre p1.b/ have the homotopy type of a CW-complex. Then E has the homotopy type of a CW-complex. Chapter 14 Bundles Bundles (also called ﬁbre bundles) are one of the main objects and tools in topology and geometry. They are locally trivial maps with some additional structure. A basic example in geometry is the tangent bundle of a smooth manifold and its associated principal bundle. They codify the global information that is contained in the transitions functions (coordinate changes). The classiﬁcation of bundles is reduced to a homotopy problem. This is achieved via universal bundles and classifying spaces. We construct for each topological group G the universal G-principal bundle EG! BG over the so-called classifying space BG. The isomorphism classes of numerable bundles over X are then in bijection with the homotopy set ŒX; BG. The classiﬁcation of vector bundles is equivalent to the classiﬁcation of their associated principal bundles. A similar equivalence holds between n-fold covering spaces and principal bundles for the symmetric group Sn. This leads to a different setting for the classiﬁcation of coverings. From the set of (complex) vector bundles over a space X and their linear algebra one constructs the Grothendieck ring K.X/. The famous Bott periodicity theorem in one of its formulations is used to make the functor K.X/ part of a cohomology theory, the so-called topological K-theory. Unfortunately lack of space prevents us from developing this very important aspect. Classifying spaces and universal bundles have other uses, and the reader may search in the literature for information. The cohomology ring H.BG/ of the classifying space BG of a discrete group G is also called the cohomology of the group. There is a purely algebraic theory which deals with such objects. If X is a G-space, one can form the associated bundle EG G X! BG. This bundle over BG can be
interpreted as an invariant of the transformation group X. The cohomology of EG G X is a module over the cohomology ring H.BG/ (Borel-cohomology). The module structure contains some information about the transformation group X, e.g., about its ﬁxed point set (see [7], [43]). 14.1 Principal Bundles Let G be a topological group. In the general theory we use multiplicative notation and denote the unit element of G by e. Let r W E G! E,.x; g/ 7! xg be a continuous right action of G on E, and p W E! B a continuous map. The pair.p; r/ is called a (right) G -principal bundle if the following axioms hold: 14.1. Principal Bundles 329 (1) For x 2 E and g 2 G we have p.xg/ D p.x/. (2) For each b 2 B there exists an open neighbourhood U of b in B and a Ghomeomorphism'W p1.U /! U G which is a trivialization of p over U with typical ﬁbre G. Here G acts on U G by..u; h/; g/ 7!.u; hg/. If we talk about a G-principal bundle p W E! B, we understand a given action of G on E. From the axioms we see that G acts freely on E. The map p factors through the orbit map q W E! E=G and induces a continuous bijection h W E=G! B. Since q and p are open maps, hence quotient maps, h is a homeomorphism. Thus G-principal bundles can be identiﬁed with suitable free right G-spaces. In contrast to an arbitrary locally trivial map with typical ﬁbre G, the local trivializations in a principal bundle have to be compatible with the group action. In a similar manner we deﬁne left principal bundles. A G-principal bundle with a discrete group G is called a G -principal covering. The continuity of the action r is in this case equivalent to the continuity of all right translations rg W E! E, x 7! xg. This is due to the fact that E G is homeomorphic to the topological sum qg2
GE fgg, ifG is discrete. Let E G! E be a free action and set C.E/ D f.x; xg/ j x 2 E; g 2 Gg. We call t D tE W C.E/! G,.x; xg/ 7! g the translation map of the action. (14.1.1) Lemma. Let p W E! E=G be locally trivial. Then the translation map is continuous. Proof. Let W D p1.U / E be a G-stable open set which admits a trivialization W U G! W. The pre-image of.W W / \ C.E/ under is f.u; g; u; h/ j u 2 U; g; h 2 Gg, and t ı. / is the continuous map.u; g; u; h/ 7! g1h. A free G-action on E is called weakly proper if the translation map is continu- ous. It is called proper if, in addition, C.E/ is closed in E E. (14.1.2) Proposition. A free action of G on E is weakly proper if and only if 0 W E G! C.E/,.x; g/ 7!.x; xg/ is a homeomorphism. Proof. The map W C.E/! E G,.x; y/ 7!.x; tE.x; y// is a set-theoretical inverse of 0. It is continuous if and only if tE is continuous. Let E carry a free right G-action and F a left G-action. We have a commutative diagram pr1 =G with orbit maps P and p and q D pr1 =G. 330 Chapter 14. Bundles (14.1.3) Proposition. A free right G action on E is weakly proper if and only if for each left G-space F the diagram is a topological pullback. Proof. We compare the diagram with the canonical pullback =G with X D f..x; f /; y/ 2.E G F / E j p.x/ D p.y/g and a D pr1, b D pr2. There exists a unique map W E F! X such that b D pr1, a D P, i.e.,.x; f /
D..x; f /; x/. The diagram in question is a pullback if and only if is a homeomorphism. Suppose this is the case for the left G-space G. The homeomorphism E G G! E,.x; g/ 7! xg transforms q into p, X into C.E/ and into.x; g/ 7!.xg; x/. The latter is, in different notation, 0. Hence 0 is a homeomorphism if the diagram is a pullback for F D G. Conversely, let the action be weakly proper. The map Q W QX D f..x; f /; y/ j p.x/ D p.y/g!E F;..x; f /; y/ 7!.y; t.x; y/1f / is continuous. One veriﬁes that Q induces a map W X! E F. The equalities.x; f / D..x; f /; x/ D.x; t 1.x; x/f / D.x; f / show that is an inverse of. (14.1.4) Proposition. Let G act freely and weakly properly on E. The sections of q W E G F! E=G correspond bijectively to the maps f W E! F with the property f.xg/ D g1f.x/; here we assign to f the section sf W x 7!.x; f.x//. Proof. It should be clear that sf is a continuous section. Conversely, let s W B! E G F be a continuous section. We use the pullback diagram displayed before (14.1.3). It yields an induced section of pr1 which is determined by the conditions pr1 ı D id and P ı D s ı p. Let f D pr2 ı W E! F. Then pr1.xg/ D xg D pr1..x/g/, since is a section and pr1 a G-map. (The right action on E F is.e; f; g/ 7!.eg; g1f /.) The equalities P.xg/ D sp.xg/ D sp.x/ D P.x/ D P..x/g/ hold, since is an induced section and P the orbit map. Since.xg
/ and.x/g have the same image under P and pr1, these elements are equal; we now apply the G-map pr2 and obtain ﬁnally f.xg/ D g1f.x/. (14.1.5) Proposition. Let the free G-action on E be weakly proper. Then the orbit map p W E! E=G D B is isomorphic to pr W B G! B, if and only if p has a section. 14.1. Principal Bundles 331 Proof. Let s be a section of p. Then B G! E,.b; g/ 7! s.b/g and E! B G, x 7!.p.x/; t.spx; x// are inverse G-homeomorphisms, compatible with the projections to B. Conversely, pr has a section and hence also the isomorphic map p. (14.1.6) Proposition. Let X and Y be free G-spaces and ˆ W X! Y a G-map. If'D ˆ=G is a homeomorphism and Y weakly proper, then ˆ is a homeomorphism. Proof. X is weakly proper, since the translation map of X is obtained from the translation map of Y by composition with ˆ ˆ. We have to ﬁnd an inverse ‰ W Y! X. By (14.1.4), it corresponds to a section of Y W.Y X/=G! Y =G. We have the section s W x 7!.x; ˆ.x// of X W.X Y /=G! X=G. Let be the inverse of '. With the interchange map W.X Y /=G!.Y X/=G we form D ı s ı. One veriﬁes that is a section of Y. Let a commutative diagram below with principal G-bundles p and q be given, and let F be a G-map. Then F or.F; f / is called a bundle map. Y q C F f X p B If f is a homeomorphism, then F is a homeomorphism (see (14.1.6)). If f is the identity, then F is called a bundle isomorphism. Given a principal bundle p W X! B and a map f
W C! B, we have a pullback diagram as above with Y D f.c; x/ j f.c/ D p.x/g C X. The maps q and F are the restrictions of the projections onto the factors. The G-action on Y is.c; x/g D.c; xg/. If p is trivial over V, then q is trivial over f 1.V /. Therefore q is a principal bundle, called the bundle induced from p by f. Also F is a bundle map. From the universal property of a pullback we see, that the bundle map diagram above is a pullback. (14.1.7) Proposition. Let U be a right G-space. The following are equivalent: (1) There exists a G-map f W U! G. (2) There exists a subset A U such that m W A G! U,.a; g/ 7! ag is a homeomorphism. (3) The orbit map p W U! U=G is G-homeomorphic over U=G to the projection pr W U=G G! U=G. (4) U is a freeG -space, p W U! U=G has a section, and tU is continuous. Proof..1/ ).2/. Let A D f 1.e/ and v W U! A,x 7! x f.x/1.x/. Then.v; f / W U! A G is an inverse of m. 332 Chapter 14. Bundles.2/ ).3/. The G-homeomorphism m induces a homeomorphism m=G of the orbit spaces. We have the homeomorphism " W A!.A G/=G, a 7!.a; e/. With these data m ı." id/ ı m=G D pr..3/ ).4/. If'W U=G G! U is a G-homeomorphism over U=G, then the G-action is free and x 7! '.x; e/ is a section of p. The translation map of U=G G is continuous and hence, via ', also tU..4/ ).1/. Let s W U=G! G be a section. Then U! G, u 7! tU.sp.u/; u/ is a G-map. A right G
-space U is called trivial if there exists a continuous G-map f W U! G into the G-space G with right translation action. A right G-space is called locally trivial if it has an open covering by trivial G-subspaces. (14.1.8) Proposition. The total space E of a G-principal bundle is locally trivial. If E is locally trivial, then E! E=G is a G-principal bundle. 14.1.9 Hopf ﬁbrations. Consider S 2n1 Cn as a free S 1-space with action induced from scalar multiplication. Let Uj be the subset of points z D.zk/ with zj 6D 0. The map z 7! zj jzj j1 shows that Uj is a trivial S 1-space. The orbit space of this action is CP n1. The S 1-principal bundle p W S 2n1! CP n1, i.e., the orbit map, is called a Hopf ﬁbration. There is a similar Z=2-principal bundle S n1! RP n1 onto the real projective space and an S 3-principal bundle Þ S 4n1! HP n1 onto the quaternionic projective space. (14.1.10) Proposition. Let f W X! Y be a G-map and pY W Y! Y =G a Gprincipal bundle. Then the diagram X pX X=G f f =G Y pY Y =G is a pullback. Proof. Let U Y be a G-set with a G-map h W U! G. Then h ı f W f 1.U /! U! G is a G-map. Hence pX is a G-principal bundle. The diagram is therefore a bundle map and hence a pullback. We say, a map f W X! Y has local sections if each y 2 Y has an open neighbourhood V and a section s W V! X of f over V ; the latter means f s.v/ D v for all v 2 V. (14.1.11) Proposition. Let G be a topological group and H a subgroup. The quotient map q W G! G=H is an H -principal bundle if and only if q has a section over some
neighbourhood of the unit coset. 14.1. Principal Bundles 333 Proof. A locally trivial map has local sections. Conversely, let s W U! G be a section of q over U. The map tG is continuous. For each k 2 G we have the H equivariant map kq1.U /! H, kg 7! s.q.g//1g. We thus can apply (14.1.8) and (14.1.7). Actions with the properties of the next proposition were called earlier properly discontinuous. (14.1.12) Proposition. Let E G! E,.x; g/ 7! xg be a free right action of the discrete group G. The following assertions are equivalent: (1) The orbit map p W E! E=G is a G-principal covering. (2) Each x 2 E has a neighbourhood U, such that U \ Ug D ; for each g 6D e. (3) The set t 1.e/ is open in C. (4) The map t is continuous. Proof. (1) ) (4) holds by (14.1.1). (4) ) (3). The set feg G is open, since G is discrete. (3) ) (2). We have t.x; x/ D e. Since t 1.e/ is open, there exists an open neighbourhood U of x in E such that.U U / \ C t 1.e/. Let U \ Ug ¤ ;, say v D ug for u; v 2 U. Then.u; v/ D.u; ug/ 2.U U / \ C, hence t.u; ug/ D g D e. (2) ) (1). Let U be open. Then U G! UG,.u; g/ 7! ug is a G-homeo- morphism, hence UG an open trivial G-subspace. (14.1.13) Example. Let G be a closed discrete subgroup of the topological group E. Then the action G E! E,.g; x/ 7! gx is free and has property (4) of the Þ previous proposition. Examples are Z R or Z C. (14.1.14) Example. The map g W R! S 1, t 7! exp.2 it/ has
kernel Z. Therefore there exists a bijective map f W R=Z! S 1 such that fp D g. Since g is an open map, g is a quotient map and therefore f a homeomorphism. By the previous example, g is therefore a covering. Similarly exp W C! C is seen to be a Þ covering. (14.1.15) Example. Let G be a Lie group and H a closed subgroup. Then the quotient map p W G! G=H is an H -principal bundle. In the chapter on differentiable manifolds we show that G=H carries the structure of a smooth manifold such that Þ p is a submersion. A submersion has always smooth local sections. We construct locally trivial bundles from principal bundles. Let p W E! B be a right G-principal bundle and F a left G-space. The projection E F! E induces, via passage to orbit spaces, q W E G F! B. The map q is locally trivial 334 Chapter 14. Bundles with typical ﬁbre F. A bundle chart'W p1.U /! U G of p yields a bundle chart q1.U / D p1.U / G F 'G id.U G/ G F Š U F for q. We call q the associated ﬁbre bundle with typical ﬁbre F, and G is said to be the structure group of this ﬁbre bundle. The structure group contains additional information: The local trivializations have the property that the transition functions are given by homeomorphisms of the ﬁbre which arise from an action of an element of the group G. Let p W Y! B be a right G-principal bundle. It may happen that there exists a right H -principal bundle q W X! Y for a subgroup H G and a G-homeomorphism W X H G! Y over B. In that case.q; / is called a reduction of the structure of p. One can consider more generally a similar problem for homomorphisms ˛ W H! G. (14.1.16) Example. Let E! B be a G-principal bundle and H G a subgroup. Then E G G=H! E=H,.x; gH / 7! xgH is a homeomorphism. Therefore
E=H! E=G, xH 7! xG is isomorphic to the associated bundle E G G=H! E=G. From a subgroup K H G we obtain in this manner G=K! G=H as a bundle with structure group H and ﬁbre H=K, ifG! G=H has local sections. If X is a G-space and H C G, then G=H acts on X=H by.xH; gH / 7! xgH. The quotient map X=H! X=G induces a homeomorphism.X=H /=.G=H / Š In particular, E=H! E=G becomes in this manner a G=H -principal X=G. Þ bundle. One can “topologise” various algebraic notions in analogy to the passage from groups to topological groups. An important notion in this respect is that of a small category. A (small) topological category C consists of an object space Ob.C / and a morphism space Mor.C / such that the structure data s W Mor.C /! Ob.C / r W Mor.C /! Ob.C / i W Ob.C /! Mor.C / c W Mor2.C /! Mor.C / (source), (range), (identity), (composition) are continuous. Here Mor2.C / D f.ˇ; ˛/ 2 Mor.C / Mor.C / j s.ˇ/ D r.˛/g carries the subspace topology of Mor.C /Mor.C /. For these data the usual axioms of a category hold. In a groupoid each morphism has an inverse. In a topological 14.2. Vector Bundles 335 groupoid we require in addition that passage to the inverse is a continuous map Mor.C /! Mor.C /. A topological group can be considered as a topological groupoid with object space a point. Principal bundles yield a parameterized version. (14.1.17) Example. Let E be a free right G-space with a weakly proper action. Let p W E! B be a map which factors over the orbit map q W E! E=G and induces a homeomorphism E=G Š B. We construct a topological groupoid with
object space B. The product p p W E E! B B factors over the orbit map Q W E E!.E E/=G,.a; b/ 7! Œa; b of the diagonal action and induces.s; r/ W.E E/=G! B B: We deﬁne.E E/=G as the morphism space of our category and s; r as source, range. The diagonal of E induces i W B Š E=G!.E E/=G, and this is deﬁned to be the identity. We deﬁne the composition by Œa; b ı Œx; y D Œx; b t.y; a/1 with the translation map t D tE of E. One veriﬁes that composition is associative and continuous. (The space Mor2 is a quotient space of E C.E/ E.) The morphism Œb; a is inverse to Œa; b, hence the inverse is continuous and we have Þ obtained a topological groupoid. Problems 1. A free action of a ﬁnite group G on a Hausdorff space E is proper. 2. The action R2 X 0 R! R2 X 0,..x; y/; t/ 7!.tx; t 1y/ is a non-trivial R-principal bundle. Determine the orbits and the orbit space. 3. Let E be a space with a free right G-action. Then the translation map tE is continuous if and only if the pullback of p along p is a trivial G-space. 4. The continuous maps E G F1! E G F2 over E=G correspond via.x; u/ 7!.x; ˛.x; u// to the continuous maps ˛ W E F1! F2 with the equivariance condition g˛.x; u/ D ˛.xg1gu/. If E is connected and F1; F2 discrete, then ˛ does not depend on x 2 E and has the form ˇ ı pr with a uniquely determined equivariant map ˇ W F1! F2. 5. The groupoid (14.1.17) has further properties. There exists at least one morph
b/ into the standard orientation of Rn. A chart with this property is called positive with respect to the given orientation. The positive charts form an orienting atlas, and for each orienting atlas there exists a unique orientation such that its charts are positive with respect to the orientation. A complex vector space has a canonical orientation. If one uses this orientation in each ﬁbre, then the bundle, considered as a real bundle, is an oriented bundle. Let W E./! B and W E./! C be real vector bundles. A bundle mor- phism! over'W B! C is a commutative diagram E./ ˆ E./ B'C with a map ˆ which is ﬁbrewise linear. If ˆ is bijective on ﬁbres, then we call the bundle morphism a bundle map. Thus we have categories of vector bundles with bundle morphisms or bundle maps as morphisms. The trivial n-dimensional bundle is the product bundle pr W B Rn! B. More generally, we call a bundle trivial if it is isomorphic to the product bundle. (14.2.1) Proposition. A bundle map over the identity is a bundle isomorphism. 14.2. Vector Bundles 337 Proof. We have to show that the inverse of ˆ is continuous. Via bundle charts this can be reduced to a bundle map between trivial bundles U Rn! U Rn;.u; v/ 7!.u; gu.v//: In that case.u; v/ 7!.u; g1 continuous. u.v// is a continuous inverse, since u 7! g1 u is (14.2.2) Proposition. If the previous diagram is a pullback in TOP and a vector bundle, then there exists a unique structure of a vector bundle on such that the diagram is a bundle map. Proof. Since ˆ is bijective on ﬁbres, we deﬁne the vector space structures in such a way that ˆ becomes ﬁbrewise linear. It remains to show the existence of bundle charts. If is the product bundle, then can be taken as product bundle. If has a bundle chart over V, then has a bundle chart over '1.V /, by transitivity of pullbacks. We call in (14.2.2) the bundle induced from along'and write occasionally D
'in this situation. The previous considerations show that a bundle map is a pullback. (Compare the analogous situation for principal bundles.) A bundle morphism over id.B/ has for each b 2 B a rank, the rank of the linear map between the ﬁbres over b. It is then clear what we mean by a bundle morphism of constant rank. A subset E 0 E of an n-dimensional real bundle p W E! B is a k-dimensional subbundle of p, if there exists an atlas of bundle charts (called adapted charts)'W p1.U /! U Rn such that '.E 0 \ p1.U // D U.Rk 0/. The restriction p W E 0! B is then a vector bundle. A vector bundle is to be considered as a continuous family of vector spaces. We can apply constructions and notions from linear algebra to these vector spaces. We begin with kernels, cokernels, and images. S S (14.2.3) Proposition. Let ˛ W 1! 2 be a bundle morphism over B of constant rank and ˛b the induced linear map on the ﬁbres over b. Then the following hold: (1) Ker ˛ D (2) Im.˛/ D (3) Suppose Coker.˛/ D b2B Ker.˛b/ E.1/ is a subbundle of 1. b2B Im.˛b/ E.2/ is a subbundle of 2. S b2B E.2/b= Im.˛b/ carries the quotient topology from E.2/. Then, with the canonical projection onto B, Coker.˛/ is a vector bundle. Proof. The problem in all three cases is the existence of bundle charts. This is a local problem. Therefore it sufﬁces to consider morphisms ˛ W B Rm! B Rn;.b; v/ 7!.b; ˛b.v// 338 Chapter 14. Bundles between trivial bundles. We write Kx D Ker.˛x/ and Lx D Im.˛x/. We ﬁx b 2 B and choose complements Rm D K ˚ K0 and Rn D L ˚ L0 for K D Kb and L D Lb. Let q
W Rm! K and p W Rn! L be the projections with Ker.q/ D K0 and Ker.p/ D L0. Then x W Rm ˚ L0! Rn ˚ K;.v; w/ 7!.˛x.v/ C w; q.v// is an isomorphism for x D b, hence also an isomorphism for x in a neighbourhood of b. Thus let us assume without essential restriction that x is always an isomorphism. Since ˛x has constant rank k, we conclude that Kx \ K0 D 0 and Lx \ L0 D 0. This fact is used to verify that B Rm! B.L K/;.x; v/ 7!.x; p˛x.v/; q.v// B.K0 L0/! B Rn;.x; v; w/ 7!.x; ˛x.v/ C w/ S are ﬁbrewise linear homeomorphisms. The ﬁrst one maps B.0 K/ and the second one B.K0 0/ onto S the second induces a bijection of B.0 L0/ with have veriﬁed (1)–(3) in the local situation. x2B fxg K x onto x2B fxg L x. Moreover, x2B fxg R n=Lx. Thus we S 14.2.4 Tangent bundle of the sphere. Let TS n D f.x; v/ j hx; v i D0g S n RnC1 with the projection p W TS n! S n onto the ﬁrst factor. The ﬁbre p1.x/ is the orthogonal complement of x in RnC1. These data deﬁne the tangent bundle of the sphere. One can apply (14.2.3) to the family ˛x W RnC1! RnC1, v 7! hx; v ix of orthogonal projections. Recall the stereographic projections '˙ W S n Xf˙enC1g!R n. The differential of'ı '1 C W Rn X 0! Rn X 0 at x is the linear map Rn! Rn;
7! kxk2 2hx; ix kxk4 : For kxk D1 we obtain the reﬂection 7! 2hx; ix at the hyperplane orthogonal to x. Let U˙ D p1.S n X fenC1g/. The differential of '˙ yields a homeomorphism which is ﬁbrewise linear and the diagram U˙ p D'˙ Rn Rn pr1 Rn: '˙ S n X f˙enC1g 14.2. Vector Bundles 339 x kxk4 kxk2 ; kxk22h x; ix If we identify in Rn Rn C Rn Rn the point.x; v/ 2.Rn X 0/ Rn in the then we obtain a vector bundle over S D ﬁrst summand with.Rn X 0 C Rn X 0/=x xkxk2 which is isomorphic to p W TS n! S n. We can simplify the situation by identifying in Dn Rn C Dn Rn the point.x; v/ 2 S n1 Rn in the ﬁrst summand with.x; v 2hx; v ix/ in the second summand. In the tangent bundle we have the subspace of tangent vectors of length 1. In our case this is the space f.u; v/ 2 S n S n j hu; v i D1g, the Stiefel manifold V2.RnC1/. We can obtain it from Dn S n1 C Dn S n1 by the identiﬁcation.x; v/.x; v 2hx; v iv/. For even n we obtain from (10.7.8) the integral homology of this Stiefel manifold. For “smallest” structure groups of the tangent bundle of S n see [36]. The vector Þ ﬁeld problem [3] is a special case of this problem. Important vector bundles in geometry are the tangent bundles of differentiable manifolds and the normal bundles of immersed manifolds. 14.2.5 Tautological bundles. Let V be an n-dimensional real vector space and Gk.V / the Grassmann manifold of the k-dimensional subspaces of V
. We set Ek.V / D f.x; v/ j x 2 Gk.V /; v 2 xg G k.V / V: We have the projections k.V / D k W Ek.V /! Gk.V /,.x; v/ 7! x, and the ﬁbre over the element x 2 Gk.V / is the subspace x. For this reason we call this bundle the tautological bundle. It remains to verify that k is locally trivial. For this purpose we recall the O.k/-principal bundle p W Sk.Rn/! Gk.Rn/ from the Stiefel manifold to the Grassmann manifold. The map Sk.Rn/ O.k/ Rk! Ek.Rn/;..v1; : : : ; vk/;.1; : : : ; k// 7!.hv1; : : : ; vki; P j vj / is a ﬁbrewise linear homeomorphism; it describes the tautological bundle as an associated ﬁbre bundle. Here is a different argument. Suppose x is spanned by.v1; : : : ; vk/ 2 Sk.Rn/; k then px W Rn! Rn, v 7! j D1hv; vj ivj is the orthogonal projection onto x. It depends continuously on.v1; : : : ; vk/ and induces a continuous map Gk.Rn/! Hom.Rn; Rn/, x 7! px, and P Gk.Rn/ Rn! Gk.Rn/ Rn;.x; v/ 7!.x; px.v// is a bundle morphism of constant rank with image Ek.Rn/. Now one can use (14.2.3). There exist analogous complex tautological bundles over the complex GrassÞ mannians. 340 Chapter 14. Bundles 14.2.6 Line bundles over CP n. Let H W CnC1 X 0! CP n be the deﬁning C-principal bundle. Let C.k/ be the one-dimensional complex C-representation C C! C,.; z/ 7! kz. We obtain the associated complex line bundle H.k/ D.CnC1 X
0/ C C.k/! CP n: Thus H.k/ is the quotient of.CnC1 X0/C under the equivalence relation.z; u/.z; ku/ for 2 C. We also have the S 1-principal bundle S 2nC1! Cn (the Hopf bundle). The inclusion S 2nC1 C!.CnC1 X 0/ C induces a homeomorphism S 2nC1 S 1 C.k/!.CnC1 X 0/ C C.k/: The inverse homeomorphism is induced by.z; u/ 7!.kzk1z; kzkku/. The unit sphere bundle is S 2nC1 S 1 S 1.k/. The assignment z 7!.z; 1/ induces a homeomorphism S 2nC1=Cjkj! S 2nC1 S 1 S 1.k/: Here Cm S 1 is the subgroup of order m (roots of unity). The bundles H.k/ over CP n exist for 1 n 1. We call the bundle H.1/ the canonical complex line bundle. For n D 1 it will serve as a universal onedimensional vector bundle. The tautological bundle over CP n D G1.CnC1/ is H.1/, since.CnC1 X 0/ C C.1/! E1.CnC1/;.x; u/ 7!.Œx; ux/ is an isomorphism. The sections of H.k/ correspond to the functions f W CnC1 X 0! C with the property f.z/ D kf.z/, they are homogenous functions of degree k. This is the Þ reason to deﬁne H.k/ with C.k/. Let q W E! B be a right G-principal bundle and V an n-dimensional representation of G. Then the associated bundle p W E G V! B is an n-dimensional vector bundle. A bundle chart'W q1.U /! U G for q induces a bundle chart q1.U / G V! U V for p, and the vector space structure on the ﬁbres of p is uniquely determined by the requirement that the bundle charts are ﬁbrewise linear. We now show that
vector bundles are always associated to principal bundles. Let p W X! B be an n-dimensional real vector bundle. Let Eb D Iso.Rn; Xb/ be the space of linear isomorphisms. The group G D GLn.R/ D Iso.Rn; Rn/ D Aut.Rn/ acts freely and transitively on Eb from the right by composition of linear maps. We have the set map q W E./ D E D b2B Eb! B; Eb! fbg ` with ﬁbrewise GLn.R/-action just explained. If'W p1.U /! U Rn, 'b W Xb! Rn is a bundle chart of p, we deﬁne Q' W q1.U / D b2U Eb! U Iso.Rn; Rn/; ˛ 2 Eb 7!.b; 'b ı ˛/ ` 14.2. Vector Bundles 341 to be a bundle chart for q. The transition function for two such charts has the form.U \ V / Aut.Rn/!.U \ V / Aut.Rn/;.b; / 7!.b; b'1 b /: This map is continuous, because b 7! b'1 is continuous. Therefore there exists a unique topology on E in which the sets q1.U / are open and the charts Q' homeomorphisms. The ﬁbrewise GLn.R/-action on E now becomes continuous and Q' is equivariant. This shows q W E! B to be a GLn.R/-principal bundle. The evaluation Iso.Rn; Xb/ Rn! Xb,.f; u/ 7! f.u/ induces an isomorphism E./ GLn.R/ Rn Š X./ of vector bundles. b (14.2.7) Theorem. The assignment which associated to a GLn.R/-principal bundle E! B the vector bundle E GLn.R/ Rn! B is an equivalence of the category of GLn.R/-principal bundles with the category of n-dimensional real vector bundles; the morphisms are in both cases the bundle maps. Proof. The construction above shows that each vector bundle is, up to isomorphism, in the image of this fun
exists a bundle map R W E! E over r W B I! B I,.b; t/ 7!.b; 1/ which is the identity on EjB 1 and the morphism.R; r/ is a pullback. Proof. (1) By (13.1.6), (3.1.4), and (13.1.8) we choose a numerable countable covering.Uj j j 2 N/ of B such that p is trivial over Uj I. We then choose a numeration.tj / of.Uj /. Let t.x/ D max.tj.x// and set uj.x/ D tj.x/=t.x/. Then the support of uj is contained in Uj and maxfuj.x/ j j 2 Ng D1 holds. Let rj W B I! B I;.x; t/ 7!.x; max.uj.x/; t//: We deﬁne over rj a bundle map Rj W E! E: It is the identity in the complement of p1.Uj I /, and over Uj I a trivialization Uj I G! EjUj I transforms 14.3. The Homotopy Theorem 343 it into.x; t; g/ 7!.x; max.uj.x/; t/; g/: Then Rj is the identity on EjB 1. From the construction we see that.Rj ; rj / is a pullback. The desired bundle map R is the composition of the Rj according to the ordering of N. This is sensible, since for each x 2 E only a ﬁnite number of Rj.x/ are different from x. The condition maxfuj.x/ j j 2 J g D1 shows that R is a map over r. If we apply the previous proof to principal bundles (to vector bundles), then.R; r/ is a bundle map in the corresponding category of bundles. Let p W E! B I be as in (14.3.1) and denote by pt W Et! B t Š B its restriction to B t. We obtain from (14.3.1) a pullback.R; r/ W p! p1. The map r induces from p the product bundle p1 id W E
1 I! B I. We conclude that there exists an isomorphism E Š E1 I of bundles which is the identity E1 D E1 1 over B 1. (14.3.2) Theorem. Under the assumptions of (14.3.1) the bundles E0 and E1 are isomorphic. Proof. We have bundle maps E0 D EjB 0 E Š E1 I pr! E1. 14.3.3 Homotopy Theorem. Let q W E! C be a numerable G-principal bundle and h W B I! C a homotopy. Then the bundles induced from p along h0 and h1 are isomorphic. A similar statement holds for vector bundles. Proof. This follows from the previous theorem, since h j q D.hq/j. 14.3.4 Homotopy lifting. Let X p B ˆ'Y q C be a bundle map between numerable G-principal bundles. Let h W B I! C be a homotopy with h0 D '. Then there exists a homotopy of bundle maps H W X I! Y with H0 D ˆ and q ı H D h ı.p id/. Proof. There exists a diagram i0 X p B Z Q B I Qh h Y q C 344 Chapter 14. Bundles with two pullback squares and hi0 D'and Qh D ˆ. There exists an isomorphism ˛ W X I! Z of p id with Q such that ˛i0 D. The desired homotopy is H D Qh ı ˛. (14.3.5) Theorem. Let q W X! C be a numerable locally trivial map.Then q is a ﬁbration. Proof. Given a homotopy h W B I! C and an initial condition a W B! X. We pull back the bundle along h. The initial condition gives a section of the pullback bundle over B 0. We have seen that the bundle over a product B I is isomorphic to a product bundle, and in a product the section has an obvious extension to B I. We have remarked earlier that the extendibility of the section is equivalent to ﬁnding a lifting of the homotopy with given initial condition. 14.4 Universal Bundles. Classifying Spaces We denote by B
.B; G/ the set of isomorphism classes of numerable G-principal bundles over B. (This is a set!) A continuous map f W B! C induces via pullback a well-deﬁned map B.f / D f W B.C; G/! B.B; G/. We thus obtain a homotopy invariant functor B.; G/. Let pG W EG! BG be a numerable G-principal bundle and ŒB; BG the set of homotopy classes B! BG. Since homotopic maps induce isomorphic bundles, we obtain a well-deﬁned map B W ŒB; BG! B.B; G/; Œf 7! Œf pG: The B constitute a natural transformation. We call the total space EG universal if each numerable free G-space E has up to G-homotopy a unique G-map E! EG. (Thus EG is a terminal object in the appropriate homotopy category.) The corresponding bundle pG W EG! BG is also called universal. Let W E./! B be a numerable G-principal bundle. Then there exists a G-map ˆ W E./! EG and an induced map xˆ W B! BG; and G-homotopic maps induce homotopic maps between the base spaces. We assign to the class Œ xˆ 2 ŒB; BG. Isomorphic bundles yield the same homotopy class. Thus we obtain a well-deﬁned map B W B.B; G/! ŒB; BG, and the B constitute a natural transformation. The compositions B B and B B are the identity. If p0 W E0G! B0G is another universal bundle, then there exist bundle maps ˇ W EG! E0G, W E0G! EG. The compositions ˇ and ˇ are homotopic to the identity as bundle maps. In particular, the spaces BG and B0G are homotopy equivalent. The space BG is called a classifying space of the group G. A map k W B! BG which induces from EG! BG a given bundle q W E! B is called a classifying map of the bundle q. Hence: 14.4. Universal Bundles. Classifying Spaces 345 (14.4.1
) Theorem (Classiﬁcation Theorem). We assign to each isomorphism class of numerable G-principal bundles the homotopy class of a classifying map and obtain a well-deﬁned bijection B.G; B/ Š ŒB; BG. The inverse assigns to k W B! BG the bundle induced by k from the universal bundle. (14.4.2) Theorem. There exist universal G-principal bundles. The proof of the theorem will be given in three steps. (1) Construction of the space EG (14.4.3). (2) Proof that any two G-maps E! EG are G-homotopic (see (14.4.4)). (3) Proof that each numerable G-space E admits a G-map (see (14.4.5)). 14.4.3 The Milnor space. We present a construction of the universal bundle which is due to Milnor [131]. It uses the notion of a join of a family of spaces. Let.Xj j j 2 J / be a family of spaces Xj. The join X D j 2J Xj is deﬁned as follows. The elements of X are represented by families.tj xj j j 2 J /; tj 2 Œ0; 1; xj 2 Xj ; P j 2J tj D 1 in which only a ﬁnite number of tj are different from zero. The families.tj xj / and.uj yj / represent the same element of X if and only if (1) tj D uj for each j 2 J, (2) xj D yj whenever tj ¤ 0. The notation tj xj is short-hand for the pair.tj ; xj /. This is suggestive, since we can replace 0xj by 0yj for arbitrary xj and yj in Xj. We therefore have coordinate maps tj W X! Œ0; 1;.ti xi / 7! tj ; pj W t 1 j 0; 1! Xj ;.ti xi / 7! xj : The Milnor topology on X shall be the coarsest topology for which all tj and pj are continuous. This topology is characterized by the following universal property: A map f W
Y! X from any space Y is continuous if and only if the maps tj f W Y! Œ0; 1 and pj f W f 1t 1 0; 1! Xj are continuous. For a ﬁnite number of spaces we use the notation X1?? Xn for their join. j If the spaces Xj are right G-spaces, then..tj xj /; g/ 7!.tj xj g/ deﬁnes a continuous action of G on X. Continuity is veriﬁed with the universal property of the join topology. The Milnor space is EG D G? G? G? ; a join of a countably inﬁnite number of copies of G. We write BG D EG=G for the orbit space and p W EG! BG for the orbit map. 346 Chapter 14. Bundles It remains to show that EG! BG is numerable. The coordinate functions tj are G-invariant and induce therefore functions j on BG. The j are a point-ﬁnite partition of unity subordinate to the open covering by the Vj =G, Vj D t 1 0; 1. The bundle is trivial over Vj =G, since we have, by construction, G-maps pj W Vj! G. Þ j (14.4.4) Proposition. Let E be a G-space. Any two G-maps f; g W E! EG are G-homotopic. Proof. We consider the coordinate form of f.x/ and g.x/,.t1.x/f1.x/; t2.x/f2.x/; : : : / and.u1.x/g1.x/; u2.x/g2.x/; : : : /; and show that f and g are G-homotopic to maps with coordinate form.t1.x/f1.x/; 0; t2.x/f2.x/; 0; : : : / and.0; u1.x/g1.x/; 0; u2.x/g2.x/; : : : / where 0 denotes an element of the form 0 y. In order to achieve this, for f say, we construct a homotopy in an inﬁ
nite number of steps. The ﬁrst step has in the homotopy parameter t the form.t1f1; t t2f2;.1 t/t2f2; t t3f3;.1 t/t3f3; : : : /: It removes the ﬁrst zero in the ﬁnal result just stated. We now iterate this process appropriately. We obtain the desired homotopy by using the ﬁrst step on the interval Œ0; 1 2 ; 3 4, and so on. The total homotopy is continuous, since in each coordinate place only a ﬁnite number of homotopies are relevant. 2, the second step on the interval Œ 1 Having arrived at the two forms above, they are now connected by the homotopy..1 t/t1f1; tu1g1;.1 t/t2f2; tu2g2; : : : / in the parameter t. (14.4.5) Proposition. Let E be a G-space. Let.Un j n 2 N/ be an open covering by G-trivial sets. Suppose there exists a point-ﬁnite partition of unity.vn j n 2 N/ by G-invariant functions subordinate to the covering.Un/. Then there exists a G-map'W E! EG. A numerable free G-space E admits a G-map E! EG. Proof. By deﬁnition of a G-trivial space, there exist G-maps 'j W Uj! G. The desired map'is now given by '.z/ D.v1.z/'1.z/; v2.z/'2.z/; : : : /. It is continuous, by the universal property of the Milnor topology. In order to apply the last result to the general case, we reduce arbitrary partitions of unity to countable ones (see (13.1.8)). (14.4.6) Proposition. The space EG is contractible. 14.4. Universal Bundles. Classifying Spaces 347 Proof. We have already seen that there exists a homotopy of the identity to the map.tj gj / 7!.t1g1; 0; t2g2; t3g3
; : : : /. The latter map has the null homotopy..1 t/t1g1; te;.1 t/t2g2; : : : /. (14.4.7) Example. The locally trivial map p W EG! BG is a ﬁbration with contractible total space by (14.3.5). We also have the path ﬁbration P! BG with contractible total space and ﬁbre BG. We can turn a homotopy equivalence EG! P into a ﬁbrewise map, and this map is then a ﬁbrewise homotopy equivalence. Hence we have a homotopy equivalence BG'G. The exact homotopy Þ sequence then yields an isomorphism @ W n.BG/ Š n1.G/. (14.4.8) Example. For a discrete group, BG is an Eilenberg–Mac Lane space of type K.G; 1/. The space BS1 is an Eilenberg–Mac Lane space K.Z; 2/. Models Þ for BZ=2 and BS1 are RP 1 and CP 1, respectively. A continuous homomorphism ˛ W K! L induces the map E.˛/ W EK! EL;.ti ki / 7!.ti ˛.ki // which is compatible with the projections to the classifying spaces. We obtain an In this manner B becomes a functor from the induced map B.˛/ W BK! BL. category of topological groups into TOP. (14.4.9) Proposition. An inner automorphism ˛ W K! K, k 7! uku1 induces a map B.˛/ which is homotopic to the identity. Proof. The map.ti ki / 7!.ti uki / is a K-map and therefore K-homotopic to the identity. The assignment.ti ki / 7!.ti uki u1/ induces the same map between the orbit spaces. (14.4.10) Proposition. Let X be a free numerable G-space. Then the join E D X? X? is a universal G-space. Proof. As in the proof of 14.4.3 we see that any two G-maps into E are G- homotopic. Since X is numerable, so is
E. (14.4.11) Corollary. Let H be a subgroup of G. Assume that G is numerable as H -space. Then EG is, considered as H -space, universal. The next theorem characterizes universal bundles so that we need not rely on a special construction. (14.4.12) Theorem. A numerable G-principal bundle q W E! B is universal if and only if E is contractible (as a space without group action). 348 Chapter 14. Bundles Proof. We know already that Milnor’s space EG is contractible. If p is universal, the G-space E is G-homotopy equivalent to EG and hence contractible. Conversely, assume that E is contractible. Then the associated ﬁbre bundle E G EG! B has a contractible ﬁbre EG and is therefore shrinkable (use (13.3.3)). Hence it has a section and any two sections are homotopic as sections. A section corresponds to a bundle map ˛ W E! EG (see (14.1.4). For the same reason there exists a bundle map ˇ W EG! E. By 14.4.3, ˛ˇ is homotopic to the identity as a bundle map. In order to see that ˇ˛ is homotopic id.E/, we use that sections are homotopic. We compare classifying spaces of different groups and discuss the functorial properties of classifying spaces. Let ˛ W K! L be a continuous homomorphism between topological groups. We denote by ˛L the K-space K L! L;.k; l/ 7! ˛.k/ l: The associated bundle E.K/ K ˛L! B.K/ inherits a right L-action and is an L-principal bundle. It has a classifying map B.˛/ W B.K/! B.L/. For the Milnor bundle the homotopy class is the same as the one already deﬁned. If ˇ W L! M is a further homomorphism, then the relation B.ˇ/B.˛/'B.ˇ˛/ is easily veriﬁed. Let i W H G be the inclusion of a subgroup
. We restrict the G-action to H and obtain a free and contractible H -space resH EG. If G! G=H is a numerable H -principal bundle, then resH EG is numerable as H -space; hence we have in this case in resH EG!.resH EG/=H as model for EH! BH. We then obtain, because of EG G H Š EG=H, a map Bi W BH D.EG/=H!.EG/=G D BG; which is a ﬁbre bundle with ﬁbre G=H. If G=H is contractible, then Bi is a numerable ﬁbration with contractible ﬁbre, hence a homotopy equivalence. This situation occurs for the inclusions O.n/! GLn.R/ and U.n/! GLn.C/, and in general for the inclusion K G of a maximal compact subgroup K of a connected Lie group G [84, p. 180]. (14.4.13) Proposition. The inclusions of subgroups induce homotopy equivalences BO.n/! BGLn.R/ and BU.n/! BGLn.C/. Let H be a normal subgroup of G. Then E.G=H / E.G/ is a numerable free G-space; hence.E.G=H / EG/=G is a model for BG. (In general, for each G-space X which is contractible, the product X EG is another model for EG.) With this model and the orbit map of the projection E.G=H / EG! E.G=H / we obtain a map p W BG! B.G=H / which is a ﬁbre bundle with structure group G=H and ﬁbre BH. In this case BH D EG=H with induced G=H -action. The map 14.4. Universal Bundles. Classifying Spaces 349 p and the inclusion i W BH! BG of a ﬁbre are induced maps of the type B for the cases i W H G and W G! G=H. Therefore we have a ﬁbre bundle BH Bi! BG Bp! B.G=H /: Principal bundles for a discrete group G are covering spaces; this holds in particular for the universal
bundle EG! BG. Since EG is simply connected, it is also the universal covering of BG. Thus two notions of “universal” meet. What is the relation between these concepts? Let B be a pathwise connected space with universal covering p W E! B, a right -principal covering for D 1.B/. Let'W! G be a homomorphism. We have as before the right G-principal covering E 'G. If'and are conjugate homomorphisms, i.e., if g'.a/g1 D.a/ for a g 2 G, then E 'G! E G;.x; h/ 7!.x; gh/ is an isomorphism of G-principal coverings. The assignment'7! E 'G is a map ˛ W Hom.; G/c! B.G; B/ from the set of conjugacy classes of homomorphisms (index c) to the set of isomorphism classes of G-principal bundles over B. (14.4.14) Proposition. The map ˛ is a bijection. (14.4.15) Example. Let G be discrete and abelian. Then conjugate homomorphisms are equal. If all coverings of B are numerable (say B paracompact), then B.B; G/ D ŒB; BG. A bijection ˛ W Hom.; G/ Š ŒB; BG D H 1.BI G/ is obtained from (14.4.14). For the last equality note that BG is an Eilenberg–Mac Þ Lane space and represents the ﬁrst cohomology. (14.4.16) Example. The ﬁbration U.n/=U.n 1/! BU.n 1/! BU.n/ and U.n/=U.n 1/ Š S 2n1 show that the map i.n/ W BU.n 1/! BU.n/ induced by the inclusion U.n 1/ U.n/ is.2n 1/-connected. The induced map i.n/ W ŒX; B.n 1/! ŒX; BU.n/ is therefore bijective (surjective) for a CWcomplex X of dimension dim X < 2n1 (dim X
2n1). So if dim X 2n 2, then a k-dimensional complex vector bundle over X is isomorphic to ˚.k n/" Þ for a unique isomorphism class of an n-dimensional bundle. Problems 1. Work out a proof of (14.4.14). 2. The canonical diagram K ˛ L'BK B.˛/'BL 350 Chapter 14. Bundles is homotopy commutative. See (14.4.7). 3. The abelianized group 1.B/ is isomorphic to the homology group H1.BI Z/. Therefore we can write (14.4.15) in the form ˛ W Hom.H1.B/; G/ Š H 1.BI G/. The classifying map f W B! BG of a G-principal bundle q W X! B induces homomorphisms f W 1.X/! 1.B/ and f W H1.B/! H1.BG/. If G is abelian and discrete, then G Š 1.BG/ Š H1.BG/. We thus obtain a map ˇ W ŒB; BG! Hom.H1.B/; G/: Under the hypotheses of (14.4.15), ˇ is inverse to ˛. 4. We give an example of a non-numerable bundle. The equation xz C y2 D 1 yields a hyperboloid Q in R3. The action of the additive group R on R3 c.x; y; z/ D.x; y C cx; z 2cy c2x/; is free on Q, and Q becomes an R-principal bundle. Numerable R-principal bundles are trivial, since R is contractible. The bundle Q is non-trivial. The orbit space is the non-Hausdorff line with two origins. If the bundle were trivial it would have a section, and this would imply that the orbit space is separated. 5. The join S m? S n is homeomorphic to S mCnC1,.t1z1; t2z2/ 7!. t2z2/ is a homeomorphism. The join of k copies S 1 is homeomorphic to S 2k1. A suitable homeomorphism respects the S 1-action,
if we let S 1 act on S 2k1 by scalar multiplication.; v/ 7! v. The Milnor construction thus yields in this case the Hopf bundle S 2k1! CP k1. 6. A suitable isomorphism n.BG/ Š n1.G/ has the following interpretation. Let p W E! S n be a G-principal bundle. Write S n D DC [ D, S n1 D DC \ D as usual. Then pjDC and pjD are trivial. Choose trivializations t˙ W p1.D˙/! D˙ G. They differ over S n1 by an automorphism t1z1; p p S n1 G! S n1 G;.x; g/ 7!.x; ˛x.g// of principal bundles. Hence ˛x.g/ D ˛x.e/g, and x 7! ˛x.e/ represents an element in n1.G/ which corresponds under the isomorphism in question to the classifying map of p. 7. The canonical map S 1! CP 1 is an S 1-principal bundle with contractible total space. Hence CP 1 is a model for BS1. This space is also an Eilenberg–Mac Lane space of type K.Z; 2/. In a similar manner one has B.Z=2/ D RP 1 D K.Z=2; 1/. 8. Suppose the Xj are Hausdorff spaces. Then their join is a Hausdorff space. 9. The map.X1? X2/? X3! X1? X2? X3 which sends.u1.t1x1; t2x2/; u2x3/ to.u1t1x1; u1t2x2; u2x3/ is a homeomorphism. Discuss in general the associativity of the join. 10. The join of a family.Xj j j 2 J / is a subspace of the product j 2J CXj of cones, when the cone CX D I X=0 X is given the Milnor topology with coordinate functions t W.x; t/ 7! t and t 10; 1! X,.x; t/ 7! x (and not the quotient topology as previously used
dles. Examples are: V V ˚ W V ˝ W ƒi V Hom.V; W / dual space of V direct sum tensor product i-th exterior power ƒi homomorphisms ˚ ˝ Hom.; / dual bundle of Whitney sum tensor product i-th exterior power homomorphism bundle Canonical isomorphisms between algebraic constructions yield canonical isomorphisms for the corresponding vector bundles. Examples are:. ˚ / ˝ Š. ˝ / ˚. ˝ / Hom.; / Š ˝ ƒk. ˚ / Š L iCj Dk.ƒi ˝ ƒj /: In the last isomorphism ƒ0 is the trivial one-dimensional bundle and ƒ1 Š. In order to prove such statements, one has to use that the constructions of linear algebra are in an appropriate sense continuous. It sufﬁces to consider an example, say the tensor product. Let W E./! B and W E./! B be real vector bundles. The total space of ˝ has the underlying set S b2B.b ˝ b/ D E. ˝ /; the disjoint union of the tensor products of the ﬁbres. Let'W 1.U /! U Rm be a bundle chart of and W 1.U /! U Rn a chart of. Then a bundle chart for ˝ over U should be S W b2U.b ˝ b/! Uj.Rm ˝ Rn/; the ﬁbre b ˝ b is mapped by the tensor product of the linear maps'over b and over b. At this point it is now important to observe that the transition maps of such charts are homeomorphisms. Therefore there exists a unique topology on E. ˝ / such that the sources of the are open and the are homeomorphisms. In 14.5. Algebra of Vector Bundles 353 this manner we have obtained the data of the bundle ˝. In dealing with tensor products one has to distinguish ˝R for real bundles and ˝C for complex bundles. If we start with bundles W E./! B and W E./! C we obtain in a similar
manner a bundle O˝ over B C with ﬁbres b ˝ c. It is called the exterior tensor product. Let B D C and let d W B B be the diagonal; then d. O˝/ D ˝. Let p W B C! B and q W B C! C be the projections; then O˝ D p ˝ q. (14.5.3) Example. Let p W E! B be an S 1-principal bundle and W E./ D E S 1 C! B the associated complex line bundle. Then p is the unit-sphere bundle of. Let Cm S 1 be the cyclic subgroup of order m. The m-fold tensor product ˝m D ˝ ˝ is E S 1 C.m/, where S 1 acts on C by.; z/ 7! mz. The unit-sphere bundle of ˝m is E=Cm! B. If we use the model S 1! BS1 for the universal S 1-principal bundle, we obtain the canonical map BCm! BS1 as the sphere bundle of the m-fold tensor product of the universal line bundle. Þ (14.5.4) Example. Let p W E! B be a right G-principal bundle. Let V; W be complex G-representations. Let pV W E G V! B be the associated complex vector bundle. Then there are canonical isomorphisms pV ˚ pW Š pV ˚W and pV ˝pW Š pV ˝W. For the bundles H.k/ over CP n the relations H.k/˝CH.l/ Š Þ H.k C l/ hold. 14.5.5 Complex vector bundles over S 2 D CP 1. We have the line bundles H.k/ over CP 1 for k 2 Z. The total space is H.k/ D.C2 X 0/ C C with equivalence relation..z0; z1/; u/..z0; z1/; ku/. Set D H.1/; then n Š H.n/. Let be an arbitrary line bundle over CP 1. We have the charts '0 W C! U0 D fŒz0; z1 j z0
6D 0g and '1 W C! U1fŒz0; z1 j z1 6D 0g. We pull back along 'j and obtain a trivial bundle. Let ˆj W C C! jUj be a trivialization. Then 1 ˆ0 W C C! C C has the form.z; u/ 7!.z1; az u/ for some map ˆ1 a W C! C. The map a is homotopic to a map z 7! zk. We use a homotopy in order to construct a bundle over CP 1 Œ0; 1 which is over CP 1 0 given by the gluing.z; u/ 7!.z1; azu/ and over CP 1 1 by the gluing.z; u/ 7!.z1; zku/. The latter gives H.k/. By the homotopy theorem we see that is isomorphic to H.k/. Let B denote the set of isomorphism classes of complex line bundles over CP 1 with tensor product as composition law (see Problem 4). We have just seen that W Z! B, k 7! H.k/ is a surjective homomorphism. We know that B Š ŒCP 1; CP 1 Š 2.CP 1/ Š Z. Therefore has a trivial kernel, because otherwise B would be a ﬁnite cyclic group. Altogether we have seen that the H.k/ represent the isomorphism classes of complex line bundles. Now we use (14.4.16) and see that a k-dimensional bundle (k 1) is isomorphic to H.n/ ˚.k 1/" for a unique n 2 Z. Bundles over CP 1 have a cancellation property: An isomorphism ˚ Š ˚ implies Š ; this is again a consequence of (14.4.16). 354 Chapter 14. Bundles For the bundles H.k/ over CP 1 the relations H.k/ ˚ H.l/ Š H.k C l/ ˚ " hold. In order to prove this relation, we construct an isomorphism of ˚ 1 to the trivial bundle. We write the bundle in the form S 3 S 1.C.1/ ˚ C.1//. A ﬁbrewise map to C
2 is given by..z0; z1/;.u0; u1// 7!.z0u0 xz1u1; z1u0 C xz0u1/: Observe that the matrix with rows.z0; xz1/;.z1; xz0/ is unitary. For u0 the image is the tautological bundle H.1/, for u1 the orthogonal complement H.1/. We show by induction k C.k 1/" D k. This relation is clear for k D 1 and follows for k D 2 from ˚ 1 D 2". Multiply k C.k 1/" D k by, add k" and cancel.k 1/; the desired relation for k C 1 drops out. Suppose k; l 2 N. Then k ˚ l ˚.k 1 C l 1/" D.k C l/ D kCl ˚.k C l 1/"; and cancellation of.k 1 C l 1/" gives k ˚ l D kCl ˚ ". We multiply this Þ relation by k; l, or.kCl/ in order to verify the remaining cases. Problems 1. Let and be vector bundles over B. An orientation of and induces an orientation of ˚, ﬁbrewise the sum orientation of the vector spaces. If two of the bundles,, and ˚ are orientable, then the third is orientable. 2. In a bundle with Riemannian metric the ﬁbrewise orthogonal complement of a subbundle is a subbundle. 3. Let p W E! B be an n-dimensional bundle with Riemannian metric. Then there exists a bundle atlas such that the transition maps have an image in the orthogonal group O.n/. The structure group is therefore reducible to O.n/. If the bundle is orientable, then the structure group is reducible to SO.n/. 4. Let and be complex line bundles over B. Then ˝C is again a line bundle. The bundle ˝C is trivial; the assignments b ˝ b! C,.x; / 7!.x/ are an isomorphism to the trivial bundle. The isomorphism classes of complex line bundles are an abelian group with composition law the tensor product. 5. Let X be
a normal space and Y X a closed subset. A section s W Y! EjY over Y of a numerable vector bundle W E! X has an extension to a section over X. 6. Let p W E! X and q W F! X be vector bundles. The bundle morphisms E! F correspond to the sections of Hom.E; F /! X. 7. Let p W E! X and q W F! X be numerable bundles over the normal space X. If f W EjY! F jY is an isomorphism over the closed set Y, then there exists an open neighbourhood U of Y and an isomorphism f W EjU! F jU which extends f over Y. 8. The map CP a CP b! CP aCb,.Œxi ; Œyj / 7! Œzk with xt D iCj Dt xi yj induces from H.1/ the exterior tensor product H.1/ O˝H.1/. In the case a D b D 1 the map is associative and deﬁnes the structure of an H -space. It induces on ŒB; CP 1 the group P 14.6. Grothendieck Rings of Vector Bundles 355 structure on the set of line bundles given by the tensor product. 9. Determine two-dimensional real bundles over RP 2 [44, p. 434]. 14.6 Grothendieck Rings of Vector Bundles Denote by V.X/ the set of isomorphism classes of complex vector bundles over X. The Whitney sum and the tensor product induce on V.X/ two associative and commutative composition laws (addition C, multiplication ), and the distributive law holds. Addition has a zero element, the 0-dimensional bundle; multiplication has a unit element, the 1-dimensional trivial bundle. A commutative monoid M is a set together with an associative and commutative composition law C with zero element. A universal group for M is a homomorphism W M! K.M / into an abelian group K.M / such that each monoidhomomorphism'W M! A into an abelian group A has a unique factorization ˆ ı D'with a homomorphism ˆ W K.M /! A. A monoid-homomorphism
f W M! N induces a homomorphism K.f / W K.M /! K.N /. Let N M be a submonoid. We deﬁne an equivalence relation on M by x y, there exist a; b 2 N such that x C a D y C b. Let p W M! M=N, x 7! Œx denote the quotient map onto the set of equivalence classes. We obtain by Œx C Œy D Œx C y a well-deﬁned composition law on M=N which is a monoid structure. In the product monoid M M we have the diagonal submonoid D.M / D f.m; m/g. We set K.M / D.M M /=D.M / and.x/ D Œx; 0. Then K.M / is an abelian group and a universal homomorphism. Since Œx; 0 C Œ0; x D Œx; x D 0, we see Œ0; x D.x/. The elements of K.M / are formal differences x y, x; y 2 M,.x/.y/ D Œx; y, and x y D x0 y0 if and only if x Cy0 Cz D x0 Cy Cz holds as equality in M for some z 2 M. We apply these concepts to M D V.X/ and write K.X/ D K.V.X//. The tensor product induces a bi-additive map V.X/ V.X/! V.X/. It induces a bi-additive map in the K-groups: (14.6.1) Proposition. Let A, B, C be abelian monoids and m W A B! C be a bi-additive map. The there exists a unique bi-additive map K.m/ such that the diagram A B AB K.A/ K.B/ m K.m/ C C K.C / is commutative. 356 Chapter 14. Bundles The bi-additive map induced by the tensor product is written as multiplication, and K.X/ becomes in this way a commutative ring. This ring is often called the Grothendieck ring of complex vector bundles. In general, the universal groups K.M / are
called Grothendieck groups. We can apply the same construction to real vector bundles and obtain the Grothendieck ring KO.X/. Other notations for these objects are KO.X/ D KR.X/ and K.X/ D KU.X/ D KC.X/. Pullback of bundles along f W X! Y induces a ring homomorphism K.f / D f W K.Y /! K.X/ and similarly for KO. Homotopic maps induce the same homomorphism. (14.6.2) Example. K.S 2/ is free abelian as an additive group with basis 1 and. The multiplicative structure is determined by 2 D 2 1. This is a consequence Þ of 14.5.5. The inclusions U.n/! U.n C 1/, A 7! are used to deﬁne U D colim U.n/, a topological group with the colimit topology. The inclusion of groups U.n/! U induces BU.n/! BU. If we compose a classifying map X! BU.n/ with this map, we call the result X! BU the stable classifying map. Bundles and ˚ a" are called stably equivalent, and they have the same stable classifying map (up to homotopy). A 0 0 1 (14.6.3) Proposition. Let X be a path connected compact Hausdorff space. Then there exists a natural bijection K.X/ Š ŒX; Z BU. Here Z carries the discrete topology. Proof. Let Œ Œ 2 K.X/. A bundle over a compact Hausdorff space has an inverse bundle (see (14.5.2)). Hence Œ ˚ Œ ˚ 2 K.X/ is the same element. Therefore each element in K.X/ can be written in the form Œ n. Suppose Œ n D Œ m. Then ˚ m" ˚ Š ˚ n" ˚ for some. We add an inverse of and arrive at a relation of the form ˚ a" Š ˚ b", i.e., and are stably equivalent. The homotopy class of a stable classifying map k W X! BU is therefore uniquely determined by the element Œ n. We de�
��ne W K.X/! ŒX; Z BU by sending Œ n to k W X!.dim n/ BU. Conversely, let f W X! Z BU be given. Since X is path connected, the image is contained in some k BU. The compactness of X is used to verify that f admits a factorization X! BU.n/! BU. We obtain a well-deﬁned inverse map ŒX; Z BU! K.X/, if we assign to f W X! BU.n/! BU the element Œf n n Œn k. For more general spaces the Grothendieck ring K.X/ can differ substantially from the homotopy group ŒX; Z BU, e.g., for X D CP 1. The latter is a kind of completion of the Grothendieck ring. 14.6. Grothendieck Rings of Vector Bundles 357 The (exterior) tensor product of bundles yields a ring homomorphism K.X/ ˝Z K.Y /! K.X Y /: A fundamental result is the periodicity theorem of Bott. One of its formulations is: For compact spaces X the tensor product yields isomorphisms K.X/ ˝ K.S 2/ Š K.X S 2/; KO.X/ ˝ KO.S 8/ Š KO.X S 8/: Starting from this isomorphism one constructs the cohomology theories which are called K-theories. For an introduction see [15], [9], [10], [12], [11], [13], [14], [102]. For the Bott periodicity see also [6], [106], [19]. Chapter 15 Manifolds This chapter contains an introduction to some concepts and results of differential topology. For more details see [30], [44], [107]. We restrict attention to those parts which are used in the proof of the so-called Pontrjagin–Thom theorem in the chapter on bordism theory. We do not summarize the results here, since the table of contents should give enough information. 15.1 Differentiable Manifolds A topological space X is n-dimensional locally Euclidean if each x 2 X has an open neighbourhood U which is homeomorphic to an open subset V of Rn. A homeomorphism h W U! V
is a chart or local coordinate system of X about x with chart domain U. The inverse h1 W V! U is a local parametrization of X about x. If h.x/ D 0, we say that h and h1 are centered at x. A set of charts is an atlas for X if their domains cover X. If X is n-dimensional locally Euclidean, we call n the dimension of X and write dim X D n. The dimension is well-deﬁned, by invariance of dimension. An n-dimensional manifold or just n-manifold is an n-dimensional locally Euclidean Hausdorff space with countable basis for its topology. Hence manifolds are locally compact. A surface is a 2-manifold. A 0-manifold is a discrete space with at most a countably inﬁnite number of points. The notation M n is used to indicate that n D dim M. Suppose.U1; h1; V1/ and.U2; h2; V2/ are charts of an n-manifold. Then we have the associated coordinate change or transition function h2h1 1 W h1.U1 \ U2/! h2.U1 \ U2/; a homeomorphism between open subsets of Euclidean spaces. Recall: A map f W U! V between open subsets of Euclidean spaces (U Rn; V Rm) is aC k-map if it is k-times continuously differentiable in the ordinary sense of analysis (1 k 1). A continuous map is also called a C 0-map. A C k-map f W U! V has a differential Df.x/ W Rn! Rm at x 2 U. If the coordinate changes h2h1 1 are C k-maps, we call the charts.U1; h1; V1/ and.U2; h2; V2/ C k-related (1 k 1). An atlas is a C k-atlas if any two of its charts are C k-related. We call C 1-maps smooth or just differentiable; similarly, we talk about a smooth or differentiable atlas. and h1h1 2 15.1. Differentiable Manifolds 359 (15.1.1) Proposition. Let A be a smooth atlas for
M. The totality of charts which are smoothly related to all charts of A is a smooth atlas D.A/. If A and B are smooth atlases, then A [ B is a smooth atlas if and only if D.A/ D D.B/. The atlas D.A/ is the uniquely determined maximal smooth atlas which contains A. A differential structure on the n-manifold M is a maximal smooth atlas D for M. The pair.M; D/ is called a smooth manifold. A maximal atlas serves just the purpose of this deﬁnition. Usually we work with a smaller atlas which then generates a unique differential structure. Usually we omit the differential structure from the notation; the charts of D are then called the charts of the differentiable manifold M. Let M and N be smooth manifolds. A map f W M! N is smooth at x 2 M if f is continuous at x and if for charts.U; h; U 0/ about x and.V; k; V 0/ about f.x/ the composition kf h1 is differentiable at h.x/. We call kf h1 the expression of f in local coordinates. The map f is smooth if it is differentiable at each point. The composition of smooth maps is smooth. Thus we have the category of smooth manifolds and smooth maps. A diffeomorphism is a smooth map which has a smooth inverse. Manifolds M and N are diffeomorphic if there exists a diffeomorphism f W M! N. Smooth manifolds M and N have a product in the category of smooth manifolds. The charts of the form.U V; f g; U 0 V 0/ for charts.U; f; U 0/ of M and.V; g; V 0/ of N deﬁne a smooth structure on M N. The projections onto the factors are smooth. The canonical isomorphisms Rm Rn D RmCn are diffeomorphisms. A subset N of an n-manifold M is a k-dimensional submanifold of M if the following holds: For each x 2 N there exists a chart h W U! U 0 of M about x such that h.U \ N / D U 0 \.Rk 0/. A chart with this property is called adapted to N. The difference n k is the codimension of N in M.
(The subspace Rk 0 of Rn may be replaced by any k-dimensional linear or afﬁne subspace if this is convenient.) If we identify Rk 0 D Rk, then.U \ N; h; U 0 \ Rk/ is a chart of N. If M is smooth, we call N a smooth submanifold of M if there exists about each point an adapted chart from the differential structure of M. The totality of charts.U \ N; h; U 0 \ Rk/ which arise from adapted smooth charts of M is then a smooth atlas for N. In this way, a differentiable submanifold becomes a smooth manifold, and the inclusion N M is a smooth map. A smooth map f W N! M is a smooth embedding if f.N / M is a smooth submanifold and f W N! f.N / a diffeomorphism. The spheres are manifolds which need an atlas with at least two charts. We have the atlas with two charts.UN ; 'N / and.US ; 'S / coming from the stereographic projection (see (2.3.2)). The coordinate transformation is 'S ı '1 N.y/ D kyk2y. The differential of the coordinate transformation at x is 7!.kxk2 2hx; ix/ kxk4. For kxk D1 we obtain the reﬂection 7! 2hx; ix in a hyperplane. We now want to construct charts for the projective space RP n. The subset 360 Chapter 15. Manifolds Ui D fŒx0; : : : ; xn j xi ¤ 0g is open. The assignment 'i W Ui! Rn; Œx0; : : : ; xn 7! x1 i.x0; : : : ; xi1; xiC1; : : : ; xn/ is a homeomorphism. These charts are smoothly related. Charts for CP n can be deﬁned by the same formulas. Note that CP n has dimension 2n as a smooth manifold. (It is n-dimensional when viewed as a socalled complex manifold.) (15.1.2) Proposition. Let M be an n-manifold and U D.Uj j j 2 J / an open covering of
M. Then there exist charts.Vk; hk; Bk j k 2 N/ of M with the following properties: (1) Each Vk is contained in some member of U. (2) Bk D U3.0/ D fx 2 Rn j kxk < 3g. (3) The family.Vk j k 2 N/ is a locally ﬁnite covering of M. In particular, each open cover has a locally ﬁnite reﬁnement, i.e., manifolds are paracompact. If M is smooth, there exists a smooth partition of unity.k j k 2 N/ subordinate to.Vk/. There also exists a smooth partition of unity.˛j j j 2 J / such that the support of ˛j is contained in Uj and at most a countable number of the ˛j are non-zero. Proof. The space M is a locally compact Hausdorff space with a countable basis. Therefore there exists an exhaustion M0 M1 M2 M D S 1 iD1 Mi by open sets Mi such that SMi is compact and contained in MiC1. Hence Ki D SMiC1XMi is compact. For each i we can ﬁnd a ﬁnite number of charts.V; h; B/, B D U3.0/, such that V Uj for some j and such that the h1 U1.0/ cover Ki and such that V MiC2 X SMi1 (M1 D ;). Then the V form a locally ﬁnite, countable covering of M, now denoted.Vk; hk; Bk j k 2 N/. The function W R! R,.t/ D 0 for t 0,.t / D exp.1=t/ for t > 0, is a C 1-function. For " > 0, the function '".t/ D.t /..t/ C." t//1 is C 1 and satisﬁes 0 '" 1, '".t/ D 0, t 0, '".t/ D 1, t ". Finally, W Rn! R, x 7! '".kxk r/ is a C 1-map which satisﬁes 0.x/ 1,.x/ D 1, x 2
Ur.0/,.x/ D 0, kxk r C ". We use these functions for r D 1 and " D 1 and deﬁne i by ı hi on Vi 1 j D1 j yield a and as zero on the complement. Then the k D s1 k with s D smooth, locally ﬁnite partition of unity subordinate to.Vk j k 2 N/. P The last statement follows from (13.1.2). Let C0 and C1 be closed disjoint subsets of the smooth manifold M. Then there exists a smooth function'W M! Œ0; 1 such that '.Cj / fj g; apply the previous proposition to the covering by the Uj D M X Cj. 15.1. Differentiable Manifolds 361 Let A be a closed subset of the smooth manifold M and U an open neighbourhood of A in M. Let f W U! Œ0; 1 be smooth. Then there exists a smooth function F W M! Œ0; 1 such that F jA D f jA. For the proof choose a partition of unity.'0; '1/ subordinate to.U; M X A/. Then set F.x/ D '0.x/f.x/ for x 2 U and F.x/ D 0 otherwise. (15.1.3) Proposition. Let M be a submanifold of N. A smooth function f W M! R has a smooth extension F W N! R. Proof. From the deﬁnition of a submanifold we obtain for each p 2 M an open neighbourhood U of p in N and a smooth retraction r W U! U \ M. Hence we can ﬁnd an open covering.Uj j j 2 J / of M in N and smooth extensions fj W Uj! R of f jUj \ M. Let.˛j j j 2 J / be a subordinate smooth partition of unity and set F.x/ D j 2J ˛j.x/fj.x/, where a summand is deﬁned to be zero if fj.x/ is not deﬁned. P (15.1.4) Proposition. Let M be a smooth manifold. There exists a smooth proper function f W M! R. S Proof. A function between Hausdorff spaces
is proper if the pre-image of a compact set is compact. We choose a countable partition of unity.k j k 2 N/ such that the kD1 k k W M! R. If functions k have compact support. Then we set f D x … j >n j.x/ and therefore f.x/ D S P n j D1 supp.j / and therefore j >n jj.x/ > n. Hence f 1Œn; n is contained in n j D1 supp.j /, then 1 D j 1 j.x/ D P P P 1 compact. In working with submanifolds we often use, without further notice, the following facts. Let M be a smooth manifold and A M. Then A is a submanifold if and only if each a 2 A has an open neighbourhood U such that A \ U is a submanifold of U. (Being a submanifold is a local property.) Let f W N1! N2 be a diffeomorphism. Then M1 N1 is a submanifold if and only if f.M1/ D M2 N2 is a submanifold. (Being a submanifold is invariant under diffeomorphisms.) Important objects in mathematics are the group objects in the smooth category. A Lie group consists of a smooth manifold G and a group structure on G such that the group multiplication and the passage to the inverse are smooth maps. The fundamental examples are the classical matrix groups. A basic result in this context says that a closed subgroup of a Lie group is a submanifold and with the induced structure a Lie group [84], [29]. Problems 1. The gluing procedure (1.3.7) can be adapted to the smooth category. The maps gj i are assumed to be diffeomorphisms, and the result will be a locally Euclidean space. Again one has to take care that the result will become a Hausdorff space. 362 Chapter 15. Manifolds 2. Let E be an n-dimensional real vector space 0 < r < n. We deﬁne charts for the Grassmann manifold Gr.E/ of r-dimensional subspaces of E. Let K be a subspace of codimension r in E. Consider the set of complements in K U.K/ D fF 2 Gr.E/ j F
˚ K D Eg: The sets are the chart domains. Let P.K/ D fp 2 Hom.E; E/ j p2 D p; p.E/ D Kg be the set of projections with image K. Then P.K/! U.K/, p 7! Ker.p/ is a bijection. The set P.K/ is an afﬁne space for the vector space Hom.E=K; K/. Let j W K E and let q W E! E=K/ be the quotient map. Then Hom.E=K; K/ P.K/! P.K/;.'; p/ 7! p C j'q is a transitive free action. We choose a base point p0 2 P.K/ in this afﬁne space and obtain a bijection U.K/ P.K/! Hom.E=K; K/; Ker.p/ Í p 7! p p0: The bijections are the charts for a smooth structure. 3. f.x; y; z/ 2 R3 j z2x3 C 3zx2 C 3x zy2 2y D 1g is a smooth submanifold of R3 diffeomorphic to R2. If one considers the set of solutions.x; y; z/ 2 C2, then one obtains a smooth complex submanifold of C3 which is contractible but not homeomorphic to C2 (see [47]). 15.2 Tangent Spaces and Differentials We associate to each point p of a smooth m-manifold M an m-dimensional real vector space Tp.M /, the tangent space of M at the point p, and to each smooth map f W M! N a linear map Tpf W Tp.M /! Tf.p/.N /, the differential of f at p, such that the functor properties hold (chain rule) Tp.gf / D Tf.p/g ı Tpf; Tp.id/ D id : The elements of Tp.M / are the tangent vectors of M at p. Since there exist many different constructions of tangent spaces, we deﬁne them by a universal property. A tangent space of the m-dimensional smooth manifold M at p consists of an m-dimensional vector space T
p.M / together with an isomorphism ik W TpM! Rm for each chart k D.U; '; U 0/ about p such that for any two such charts k and l D.V; ; V 0/ the isomorphism i 1 ik is the differential of the coordinate change '1 at '.p/. k ı i 0 k W T 0 pM! TpM is independent of the choice of k. Thus a tangent space is determined, up to unique isomorphism, by the universal property. If we ﬁx a chart k, an arbitrary m-dimensional vector space TpM, and an isomorphism ik W TpM! Rm, then there exists a unique tangent space with underlying vector space TpM and k/ is another tangent space, then p D i 1 pM; i 0 If.T 0 l 15.2. Tangent Spaces and Differentials 363 isomorphism ik; this follows from the chain rule of calculus. Often we talk about the tangent space TpM and understand a suitable isomorphism ik W TpM! Rm as structure datum. Let f W M m! N n be a smooth map. Choose charts k D.U; '; U 0/ about p 2 M and l D.V; ; V 0/ about f.p/ 2 N. There exists a unique linear map Tpf which makes the diagram TpM ik Rm Tp f Tf.p/N il D. f '1/ Rn commutative; the morphism at the bottom is the differential of f '1 at '.p/. Again by the chain rule, Tpf is independent of the choice of k and l. Differentials, deﬁned in this manner, satisfy the chain rule. This deﬁnition is also compatible with the universal maps p for different choices of tangent spaces Tpf ıp D f.p/ ıT 0 pf. In abstract terms: Make a choice of Tp.M / for each pair p 2 M. Then the TpM and the Tpf constitute a functor from the category of pointed smooth manifolds and pointed smooth maps to the category of real vector spaces. Different choices of tangent spaces yield isomorphic functors. The purpose of tangent spaces is to allow the deﬁ
nition of differentials. The actual vector spaces are adapted to the situation at hand and can serve other geometric purposes (e.g., they can consist of geometric tangent vectors). We call a smooth map f an immersion if each differential Txf is injective and a submersion if each differential Txf is surjective. The point x 2 M is a regular point of f if Txf is surjective. A point y 2 N is a regular value of f if each x 2 f 1.y/ is a regular point, and otherwise a singular value. If f 1.y/ D ;, then y is also called a regular value. (15.2.1) Rank Theorem. Let f W M! N be a smooth map from an m-manifold into an n-manifold..1/ If Taf is bijective, then there exist open neighbourhoods U of a and V of f.a/, such that f induces a diffeomorphism f W U! V..2/ If Taf is injective, then there exist open neighbourhoods U of a, V of f.a/, W of 0 2 Rnm and a diffeomorphism F W U W! V such that F.x; 0/ D f.x/ for x 2 U..3/ If Taf is surjective, then there exist open neighbourhoods U of a, V of f.a/, W of 0 2 Rmn and a diffeomorphism F W U! V W such that prV F.x/ D f.x/ for x 2 U with the projection prV W V W! V..4/ Suppose Txf has rank r for all x 2 M. Then for each a 2 M there exist open neighbourhoods U of a, V of f.a/ and diffeomorphisms'W U! U 0, W V! V 0 onto open sets U 0 Rm, V 0 Rn such that f.U / V and f '1.x1; : : : ; xm/ D.x1; : : : ; xr ; 0; : : : ; 0/ for all.x1; : : : ; xm/ 2 U 0. 364 Chapter 15. Manifolds Proof. The assertions are of a local nature. Therefore we can, via local charts, reduce to the case that M and N are open subsets of Euclidean spaces
. Then these assertions are known from calculus. (15.2.2) Proposition. Let y be a regular value of the smooth map f W M! N. Then P D f 1.y/ is a smooth submanifold of M. For each x 2 P, we can identify TxP with the kernel of Txf. Proof. Let x 2 P. The rank theorem (15.2.1) says that f is in suitable local coordinates about x and f.x/ a surjective linear map; hence P is locally a submanifold. The differential of a constant map is zero. Hence TxP is contained in the kernel of Txf. For reasons of dimension, the spaces coincide. (15.2.3) Example. The differentials of the projections onto the factors yield an isomorphism T.x;y/.M N / Š Tx.M / Ty.N / which we use as an identiﬁcation. With these identiﬁcations, T.x;y/.f g/ D Txf Tyg for smooth maps f and g. Let h W M N! P be a smooth map. Then T.x;y/h, being a linear map, is determined by the restrictions to TxM and to TyN, hence can be computed from the differentials of the partial maps h1 W a 7! h.a; y/ and h2 W b 7! h.x; b/ via Þ T.x;y/h.u; v/ D Txh1.u/ C Tyh2.v/. (15.2.4) Proposition. Suppose f W M! N is an immersion which induces a homeomorphism M! f.M /. Then f is a smooth embedding. Proof. We ﬁrst show that f.M / is a smooth submanifold of N of the same dimension as M. It sufﬁces to verify this locally. Choose U; V; W and F according to (15.2.1). Since U is open and M! f.M / a homeomorphism, the set f.U / is open in f.M /. Therefore f.U / D f.M / \ P, with some open set P N. The set R D V \ P is an open neighbourhood of b in N, and R \
f.M / D f.U / holds by construction. It sufﬁces to show that f.U / is a submanifold of R. We set Q D F 1R, and have a diffeomorphism F W Q! R which maps U 0 bijectively onto f.U /. Since U 0 is a submanifold of U W, we see that f.U / is a submanifold. By assumption, f W M! f.M / has a continuous inverse. This inverse is smooth, since f W M! f.M / has an injective differential, hence bijective for dimensional reasons, and is therefore a local diffeomorphism. (15.2.5) Proposition. Let f W M! N be a surjective submersion and g W N! P a set map between smooth manifolds. If gf is smooth, then g is smooth. Proof. Let f.x/ D y. By the rank theorem, there exist chart domains U about x and V about y such that f.U / D V and f W U! V has, in suitable local coordinates, the form.x1; : : : ; xm/ 7!.x1; : : : ; xn/. Hence there exists a smooth map s W V! U such that f s.z/ D z for all z 2 V. Then g.z/ D gf s.z/, and gf s is smooth. (The map s is called a local section of f about y.) 15.2. Tangent Spaces and Differentials 365 It is an important fact of analysis that most values are regular. A set A N in the n-manifold N is said to have (Lebesgue) measure zero if for each chart.U; h; V / of N the subset h.U \ A/ has measure zero in Rn. A subset of Rn has measure zero if it can be covered by a countable number of cubes with arbitrarily small total volume. We use the fact that a diffeomorphism (in fact a C 1-map) sends sets of measure zero to sets of measure zero. An open (non-empty) subset of Rn does not have measure zero. The next theorem is a basic result for differential topology; in order to save space we refer for its proof to the literature [136], [30
], [177]. (15.2.6) Theorem (Sard). The set of singular values of a smooth map has measure zero, and the set of regular values is dense. Problems P 1. An injective immersion of a compact manifold is a smooth embedding. 2. Let f W M! N be a smooth map which induces a homeomorphism M! f.M /. If the differential of f has constant rank, then f is a smooth embedding. By the rank theorem, f has to be an immersion, since f is injective. 3. Let M be a smooth m-manifold and N M. The following assertions are equivalent: (1) N is a k-dimensional smooth submanifold of M. (2) For each a 2 N there exist an open neighbourhood U of a in M and a smooth map f W U! Rmk such that the differential Df.u/ has rank m k for all u 2 U and such that N \ U D f 1.0/. (Submanifolds are locally solution sets of “regular” equations.) 4. f W RnC1! R,.x0; : : : ; xn/ 7! x2 i D kxk2 has, away from the origin, a nonzero differential. The sphere S n.c/ D f 1.c2/ D fx 2 RnC1 j c D kxkg of radius c > 0 is therefore a smooth submanifold of RnC1. From Proposition (15.2.2) we obtain TxS n.c/ D fv 2 RnC1 j x? vg. 5. Let M.m; n/ be the vector space of real.m; n/-matrices and M.m; nI k/ for 0 k min.m; n/ the subset of matrices of rank k. Then M.m; nI k/ is a smooth submanifold of M.m; n/ of dimension k.m C n k/. 6. The subset Sk.Rn/ D f.x1; : : : ; xk/ j xi 2 RnI x1; : : : ; xk linearly independent g of the k-fold product of Rn is called the Stiefel manifold of k-frames in Rn. It can be identiﬁed with
M.k; nI k/ and carries this structure of a smooth manifold. 7. The group O.n/ of orthogonal.n; n/-matrices is a smooth submanifold of the vector space Mn.R/ of real.n; n/-matrices. Let Sn.R/ be the subspace of symmetric matrices. The map f W Mn.R/! Sn.R/, B 7! B t B is smooth, O.n/ D f 1.E/, and f has surjective differential at each point A 2 O.n/. The derivative at s D 0 of s 7!.At C sX t /.A C sX/ is At X C X t A; the differential of f at A is the linear map Mn.R/! Sn.R/, X 7! At X C X t A. It is surjective, since the symmetric matrix S is the image of X D 1 2 AS. From (15.2.2) we obtain TAO.n/ D fX 2 Mn.R/ j At X C X t A D 0g; and in particular for the unit matrix E, TE O.n/ D fX 2 Mn.R/ j At C A D 0g, the space of skew-symmetric matrices. A local parametrization of O.n/ about E can be obtained from 0 X k=kŠ. Group multiplication the exponential map TE O.n/! O.n/, X 7! exp X D P1 366 Chapter 15. Manifolds and passage to the inverse are smooth maps. 8. Make a similar analysis of the unitary group U.n/. 9. The Stiefel manifolds have an orthogonal version which generalizes the orthogonal group, the Stiefel manifold of orthonormal k-frames. Let Vk.Rn/ be the set of orthonormal ktuples.v1; : : : ; vk/, vj 2 Rn. If we write vj as row vector, then Vk.Rn/ is a subset of the vector space M D M.k; nI R/ of real.k; n/-matrices. Let S D Sk.R/ again be the vector space of symmetric.k; k/-matrices. Then f W M! S, A 7! A At has
the pre-image f 1.E/ D Vk.Rn/. The differential of f at A is the linear map X 7! XAt C AX t and it is surjective. Hence E is a regular value. The dimension of Vk.Rn/ is.n k/k C 1 2 k.k 1/. 10. The deﬁning map RnC1X0! RP n is a submersion. Its restriction to S n is a submersion and an immersion (a 2-fold regular covering). 11. The graph of a smooth function f W Rn! R is a smooth submanifold of RnC1. 12. Let Y be a smooth submanifold of Z and X Y. Then X is a smooth submanifold of Y if and only if it is a smooth submanifold of Z. If X is a smooth submanifold, then there exists about each point x 2 X a chart.U; '; V / of Z such that '.U \ X/ as well as '.U \ Y / are intersections of V with linear subspaces. (Charts adapted to X Y Z. Similarly for inclusions of submanifolds X1 X2 Xr.) 13. Let ƒk.Rn/ be the k-th exterior power of Rn. The action of O.n/ on Rn induces an action on ƒk.Rn/, a smooth representation. If we assign to a basis x.1/; : : : ; x.k/ of a k-dimensional subspace the element x.1/ ^ ^ x.k/ 2 ƒk.Rn/, we obtain a well-deﬁned, injective, O.n/-equivariant map j W Gk.Rn/! P.ƒkRn/ (Plücker coordinates). The image of j is a smooth submanifold of P.ƒkRn/, i.e., j is an embedding of the Grassmann manifold Gk.Rn/. 14. The Segre embedding is the smooth embedding RP m RP n! RP.mC1/.nC1/1;.Œxi ; Œyj / 7! Œxi yj : For m D n D 1 the image is the quadric fŒs0;
s1; s2; s3 j s0s3 s1s2 D 0g. 15. Let h W RnC1 RnC1! RnCkC1 be a symmetric bilinear form such that x ¤ 0; y ¤ 0 implies h.x; y/ ¤ 0. Let g W S n! S nCk, x 7! h.x; x/=jh.x; x/j. If g.x/ D g.y/, hence h.x; x/ D t 2h.y; y/ with some t 2 R, then h.x C ty; x ty/ D 0 and therefore x C ty D 0 or x ty D 0. Hence g induces a smooth embedding RP n! S nCk. The bilinear form h.x0; : : : ; xn; y0; : : : ; yn/ D.z0; : : : ; z2n/ with zk D iCj Dk xi yj yields an embedding RP n! S 2n [89], [95]. 16. Remove a point from S 1 S 1 and show (heuristically) that the result has an immersion into R2. (Removing a point is the same as removing a big 2-cell!). P 15.3 Smooth Transformation Groups Let G be a Lie group and M a smooth manifold. We consider smooth action G M! M of G on M. The left translations lg W M! M, x 7! gx are then diffeomorphisms. The map ˇ W G! M, g 7! gx is a smooth G-map with image the orbit B D Gx through x. We have an induced bijective G-equivariant set map W G=Gx! B. The map ˇ has constant rank; this follows from the equivariance. If Lg W G! G and lg W M! M denote the left translations by g, 15.3. Smooth Transformation Groups 367 then lg ˇ D ˇLg, and since Lg and lg are diffeomorphisms, we see that Teˇ and Tg ˇ have the same rank. (15.3.1) Proposition. Suppose the orbit B D Gx is a smooth submanifold of M. Then: (
1) ˇ W G! B is a submersion. (2) Gx D ˇ1.x/ is a closed Lie subgroup of G. (3) There exists a unique smooth structure on G=Gx such that the quotient map G! G=Gx is a submersion. The induced map W G=Gx! B is a diffeomorphism. Proof. If ˇ would have somewhere a rank less than the dimension of B, the rank would always be less than the dimension, by equivariance. This contradicts the theorem of Sard. We transport via the smooth structure from B to G=Gx. The smooth structure is unique, since G! G=Gx is a submersion. The pre-image Gx of a regular value is a closed submanifold. The previous proposition gives us Gx as a closed Lie subgroup. We need not use the general theorem about closed subgroups being Lie subgroups. (15.3.2) Example. The action of SO.n/ on S n1 by matrix multiplication is smooth. We obtain a resulting equivariant diffeomorphism S n1 Š SO.n/=SO.n 1/. In a similar manner we obtain equivariant diffeomorphisms S 2n1 Š U.n/=U.n1/ Š Þ SU.n/=SU.n 1/. (15.3.3) Theorem. Let M be a smooth n-manifold. Let C M M be the graph of an equivalence relation R on M, i.e., C D f.x; y/ j x yg. Then the following are equivalent: (1) The set of equivalence classes N D M=R carries the structure of a smooth manifold such that the quotient map p W M! N is a submersion. (2) C is a closed submanifold of M M and pr1 W C! M is a submersion. (15.3.4) Theorem. Let M be a smooth G-manifold with free, proper action of the Lie group G. Then the orbit space M=G carries a smooth structure and the orbit map p W M! M=G is a submersion. Proof. We verify the hypothesis of the quotient theorem (15.3.3). We have to show that C is
a closed submanifold. The set C is homeomorphic to the image of the map ‚ W G M! M M,.g; x/ 7!.x; gx/, since the action is proper. We show that ‚ is a smooth embedding. It sufﬁces to show that ‚ is an immersion (see (15.2.4)). The differential T.g;x/‚ W Tg G TxM! TxM TgxM 368 Chapter 15. Manifolds will be decomposed according to the two factors T.g;x/‚.u; v/ D Tg ‚.‹; x/u C Tx‚.g; ‹/v: The ﬁrst component of Tg ‚.‹; x/u is zero, since the ﬁrst component of the partial map is constant. Thus if T.g;x/.u; v/ D 0, the component of Tx‚.g; ‹/v in TxM is zero; but this component is v. It remains to show that Tg f W Tg G! TgxM is injective for f W G! M, g 7! gx. Since the action is free, the map f is injective; and f has constant rank, by equivariance. An injective map of constant rank has injective differential, by the rank theorem. Thus we have veriﬁed the ﬁrst hypothesis of (15.3.3). The second one holds, since pr1 ı‚ D pr2 shows that pr1 is a submersion. (15.3.5) Example. The cyclic group G D Z=m S 1 acts on Cn by Z=m Cn! Cn;.;.z1; : : : ; zn// 7!.r1z1; : : : ; rnzn/ where rj 2 Z. This action is a smooth representation. Suppose the integers rj are coprime to m. The induced action on the unit sphere is then a free G-manifold S.r1; : : : ; rn/; the orbit manifold L.r1; : : : ; rn/ is called a (generalized) lens space. Þ (15.3
.6) Example. Let H be a closed Lie subgroup of the Lie group G. The H -action on G by left translation is smooth and proper. The orbit space H nG carries a smooth structure such that the quotient map G! H nG is a submersion. The G-action on H nG is smooth. One can consider the projective spaces, Stiefel manifolds and Grassmann manifolds as homogeneous spaces from this view-point. Þ (15.3.7) Theorem. Let M be a smooth G-manifold. Then: (1) An orbit C M is a smooth submanifold if and only if it is a locally closed subset. (2) If the orbit C is locally closed and x 2 C, then there exists a unique smooth structure on G=Gx such that the orbit map G! G=Gx is a submersion. The map G=Gx! C, gGx 7! gx is a diffeomorphism. The G-action on G=Gx is smooth. (3) If the action is proper, then.1/ and.2/ hold for each orbit. Proof. (1) ˇ W G! C, g 7! gx has constant rank by equivariance. Hence there exists an open neighbourhood of e in G such that ˇ.U / is a submanifold of M. Since C is locally closed in the locally compact space M, the set C is locally compact and therefore ˇ W G! C is an open map (see (1.8.6)). Hence there exists an open set W in M such that C \ W D ˇ.U /. Therefore C is a submanifold in a neighbourhood of x and, by equivariance, also globally a submanifold. 15.4. Manifolds with Boundary 369 (2) Since C is locally closed, the submanifold C has a smooth structure. The map ˇ has constant rank and is therefore a submersion. We now transport the smooth structure from C to G=Gx. (3) The orbits of a proper action are closed. 15.4 Manifolds with Boundary We now extend the notion of a manifold to that of a manifold with boundary. A typical example is the n-dimensional disk Dn D fx 2 Rn j kx
k 1g. Other examples are half-spaces. Let W Rn! R be a non-zero linear form. We use the corresponding half-space H./ D fx 2 Rn j.x/ 0g. Its boundary @H./ is the kernel of. Typical half-spaces are Rn ˙ D f.x1; : : : ; xn/ 2 Rn j ˙x1 0g. If A Rm is any subset, we call f W A! Rn differentiable if for each a 2 A there exists an open neighbourhood U of a in Rm and a differentiable map F W U! Rn such that F jU \ A D f jU \ A. We only apply this deﬁnition to open subsets A of half-spaces. In that case, the differential of F at a 2 A is independent of the choice of the extension F and will be denoted Df.a/. Let n 1 be an integer. An n-dimensional manifold with boundary or @-manifold is a Hausdorff space M with countable basis such that each point has an open neighbourhood which is homeomorphic to an open subset in a half-space of Rn. A homeomorphism h W U! V, U open in M, V open in H./ is called a chart about x 2 U with chart domain U. With this notion of chart we can deﬁne the notions: C k-related, atlas, differentiable structure. An n-dimensional smooth manifold with boundary is therefore an n-dimensional manifold M with boundary together with a (maximal) smooth C 1-atlas on M. Let M be a manifold with boundary. Its boundary @M is the following subset: The point x is contained in @M if and only if there exists a chart.U; h; V / about x such that V H./ and h.x/ 2 @H./. The complement M X @M is called the interior In.M / of M. The following lemma shows that specifying a boundary point does not depend on the choice of the chart (invariance of the boundary). (15.4.1) Lemma. Let'W V! W be a diffeomorphism between open subsets V H./ and W H./ of half-spaces in Rn. Then '.V \@H.// D W \
@H./. (15.4.2) Proposition. Let M be an n-dimensional smooth manifold with boundary. Then either @M D ; or @M is an.n 1/-dimensional smooth manifold. The set M X @M is a smooth n-dimensional manifold with empty boundary. The boundary of a manifold can be empty. Sometimes it is convenient to view the empty set as an n-dimensional manifold. If @M D ;, we call M a manifold without boundary. This coincides then with the notion introduced in the ﬁrst section. In order to stress the absence of a boundary, we call a compact manifold without boundary a closed manifold. 370 Chapter 15. Manifolds A map f W M! N between smooth manifolds with boundary is called smooth if it is continuous and C 1-differentiable in local coordinates. Tangent spaces and the differential are deﬁned as for manifolds without boundary. Let x 2 @M and k D.U; h; V / be a chart about x with V open in Rn. Then the pair.k; v/, v 2 Rn represents a vector w in the tangent space TxM. We say that w is pointing outwards (pointing inwards, tangential) to@M if v1 > 0 (v1 < 0, v1 D 0, respectively). One veriﬁes that this disjunction is independent of the choice of charts. (15.4.3) Proposition. The inclusion j W @M M is smooth and the differential Txj W Tx.@M /! TxM is injective. Its image consists of the vectors tangential to @M. We consider Txj as an inclusion. The notion of a submanifold can have different meanings for manifolds with boundary. We deﬁne therefore submanifolds of type I and type II. Let M be a smooth n-manifold with boundary. A subset N M is called a k-dimensional smooth submanifold (of type I) if the following holds: For each x 2 N there exists a chart.U; h; V /, V Rn open, of M about x such that h.U \ N / D V \.Rk 0/. Such charts of M are adapted to N. The set V \.Rk 0/ Rk. A diffeomorphism onto a submanifold
of type I is an embedding of type I. From this deﬁnition we draw the following conclusions. is open in Rk 0 D Rk (15.4.4) Proposition. Let N M be a smooth submanifold of type I. The restrictions h W U \ N! h.U \ N / of the charts.U; h; V / adapted to N form a smooth atlas for N which makes N into a smooth manifold with boundary. The relation N \ @M D @N holds, and @N is a submanifold of @M. Let M be a smooth n-manifold without boundary. A subset N M is a k-dimensional smooth submanifold (of type II) if the following holds: For each x 2 N there exists a chart.U; h; V / of M about x such that h.U \ N / D V \.Rk 0/. Such charts are adapted to N. The intersection of Dn with Rk 0 is a submanifold of type I (k < n). The subset Dn is a submanifold of type II of Rn. The next two propositions provide a general means for the construction of such submanifolds. (15.4.5) Proposition. Let M be a smooth n-manifold with boundary. Let f W M! R be smooth with regular value 0. Then f 1Œ0; 1Œ is a smooth submanifold of type II of M with boundary f 1.0/. Proof. We have to show that for each x 2 f 1Œ0; 1Œ there exists a chart which is adapted to this set. If f.x/ > 0, then x is contained in the open submanifold f 10; 1Œ; hence the required charts exist. Let therefore f.x/ D 0. By the rank theorem (15.2.1), f has in suitable local coordinates the form.x1; : : : ; xn/ 7! x1. From this fact one easily obtains the adapted charts. 15.4. Manifolds with Boundary 371 (15.4.6) Proposition. Let f W M! N be smooth and y 2 f.M / \.N X @N / be a regular value of f and f j@M. Then P D f 1.y/ is a smooth sub
ifold. A smooth function f W @B! R has a smooth extension to B. A smooth function g W A! R from a submanifold A of type I or of type II of B has a smooth extension to B. 3. Verify the invariance of the boundary for topological manifolds (use local homology groups). 4. A @-manifold M is connected if and only if M X @M is connected. 5. Let M be a @-manifold. There exists a smooth function f W M! Œ0; 1Œ such that f.@M / D f0g and Txf 6D 0 for each x 2 @M. 6. Let f W M! Rk be an injective immersion of a compact @-manifold. Then the image is a submanifold of type II. 7. Verify that “pointing inwards” is well-deﬁned, i.e., independent of the choice of charts. 8. Unfortunately is not quite trivial to classify smooth 1-dimensional manifolds by just 372 Chapter 15. Manifolds starting from the deﬁnitions. The reader may try to show that a connected 1-manifold without boundary is diffeomorphic to R1 or S 1; and a @-manifold is diffeomorphic to Œ0; 1 or Œ0; 1Œ. 15.5 Orientation Let V be an n-dimensional real vector space. We call ordered bases b1; : : : ; bn and c1; : : : ; cn of V positively related if the determinant of the transition matrix is positive. This relation is an equivalence relation on the set of bases with two equivalence classes. An equivalence class is an orientation of V. We specify orientations by their representatives. The standard orientation of Rn is given by the standard basis e1; : : : ; en, the rows of the unit matrix. Let W be a complex vector space with complex basis w1; : : : ; wn. Then w1; iw1; : : : ; wn; iwn deﬁnes an orientation of the underlying real vector space which is independent of the choice of the basis. This is the orientation induced by the complex structure. Let u1; : : : ; um be a basis of U and w1; : : : ; w
n a basis of W. In a direct sum U ˚ W we deﬁne the sum orientation by u1; : : : ; um; w1; : : : ; wn. If o.V / is an orientation of V, we denote the opposite orientation (the occidentation) by o.V /. A linear isomorphism f W U! V between oriented vector spaces is called orientation preserving or positive if for the orientation u1; : : : ; un of U the images f.u1/; : : : ; f.un/ yield the given orientation of V. Let M be a smooth n-manifold with or without boundary. We call two charts positively related if the Jacobi matrix of the coordinate change has always positive determinant. An atlas is called orienting if any two of its charts are positively related. We call M orientable, if M has an orienting atlas. An orientation of a manifold is represented by an orienting atlas; and two such deﬁne the same orientation if their union contains only positively related charts. If M is oriented by an orienting atlas, we call a chart positive with respect to the given orientation if it is positively related to all charts of the orienting atlas. These deﬁnitions apply to manifolds of positive dimension. An orientation of a zero-dimensional manifold M is a function W M! f˙1g. Let M be an oriented n-manifold. There is an induced orientation on each of its tangent spaces TxM. It is speciﬁed by the requirement that a positive chart.U; h; V / induces a positive isomorphism Txh W TxM! Th.x/V D Rn with respect to the standard orientation of Rn. We can specify an orientation of M by the corresponding orientations of the tangent spaces. If M and N are oriented manifolds, the product orientation on M N is speciﬁed by declaring the products.U V; k l; U 0 V 0/ of positive charts.U; k; U 0/ of M and.V; l; V 0/ of N as positive. The canonical isomorphism T.x;y/.M N / Š TxM ˚ TyN is then compatible with the sum orientation of vector spaces. If N is a point, then
the canonical identiﬁcation M N Š M is orientation preserving if and only if.N / D 1. If M is oriented, then we denote the manifold with the opposite orientation by M. 15.5. Orientation 373 Let M be an oriented manifold with boundary. For x 2 @M we have a direct decomposition Tx.M / D Nx ˚ Tx.@M /. Let nx 2 Nx be pointing outwards. The boundary orientation of Tx.@M / is deﬁned by that orientation v1; : : : ; vn1 for which nx; v1; : : : ; vn1 is the given orientation of Tx.M /. These orientations correspond to the boundary orientation of @M ; one veriﬁes that the restriction of positive charts for M yields an orienting atlas for @M. In Rn, the boundary @Rn D 0 Rn1 inherits the orientation deﬁned by e2; : : : ; en. Thus positive charts have to use Rn. Let D2 R2 carry the standard orientation of R2. Consider S 1 as boundary of D2 and give it the boundary orientation. An orienting vector in TxS 1 is then the velocity vector of a counter-clockwise rotation. This orientation of S 1 is commonly In general if M R2 is a two-dimensional known as the positive orientation. submanifold with boundary with orientation induced from the standard orientation of R2, then the boundary orientation of the curve @M is the velocity vector of a movement such that M lies “to the left”. Let M be an oriented manifold with boundary and N an oriented manifold without boundary. Then product and boundary orientation are related as follows o.@.M N // D o.@M N /; o.@.N M // D.1/dim N o.N @M /: The unit interval I D Œ0; 1 is furnished with the standard orientation of R. Since the outward pointing vector in 0 yields the negative orientation, we specify the orientation of @I by.0/ D 1;.1/ D 1. We have @.I M / D 0M [1M. The boundary orientation of 0 M Š M is opposite to the original one and the boundary orientation of 1 M Š M is the original one, if I M carries
the product orientation. We express these facts by the suggestive formula @.These conventions suggest that homotopies should be deﬁned with the cylinder I X.) A diffeomorphism f W M! N between oriented manifolds respects the orientation if Txf is for each x 2 M orientation preserving. If M is connected, then f respects or reverses the orientation. Problems 1. Show that a 1-manifold is orientable. 2. Let f W M! N be a smooth map and let A be the pre-image of a regular value y 2 N. Suppose M is orientable, then A is orientable. We specify an orientation as follows. Let M and N be oriented. We have an exact.1/! TaM.2/ sequence 0! TaA! TyN! 0, with inclusion (1) and differential Taf at (2). This orients TaA as follows: Let v1; : : : ; vk be a basis of TaA, w1; : : : ; wl a basis of TyN, and u1; : : : ; ul be pre-images in TaM ; then v1; : : : ; vk; u1; : : : ; ul is required to be the given orientation of TaM. These orientations induce an orientation of A. This orientation 374 Chapter 15. Manifolds of A is called the pre-image orientation. i and S n1 D f 1.1/. Then the pre-image orientation 3. Let f W Rn! R,.xi / 7! x2 coincides with the boundary orientation with respect to S n1 Dn. P 15.6 Tangent Bundle. Normal Bundle The notions and concepts of bundle theory can now be adapted to the smooth category. A smooth bundle p W E! B has a smooth bundle projection p and the bundle charts are assumed to be smooth. A smooth subbundle of a smooth vector bundle has to be deﬁned by smooth bundle charts. Let ˛ W 1! 2 be a smooth bundle morphism of constant rank; then Ker ˛ and Im ˛ are smooth subbundles. The proof of (14.2.3) can also be used in this situation. A smooth vector bundle has a smooth Riemannian metric; for the existence proof one uses a smooth partition of unity and proceeds as in (14.5.1). Let be a
smooth subbundle of the smooth vector bundle with Riemannian metric; then the orthogonal complement of in is a smooth subbundle. Let M be a smooth n-manifold. Denote by TM the disjoint union of the tangent spaces Tp.M /, p 2 M. We write a point of Tp.M / TM in the form.p; v/ with v 2 Tp.M /, for emphasis. We have the projection M W TM! M,.p; v/ 7! p. Each chart k D.U; h; V / of M yields a bijection 'k W TU D S p2U Tp.M /! U Rn;.p; v/ 7!.p; ik.v//: Here ik is the morphism which is part of the deﬁnition of a tangent space. The map 'k is a map over U and linear on ﬁbres. The next theorem is a consequence of the general gluing procedure. (15.6.1) Theorem. There exists a unique structure of a smooth manifold on TM such that the.TU; 'k; U Rn/ are charts of the differential structure. The projection M W TM! M is then a smooth map, in fact a submersion. The vector space structure on the ﬁbres of M give M the structure of an n-dimensional smooth real vector bundle with the 'k as bundles charts. The vector bundle M W TM! M is called the tangent bundle of M. A smooth map f W M! N induces a smooth ﬁbrewise map Tf W TM! TN,.p; v/ 7!.f.p/; Tpf.v//. (15.6.2) Proposition. Let M Rq be a smooth n-dimensional submanifold. Then TM D f.x; v/ j x 2 M; v 2 TxM g R q Rq is a 2n-dimensional smooth submanifold. 15.6. Tangent Bundle. Normal Bundle 375 Proof. Write M locally as h1.0/ with a smooth map h W U! Rqn of constant rank q n. Then TM is locally the pre-image of zero under U Rq! Rqn Rqn;.u; v/ 7!.h
.u/; Dh.u/.v//; and this map has constant rank 2.qn/; this can be seen by looking at the restrictions to U 0 and u Rq. We can apply (15.6.2) toS n RnC1 and obtain the model of the tangent bundle of S n, already used at other occasions. Let the Lie group G act smoothly on M. We have an induced action G TM! TM;.b; v/ 7!.T lg /v: This action is again smooth and the bundle projection is equivariant, i.e., TM! M is a smooth G-vector bundle. (15.6.3) Proposition. Let W E! M be a smooth G-vector bundle. Suppose the action on M is free and proper. Then the orbit map E=G! M=G is a smooth vector bundle. We have an induced bundle map! =G. The differential Tp W TM! T.M=G/ of the orbit map p is a bundle morphism which factors over the orbit map TM!.TM/=G and induces a bundle morphism q W.TM/=G! T.M=G/ over M=G. The map is ﬁbrewise surjective. If G is discrete, then M and M=G have the same dimension, hence q is an isomorphism. (15.6.4) Proposition. For a free, proper, smooth action of the discrete group G on M we have a bundle isomorphism.TM/=G Š T.M=G/ induced by the orbit map M! M=G. (15.6.5) Example. We have a bundle isomorphism TS n˚" Š.nC1/". If G D Z=2 acts on TS n via the differential of the antipodal map and trivially on ", then the said isomorphism transforms the action into S n RnC1! S n RnC1,.x; v/ 7!.x; v/. We pass to the orbit spaces and obtain an isomorphism T.RP n/ ˚ " Š.n C 1/ with the tautological line bundle over RP n. Þ In the general case the map q W.TM/=G! T.M=G/ has a kernel, a bundle K! M=G
with ﬁbre dimension dim G. See [44, IX.6] for details. (15.6.6) Example. The deﬁning map CnC1 X 0!.CnC1 X 0/=C D CP n yields a surjective bundle map q W T.CnC1 X 0/=C! T.CP n/. The source of q is the.n C 1/-fold Whitney sum.n C 1/ where E./ is the quotient of.CnC1 X 0/ C with respect to.z; x/.z; x/ for 2 C and.z; x/ 2.CnC1 X 0/ C. The kernel bundle of q is trivial: We have a canonical section of.n C 1/ CP n!..CnC1 X 0/ CnC1/=C; Œz 7!.z; z/= ; 376 Chapter 15. Manifolds and the subbundle generated by this section is contained in the kernel of q. Hence the complex tangent bundle of CP n satisﬁes T.CP n/ ˚ " Š.n C 1/. For see Þ H.1/ in (14.2.6). Let p W E! M be a smooth vector bundle. Then E is a smooth manifold and we can ask for its tangent bundle. (15.6.7) Proposition. There exists a canonical exact sequence 0! pE ˛! TE ˇ! pTM! 0 of vector bundles over E, written in terms of total spaces. Proof. The differential of p is a bundle morphism Tp W TE! TM, and it induces a bundle morphism ˇ W TE! pTM which is ﬁbrewise surjective, since p is a submersion. We consider the total space of pE! E as E ˚ E and the projection onto the ﬁrst summand is the bundle projection. Let.v; w/ 2 Ex ˚ Ex. We deﬁne ˛.v; w/ as the derivative of the curve t 7! v C tw at t D 0. The bundle morphism ˛ has an image contained in the kernel of ˇ and is ﬁbrewise injective. Thus, for reasons of dimension, the sequence
15. Manifolds We deﬁne M locally as the solution set: Suppose U Rn is open,'W U! Rk a submersion, and '1.0/ D U \ M D W. We set N.M / \.W Rn/ D N.W /. The smooth maps ˆ W W Rn! W Rk; ‰ W W Rk! W Rn;.x; v/ 7!.x; Tx'.v//;.x; v/ 7!.x;.Tx'/t.v// satisfy N.W / D Im ‰; T.W / D Ker ˆ: The composition ˆ‰ is a diffeomorphism: it has the form.w; v/ 7!.w; gw.v// with a smooth map W! GLk.R/, w 7! gw and therefore.w; v/ 7!.w; g1 w.v// is a smooth inverse. Hence ‰ is a smooth embedding with image N.W / and ‰1jN.W / is a smooth bundle chart. (15.6.11) Proposition. The map a W N.M /! Rn,.x; v/ 7! x C v is a tubular map for M Rn. Proof. We show that a has a bijective differential at each point.x; 0/ 2 N.M /. Let NxM D TxM?. Since M Rn we consider TxM as a subspace of Rn. Then T.x;0/N.M / is the subspace TxM NxM T.x;0/.M Rn/ D TxM Rn. The differential T.x;0/a is the identity on each of the subspaces TxM and NxM. Therefore we can consider this differential as the map.u; v/ 7! u C v, i.e., essentially as the identity. It is now a general topological fact (15.6.13) that a embeds an open neighbourhood of the zero section. Finally it is not difﬁcult to verify property (3) of a tubular map. (15.6.12) Corollary. If we transport the bundle projection N.M /! M via the embedding
a we obtain a smooth retraction r W U! M of an open neighbourhood U of M Rn. (15.6.13) Theorem. Let f W X! Y be a local homeomorphism. Let A X and f W A! f.A/ D B be a homeomorphism. Suppose that each neighbourhood of B in Y contains a paracompact neighbourhood. Then there exists an open neighbourhood U of A in X which is mapped homeomorphically under f onto an open neighbourhood V of B in Y (see [30, p. 125]). For embeddings of compact manifolds and their tubular maps one can apply another argument as in the following proposition. (15.6.14) Proposition. Let ˆ W X! Y be a continuous map of a locally compact space into a Hausdorff space. Let ˆ be injective on the compact set A X. Suppose that each a 2 A has a neighbourhood Ua in X on which ˆ is injective. Then there exists a compact neighbourhood V of A in X on which ˆ is an embedding. 15.7. Embeddings 379 Proof. The coincidence set K D f.x; y/ 2 X X j ˆ.x/ D ˆ.y/g is closed in X X, since Y is a Hausdorff space. Let D.B/ be the diagonal of B X. If ˆ is injective on Ua, then.Ua Ua/ \ K D D.Ua/. Thus our assumptions imply that D.X/ is open in K and hence W D X X n.K n D.X// open in X X. By assumption, A A is contained in W. Since A A is compact and X locally compact, there exists a compact neighbourhood V of A such that V V W. Then ˆjV is injective and, being a map from a compact space into a Hausdorff space, an embedding. (15.6.15) Proposition. A submanifold M N has a tubular map. Proof. We ﬁx an embedding of N Rn. By (15.6.12) there exists an open neighbourhood W of V in Rn and a smooth retraction r W W! V. The standard inner product on Rn induces a Riemannian metric on TN. We use as normal bundle
for M N the model E D f.x; v/ 2 M Rn j v 2.TxM /? \ TxN g: Again we use the map f W E! Rn,.x; v/ 7! x C v and set U D f 1.W /. Then U is an open neighbourhood of the zero section of E. The map g D rf W U! N is the inclusion when restricted to the zero section. We claim that the differential of g at points of the zero section is the identity, if we use the identiﬁcation T.x;0/E D TxM ˚ Ex D TxN. On the summand TxM the differential T.x;0/g is obviously the inclusion TxM TxV. For.x; v/ 2 Ex the curve t 7!.x; tv/ in E has.x; v/ as derivative at t D 0. Therefore we have to determine the derivative of t 7! r.x C tv/ at t D 0. The differential of r at.x; 0/ is the orthogonal projection Rn! TxN, if we use the retraction r in (15.6.12). The chain rule tells us that the derivative of t 7! r.x C tv/ at t D 0 is v. We now apply again (15.6.13). One veriﬁes property (3) of a tubular map. 15.7 Embeddings This section is devoted to the embedding theorem of Whitney: (15.7.1) Theorem. A smooth n-manifold has an embedding as a closed submanifold of R2nC1. We begin by showing that a compact n-manifold has an embedding into some Euclidean space. Let f W M! Rt be a smooth map from an n-manifold M. Let.Uj ; j ; U3.0//, j 2 f1; : : : ; kg be a ﬁnite number of charts of M (see (15.1.2) for the deﬁnition of U3.0/). Choose a smooth function W Rn! Œ0; 1 such that.x/ D 0 for kxk 2 and.x/ D 1 for kxk 1. De�
�ne j W M! R by j.x/ D 0 for x … Uj and by j.x/ D j.x/ for x 2 Uj ; then j is a smooth function on M. With the help of these functions we deﬁne ˆ W M! Rt.R Rn/.R Rn/ D Rt RN 380 Chapter 15. Manifolds ˆ.x/ D.f.x/I 1.x/; 1.x/1.x/I : : :I k.x/; k.x/k.x//; (k factors R Rn), where j.x/j.x/ should be zero if j.x/ is not deﬁned. The differential of this map has the rank n on Wj D 1.U1.0//, as ˆ.Wj / Vj D f.zI a1; x1I : : :I ak; xk/ j aj 6D 0g, and the composition of ˆjWj with Vj! Rn, k.zI a1; x1I : : : / 7! a1 j D1 Wj, since ˆ.a/ D ˆ.b/ implies j.a/ D j.b/ for each j, and then i.a/ D i.b/ holds for some i. Moreover, ˆ is equal to f composed with Rt Rt RN on the complement of the 1 j U2.0/. Hence if f is an (injective) immersion on the open set U, then ˆ is an (injective) immersion on U [ W. In particular, if M is compact, we can apply this argument to an arbitrary map f and M D W. Thus we have shown: j xj is j. By construction, ˆ is injective on W D S j (15.7.2) Note. A compact smooth manifold has a smooth embedding into some Euclidean space. We now try to lower the embedding dimension by applying a suitable parallel projection. Let Rq1 D Rq1 0 Rq. For v 2 Rq n Rq1 we consider the projection pv W Rq! Rq1 with direction v, i.e., for x D x0 C v with x0 2
Rq1 and 2 R we set pv.x/ D x0. In the sequel we only use vectors v 2 S q1. Let M Rq. We remove the diagonal D and consider W M M n D! S q1,.x; y/ 7! N.x y/ D.x y/=kx yk. (15.7.3) Note. 'v D pvjM is injective if and only if v is not contained in the image of. Proof. The equality 'v.x/ D 'v.y/; x ¤ y and x D x0 C v; y D y0 C v imply x y D. /v ¤ 0, hence v D ˙N.x y/. Note.x; y/ D.y; x/. Let now M be a smooth n-manifold in Rq. We use the bundle of unit vectors STM D f.x; v/ j v 2 TxM; kvk D1g M S q1 and its projection to the second factor D pr2 jSTM W STM! S q1: The function.x; v/ 7! kvk2 on TM Rq Rq has 1 as regular value with pre-image STM, hence STM is a smooth submanifold of the tangent bundle TM. (15.7.4) Note. 'v is an immersion if and only if v is not contained in the image of. Proof. The map 'v is an immersion if for each x 2 M the kernel of Txpv has trivial intersection with TxM. The differential of pv is again pv. Hence 0 6D z D pv.z/ C v 2 TxM is contained in the kernel of Txpv if and only if z D v and hence v is a unit vector in TxM. 15.7. Embeddings 381 (15.7.5) Theorem. Let M be a smooth compact n-manifold. Let f W M! R2nC1 be a smooth map which is an embedding on a neighbourhood of a compact subset A M. Then there exists for each " > 0 an embedding g W M! R2nC1 which coincides on A with f and satisﬁes kf.x/ g.x/k < " for x 2
M. Proof. Suppose f embeds the open neighbourhood U of A and let V U be a compact neighbourhood of A. We apply the construction in the beginning of this section with chart domains Uj which are contained in M n V and such that the sets Wj cover M X U. Then ˆ is an embedding on some neighbourhood of M n U and ˆ W M! R2nC1 ˚ RN D Rq; x 7!.f.x/; ‰.x// is an embedding which coincides on V with f (up to composition with the inclusion R2nC1 Rq). For 2n < q 1 the image of is nowhere dense and for 2n 1 < q 1 the image of is nowhere dense (theorem of Sard). Therefore in each neighbourhood of w 2 S q1 there exist vectors v such that pv ı ˆ D ˆv is an injective immersion, hence an embedding since M is compact. By construction, ˆv coincides on V with f. If necessary, we replace ‰ with s‰ (with small s) such that kf.x/ ˆ.x/k "=2 holds. We can write f as composition of ˆ with projections Rq! Rq1!! R2nC1 along the unit vectors.0; : : : ; 1/. Sufﬁciently small perturbations of these projections applied to ˆ yield a map g such that kf.x/ g.x/k < ", and, by the theorem of Sard, we ﬁnd among these projections those for which g is an embedding. The preceding considerations show that we need one dimension less for immer- sions. (15.7.6) Theorem. Let f W M! R2n be a smooth map from a compact n-manifold. Then there exists for each " > 0 an immersion h W M! R2n such that kh.x/ f.x/k < " for x 2 M. If f W M! R2nC1 is a smooth embedding, then the vectors v 2 S 2n for which the projection pv ı f W M! R2n is an immersion are dense in S 2n. Let f W M! R be a smooth proper function from an n-manifold without boundary. Let
t 2 R be a regular value and set A D f 1.t/. The manifold A is compact. There exists an open neighbourhood U of A in M and a smooth retraction r W U! A. (15.7.7) Proposition. There exists an " > 0 and open neighbourhood V U of A such that.r; f / W V! A t "; t C "Œ is a diffeomorphism. Proof. The map.r; f / W U! A R has bijective differential at points of A. Hence there exists an open neighbourhood W U of A such that.r; f / embeds W onto an open neighbourhood of A ftg in A R. Since f is proper, each neighbourhood W of A contains a set of the form V D f 1 t "; t C "Œ. The restriction of.r; f / to V has the required properties. 382 Chapter 15. Manifolds In a similar manner one shows that a proper submersion is locally trivial (theorem of Ehresmann). We now show that a non-compact n-manifold M has an embedding into R2nC1 as a closed subset. For this purpose we choose a proper smooth function f W M! RC. We then choose a sequence.tk j k 2 N/ of regular values of f such that tk < tkC1 and limk tk D 1. Let Ak D f 1.tk/ and Mk D f 1Œtk; tkC1. Choose "k > 0 small enough such that the intervals Jk D tk "k; tk C "kŒ are disjoint and such that we have diffeomorphisms f 1.Jk/ Š Ak Jk of the type (15.7.7). We then use (15.7.7) in order to ﬁnd embeddings ˆk W f 1.Jk/! R2n Jk which have f as their second component. We then use the method of (15.7.5) to ﬁnd an embedding Mk! R2n Œtk; tkC1 which extends the embeddings ˆk and ˆkC1 in a neighbourhood of Mk C MkC1. All these embeddings ﬁt together and yield an embedding of M as a closed subset of
R2nC1. A collar of a smooth @-manifold M is a diffeomorphism W @M Œ0; 1Œ! M onto an open neighbourhood U of @M in M such that.x; 0/ D x. Instead of Œ0; 1Œ one can also use R˙. (15.7.8) Proposition. A smooth @-manifold M has a collar. Proof. There exists an open neighbourhood U of @M in M and a smooth retraction r W U! @M. Choose a smooth function f W M! RC such that f.@M / D f0g and the derivative of f at each point x 2 @M is non-zero. Then.r; f / W U! @M RC has bijective differential along @M. Therefore this map embeds an open neighbourhood V of @M onto an open neighbourhood W of @M 0. Now choose a smooth function " W @M! RC such that fxg Œ0; ".x/Œ W for each x 2 @M. Then compose @M Œ0; 1Œ! @M RC,.x; s/ 7!.x; ".x/s/ with the inverse of the diffeomorphism V! W. (15.7.9) Theorem. A compact smooth n-manifold B with boundary M has a smooth embedding of type I into D2nC1. Proof. Let j W M! S 2n be an embedding. Choose a collar k W M Œ0; 1Œ! U onto the open neighbourhood U of M in B, and let l D.l1; l2/ be its inverse. We use the collar to extend j to f W B! D2nC1 ( f.x/ D max.0; 1 2l2.x//j.l1.x//; x 2 U; x … U: 0; Then f is a smooth embedding on k.M Œ0; 1 2 Œ /. As in the proof of (15.7.4) we approximate f by a smooth embedding g W B! D2nC1 which coincides with f on k.M Œ0; 1 4 Œ / and which maps B X M into the interior of D2nC1. The image of g is then a subman
ifold of type I of D2nC1. 15.8. Approximation 383 15.8 Approximation Let M and N be smooth manifolds and A M a closed subset. We assume that N Rp is a submanifold and we give N the metric induced by this embedding. (15.8.1) Theorem. Let f W M! N be continuous and f jA smooth. Let ı W M! 0; 1Œ be continuous. Then there exists a smooth map g W M! N which coincides on A with f and satisﬁes kg.x/ f.x/k < ı.x/ for x 2 M. Proof. We start with the special case N D R. The fact that f is smooth at x 2 A means, by deﬁnition, that there exists an open neighbourhood Ux of x and a smooth function fx W Ux! R which coincides on Ux \ A with f. Having chosen fx, we shrink Ux, such that for y 2 Ux the inequality kfx.y/ f.y/k < ı.y/ holds. Fix now x 2 M X A. We choose an open neighbourhood Ux of x in M X A such that for y 2 Ux the inequality kf.y/ f.x/k < ı.y/ holds. We deﬁne fx W Ux! R in this case by fx.y/ D.x/. Let.x j x 2 M / be a smooth partition of unity subordinate to.Ux j x 2 M /. P The function g.y/ D x2M x.y/fx.y/ now has the required property. From the case N D R one immediately obtains a similar result for N D Rp. The general case will now be reduced to the special case N D Rp. For this purpose we choose an open neighbourhood U of N in Rp together with a smooth retraction r W U! N. We show in a moment: (15.8.2) Lemma. There exists a continuous function " W M! 0; 1Œ with the properties: (1) Ux D U".x/.f.x// U for each x 2 M. (2) For each x 2 M the diameter of r.Ux/ is smaller than ı.x/.
Assuming this lemma, we apply (15.8.1) toN D Rp and " instead of ı. This provides us with a map g1 W M! Rp which has an image contained in U. Then g D r ı g1 has the required properties. Proof. We ﬁrst consider the situation locally. Let x 2 M be ﬁxed. Choose.x/ > 0 and a neighbourhood Wx of x such that ı.x/ 2.x/ for y 2 Wx. Let Vx D r 1.U.x/=2.f.x// \ N /: The distance.x/ D d.f.x/; Rp X Vx/ is greater than zero. We shrink Wx to a neighbourhood Zx such that kf.x/ f.y/k < 1 4.x/ for y 2 Zx. The function f jZx satisﬁes the lemma with the constant function " D "x W y 7! 4.x/, i.e., z 2 Uy. 2.x/, and hence, by our choice of 4.x/. In order to see this, let y 2 Zx and kz f.y/k < 1 Then, by the triangle inequality, kz f.x/k < 1.x/, 1 z 2 Vx U; r.z/ 2 U.x/=2.f.x//: 384 Chapter 15. Manifolds If z1; z2 2 Uy, then the triangle inequality yields kr.z1/ r.z2/k <.x/ 1 Therefore the diameter of r.Uy/ is smaller than ı.y/. 2 ı.x/. After this local consideration we choose a partition of unity.x j x 2 M / subordinate to.Zx j x 2 M /. Then we deﬁne " W M! 0; 1Œ as ".x/ D P 1 4 a.x/.a/. This function has the required properties. a2M (15.8.3) Proposition. Let f W M! N be continuous. For each continuous map ı W M! 0; 1Œ there exists a continuous map " W M! 0; 1Œ with the following property: Each continuous map g W M! N with kg.x/ f.x/k
< ".x/ and f jA D gjA is homotopic to f by a homotopy F W M Œ0; 1! N such that F.a; t/ D f.a/ for.a; t/ 2 A Œ0; 1 and kF.x; t/ f.x/k < ı.x/ for.x; t/ 2 M Œ0; 1. Proof. We choose r W U! N and " W M! 0; 1Œ as in (15.8.1) and (15.8.2). For.x; t/ 2 M Œ0; 1 we set H.x; t/ D t g.x/ C.1 t/ f.x/ 2 U".x/.f.x//. The composition F.x; t/ D rH.x; t/ is then a homotopy with the required properties. (15.8.4) Theorem..1/ Let f W M! N be continuous and f jA smooth. Then f is homotopic relative to A to a smooth map. If f is proper and N closed in Rp, then f is properly homotopic relative to A to a smooth map..2/ Let f0; f1 W M! N be smooth maps. Let ft W M! N be a homotopy which is smooth on B D M Œ0; "Œ [M 1 "; 1 [ A Œ0; 1. Then there exists a smooth homotopy gt from f0 to f1 which coincides on A Œ0; 1 with f. If ft is a proper homotopy and N closed in Rp, then gt can be chosen as a proper homotopy. Proof. (1) We choose ı and " according to (15.8.3) and apply (15.8.1). Then (15.8.3) yields a suitable homotopy. If f is proper, ı bounded, and if kg.x/ f.x/k < ı.x/ holds, then g is proper. (2) We now consider M 0; 1Œ instead of M and its intersection with B instead of A and proceed as in (1). 15.9 Transversality Let f W A! M and g W B! N be smooth maps. We form
the pullback diagram F f C G A B g M with C D f.a; b/ j f.a/ D g.b/g A B. If g W B M, then we identify C with f 1.B/. If also f W A M, then f 1.B/ D A \ B. The space C can also 15.9. Transversality 385 be obtained as the pre-image of the diagonal of M M under f g. The maps f and g are said to be transverse in.a; b/ 2 C if Taf.TaA/ C Tbg.TbB/ D TyM; y D f.a/ D g.b/. They are called transverse if this condition is satisﬁed for all points of C. If g W B M is the inclusion of a submanifold and f.a/ D b, then we say that f is transverse to B in a if Taf.TaM / C TbB D TbM holds. If this holds for each a 2 f 1.B/, then f is called transverse to B. We also use this terminology if C is empty, i.e., we also call f and g transverse in this case. In the case that dim A C dim B < dim M, the transversality condition cannot hold. Therefore f and g are then transverse if and only if C is empty. A submersion f is transverse to every g. In the special case B D fbg the map f is transverse to B if and only if b is a regular value of f. We reduce the general situation to this case. We use a little linear algebra: Let a W U! V be a linear map and W V a linear subspace; then a.U / C W D V if and only if the composition of a with the canonical projection p W V! V =W is surjective. Let B M be a smooth submanifold. Let b 2 B and suppose p W Y! Rk is a smooth map with regular value 0, deﬁned on an open neighbourhood Y of b in M such that B \ Y D p1.0/. Then: (15.9.1) Note. f W A! M is transverse to B in a 2 A if and only if a is a regular value of p ı f W f 1
.Y /! Y! Rk. Proof. The space TbB is the kernel of Tbp. The composition of Taf W TaA! TbM=TbB with the isomorphism TbM=TbB Š T0Rk induced by Tb W TbM! T0Rk is Ta.p ı f /. Now we apply the above remark from linear algebra. (15.9.2) Proposition. Let f W A! M and f j@A be smooth and transverse to the submanifold B of M of codimension k. Suppose B and M have empty boundary. Then C D f 1.B/ is empty or a submanifold of type I of A of codimension k. The equality TaC D.Taf /1.Tf.a/B/ holds. Let, in the situation of the last proposition,.C; A/ and.B; M / be the normal bundles. Then Tf induces a smooth bundle map.C; A/!.B; M /; for, by deﬁnition of transversality, Taf W TaA=TaC! Tf.a/=Tf.a/B is surjective and then bijective for reasons of dimension. From (15.9.1) we see that transversality is an “open condition”: If f W A! M is transverse in a to B, then it is transverse in all points of a suitable neighbourhood of a, since a similar statement holds for regular points. 386 Chapter 15. Manifolds (15.9.3) Proposition. Let f W A! M and g W B! M be smooth and let y D f.a/ D g.b/. Then f and g are transverse in.a; b/ if and only if f g is transverse in.a; b/ to the diagonal of M M. Proof. Let U D Taf.TaA/, V D Tbg.TbB/, W D TyM. The statement amounts to: U C V D W and.U ˚ V / C D.W / D W ˚ W are equivalent relations (D.W / diagonal). By a small argument from linear algebra one veriﬁes this equivalence. (15.9.4) Corollary.
Suppose f and g are transverse. Then C is a smooth submanifold of A B. Let c D.a; b/ 2 C. We have a diagram TcC TF TbB T G T g TaA Tf TyM. It is bi-cartesian, i.e., hTf; T g i is surjective and the kernel is TcC. Therefore the diagram induces an isomorphism of the cokernels of T G and T g (and similarly of TF and Tf ). (15.9.5) Corollary. Let a commutative diagram of smooth maps be given Let f be transverse to g and C as above. Then h is transverse to G if and only if f h is transverse to g. Proof. The uses the isomorphisms of cokernels in (15.9.4). (15.9.6) Corollary. We apply (15.9.5) to the diagram fsg M is W f M S pr S and obtain: f is transverse to is W x 7!.x; s/ if and only if s is a regular value of pr ıf. Let F W M S! N be smooth and Z N a smooth submanifold. Suppose S, Z, and N have no boundary. For s 2 S we set Fs W M! N, x 7! F.x; s/. We consider F as a parametrized family of maps Fs. Then: 15.9. Transversality 387 (15.9.7) Theorem. Suppose F W M S! N and @F D F j.@M S/ are transverse to Z. Then for almost all s 2 S the maps Fs and @Fs are both transverse to Z. Proof. By (15.9.2), W D F 1.Z/ is a submanifold of M S with boundary @W D W \ @.M S/. Let W M S! S be the projection. The theorem of Sard yields the claim if we can show: If s 2 S is a regular value of W W! S, then Fs is transverse to Z, and if s 2 S is a regular value of @ W @W! S, then @Fs is transverse to Z. But this follows from (15.9.6). (15.9.8) The
orem. Let f W M! N be a smooth map and Z N a submanifold. Suppose Z and N have no boundary. Let C M be closed. Suppose f is transverse to Z in points of C and @f transverse to Z in points of @M \ C. Then there exists a smooth map g W M! N which is homotopic to f, coincides on C with f and is on M and @M transverse to Z. Proof. We begin with the case C D ;. We use the following facts: N is diffeomorphic to a submanifold of some Rk; there exists an open neighbourhood U of N in Rk and a submersion r W U! N with rjN D id. Let S D Ek Rk be the open unit disk and set F W M S! N;.x; s/ 7! r.f.x/ C ".x/s/: Here " W M! 0; 1Œ is a smooth function for which this deﬁnition of F makes sense. We have F.x; 0/ D f.x/. We claim: F and @F are submersions. For the proof we consider for ﬁxed x the map S! U".f.x//; s 7! f.x/ C ".x/sI it is the restriction of an afﬁne automorphism of Rk and hence a submersion. The composition with r is then a submersion too. Therefore F and @F are submersions, since already the restrictions to the fxg S are submersions. By (15.9.7), for almost all s 2 S the maps Fs and @Fs are transverse to Z. A homotopy from Fs to f is M I! N,.x; t/ 7! F.x; st/. Let now C be arbitrary. There exists an open neighbourhood W of C in M such that f is transverse to Z on W and @f transverse to Z on W \ @M. We choose a set V which satisﬁes C V ı xV W ı and a smooth function W M! Œ0; 1 such that M n W 1.1/; V 1.0/. Moreover we set D 2. Then Tx D 0, whenever.x/ D 0. We now
modify the map F from the ﬁrst part of the proof G W M S! N;.x; s/ 7! F.x;.x/s/ and claim: G is transverse to Z. For the proof we choose.x; s/ 2 G1.Z/. 7! G.x; t / is, as a Suppose, to begin with, that.x/ ¤ 0. Then S! N, t composition of a diffeomorphism t 7!.x/t with the submersion t 7! F.x; t/, also a submersion and therefore G is regular at.x; s/ and hence transverse to Z. 388 Chapter 15. Manifolds Suppose now that.x/ D 0. We compute T.x;s/G at.v; w/ 2 TxM TsS D TxX Rm. Let m W M S! M S;.x; s/ 7!.x;.x/s/: Then T.x;s/m.v; w/ D.v;.x/w C Tx.v/s/: The chain rule, applied to G D F ı m, yields T.x;s/G.v; w/ D Tm.x;s/F ı T.x;s/m.v; w/ D T.x;0/F.v; 0/ D Txf.v/; since.x/ D 0; Tx D 0 and F.x; 0/ D f.x/. Since.x/ D 0, by choice of W and, f is transverse to Z in x, hence – since T.x;s/G and Txf have the same image – also G is transverse to Z in.x; s/. A similar argument is applied to @G. Then one ﬁnishes the proof as in the case C D ;. 15.10 Gluing along Boundaries We use collars in order to deﬁne a smooth structure if we glue manifolds with boundaries along pieces of the boundary. Another use of collars is the deﬁnition of a smooth structure on the product of two manifolds with boundary (smoothing of corners). 15.10.1 Gluing along boundaries. Let M1 and M2 be @-manifolds.
Let Ni @Mi be a union of components of @Mi and let'W N1! N2 be a diffeomorphism. We denote by M D M1 [' M2 the space which is obtained from M1 C M2 by the identiﬁcation of x 2 N1 with '.x/ 2 N2. The image of Mi in M is again denoted by Mi. Then Mi M is closed and Mi X Ni M open. We deﬁne a smooth structure on M. For this purpose we choose collars ki W R Ni! Mi with open image Ui Mi. The map k W R N1! M;.t; x/ 7! ( k1.t; x/; k2.t; '.x//; t 0; t 0; is an embedding with image U D U1 [' U2. We deﬁne a smooth structure (depending on k) by the requirement that Mi X Ni! M and k are smooth embeddings. Þ This is possible since the structures agree on.Mi X Ni / \ U. 15.10.2 Products. Let M1 and M2 be smooth @-manifolds. We impose a canonical smooth structure on M1 M2 X.@M1 @M2/ by using products of charts for Mi as charts. We now choose collars ki W R@Mi! Mi and consider the composition, 15.10. Gluing along Boundaries 389 R2 @M1 @M2 id R R @M1 @M2.1/ M1 M2 k1k2.R @M1/.R @M2/: R1! R1,.r; '/ 7!.r; 1 Here W R2 4 /, written in polar coordinates.r; '/, and (1) interchanges the second and third factor. There exists a unique smooth structure on M1 M2 such that M1 M2 X.@M1 @M2/ M1 M2 Þ and are diffeomorphisms onto open parts of M1 M2. 2'C 3 15.10.3 Boundary pieces. Let B and C be smooth n-manifolds with boundary. Let M be a smooth.n 1/-manifold with boundary and suppose that 'B W M! @B; 'C W M! @C are smooth embeddings. We identify in B
C C the points 'B.m/ with 'C.m/ for each m 2 M. The result D carries a smooth structure with the following properties: (1) B X 'B.M / D is a smooth submanifold. (2) C X 'C.M / D is a smooth submanifold. (3) W M! D, m 7! 'B.m/ 'C.m/ is a smooth embedding as a submanifold of type I. (4) The boundary of D is diffeomorphic to the gluing of @B X 'B.M /ı with @C X 'C.M /ı via 'B.m/ 'C.m/; m 2 @M. The assertions (1) and (2) are understood with respect to the canonical embeddings B D C. We have to deﬁne charts about the points of.M /, since the conditions (1) and (2) specify what happens about the remaining points. For points of.M X @M / we use collars of B and C and proceed as in 15.10.1. For.@M / we use the following device. Choose collars B W R @B! B and W R @M! M and an embedding B W R @M! @B such that the next diagram commutes, R @M B @B [ R @M 'B M. Here B can essentially be considered as a tubular map, the normal bundle of '.@M / in @B is trivial. And is “half” of this normal bundle. Then we form ˆB D B ı.id B / W R R @M! B. For C we choose in a similar manner C and C, but we require 'C ı D C where.m; t/ D 390 Chapter 15. Manifolds.m; t/. Then we deﬁne ˆC from C and C. The smooth structure in a neighbourhood of.@M / is now deﬁned by the requirement that ˛ W R R @M! D is a smooth embedding where ( ˛.r; ; m/ D ˆB.r; 2 =2; m/; 2 ; ˆC.r; 2 3=2; m/; 3 2 ; with the usual polar coordinates
.r; / in R R. Þ 15.10.4 Connected sum. Let M1 and M2 be n-manifolds. We choose smooth embeddings si W Dn! Mi into the interiors of the manifolds. In M1 X s1.En/ C M2 X s2.En/ we identify s1.x/ with s2.x/ for x 2 S n1. The result is a smooth manifold (15.10.1). We call it the connected sum M1#M2 of M1 and M2. Suppose M1; M2 are oriented connected manifolds, assume that s1 preserves the orientation and s2 reverses it. Then M1#M2 carries an orientation such that the Mi X si.En/ are oriented submanifolds. One can show by isotopy theory that the oriented difÞ feomorphism type is in this case independent of the choice of the si. 15.10.5 Attaching handles. Let M be an n-manifold with boundary. Furthermore, let s W S k1 Dnk! @M be an embedding and identify in M C Dk Dnk the points s.x/ and x. The result carries a smooth structure (15.10.3) and is said to be obtained by attaching a k-handle to M. Attaching a 0-handle is the disjoint sum with Dn. Attaching an n-handle means that a “hole” with boundary S n1 is closed by inserting a disk. A fundamental result asserts that each (smooth) manifold can be obtained by successive attaching of handles. A proof uses the so-called Morse theory (see e.g., [134], [137]). A handle decomposition of a manifold replaces a cellular decomposition, the advantage is Þ that the handles are themselves n-dimensional manifolds. If M 0 arises from M by attaching a k-handle, 15.10.6 Elementary surgery. then @M 0 is obtained from @M by a process called elementary surgery. Let h W S k1 Dnk! X be an embedding into an.n 1/-manifold with image U. Then X X U ı has a piece of the boundary which is via h diffeomorphic to S k1 S nk1. We glue the boundary of Dk S nk1 with h; in

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