Math 396. Bundle pullback and transition matrices

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1 Math 396. Bundle pullback and transton matrces 1. Motvaton Let f : X X be a C p mappng between C p premanfolds wth corners (X, O) and (X, O ), 0 p. Let π : E X and π : E X be C p vector bundles. Consder a C p bundle morphsm E T E π X f so for each x X we get R-lnear maps T x : E (x) E(f(x )). A basc example s the case when X, X, and f are of class C p+1 and T = df : T X T X s the nduced total dervatve mappng that s the old tangent map df(x) : T x (X ) T f(x )(X) on fbers. For each x X we may use E and f to obtan a vector space E(f(x )) determned by f and E, and t s natural to ask f these can be glued together to be the fbers of some C p vector bundle f (E) X equpped wth a bundle morphsm f : f E E over f : X X that s the dentty map (f E)(x ) = E(f(x )) E(f(x )) on fbers. (Strctly speakng, the notaton f s abusve snce t depends on E and not just f; hopefully ths wll not cause confuson.) Such a par (f E X, f) wll be called a pullback bundle when t satsfes the unversal property that for any C p bundle morphsm T : E E over f : X X there s a unque C p vector bundle morphsm T : E f E over X gvng a factorzaton X π E f E T π T X X f f E X π so T x : E (x ) (f E)(x ) = E(f(x )) s exactly the map T x. (The content s that the settheoretc mappng T s a C p mappng.) In other words, the pullback bundle should promote bundle morphsms T between bundles over dfferent base spaces to bundle morphsms T between bundles over a common base space. Note that t does not make sense to go n the reverse drecton: f we are gven a vector bundle on X then most ponts n X are ether not ht by any pont of X or are ht by more than one pont (or perhaps nfntely many ponts) of X, so there s no reasonable way to use f to assocate vector spaces to ponts of X by means of a vector bundle on X. Our frst am n ths handout s to develop the pullback constructon, and to gve some examples. The second am of ths handout s to provde a very classcal descrpton of C p vector bundles of constant rank n terms of what are called transton matrces. Ths s used qute a lot n the more advanced study of vector bundles, and s convenefnt for explctly descrbng many lnear algebra bundle constructons va operatons on matrces. 2. Pullback of bundles Let f : X X be a C p mappng between C p premanfolds wth corners, 0 p. Let π : E X be a C p vector bundle. We want to construct a C p vector bundle f (π) : f E X 1

2 2 equpped wth a C p vector bundle morphsm f E f (π) X f f E X π that s unversal n the followng sense: for any C p vector bundle π : E X and any bundle morphsm T : E E over f : X X there s a unque way to fll n a commutatve dagram E f E T π T f (π) X X f f E X π wth T a C p bundle morphsm over X. We defne f E to be the dsjont unon of sets f E = x X E(f(x ))). A typcal pont n f E s denoted (x, v) wth v E(f(x )). There may be many ponts x X wth the same mage x X, and the assocated vector space E(f(x )) sttng n f E s abstractly the same for all such x f 1 (x). For ths reason, we must keep track of the ndexng parameter x X and not just the bare vector space E(f(x )) when descrbng operatons on ponts n f E. We defne the map f (π) : f E X by (x, v) x X. We need to topologze f E and gve t a C p -structure such that () f (π) : f E X s a C p map, () the lnear structure on the fbers (f (π)) 1 (x ) = E(f(x )) satsfes the local trvalty condton to make f E a C p bundle over X, and () the proposed unversal property holds. Our procedure wll be very smlar to the method used n an earler handout to defne the vector bundle V M assocated to a locally free fnte-rank O-module M (by puttng a sutable topology and C p - structure on the dsjont unon x X M (x)). Before we get nto the detals, we wsh to emphasze that the local pcture wll be qute smple: f {U } s an open coverng of X for whch there are C p bundle somorphsms E U U R n correspondng to a trvalzng frame {s () k } n E(U ), then for the opens U = f 1 (U ) that cover X the restrcton (f E) U has a C p trvalzaton gven by the sectons u s() k (f(u )) E(f(u )) = (f E)(u ) to f (π) : f E X over U X. Roughly speakng, the only problem s to cleanly buld the rght bundle f E over X gvng rse to such local trvalzatons on fbers over f 1 (U ) s. Ths constructon problem can be solved n several ways (all of whch gve answers that are unquely somorphc n accordance wth the unversal property), and n what follows we have chosen the constructon that seems most elegant (n terms of mnmzng non-canoncal choces and the nterventon of matrces) n vew of our present knowledge. To defne the topology on f E we wll use the method of glung topologes. Consder pars (φ, U) where U X s a non-empty open set and φ : E U U R n s a C p somorphsm of bundles. (Of course, n may depend on U.) For the open set U = f 1 (U) n X we get a bjecton φ : (f (π)) 1 (U ) U R n over U by usng the lnear somorphsm (f (π)) 1 (u ) = (f E)(u ) = E(f(u )) R n defned by φ f(u ) over each u U. We wsh to use φ to transfer the topology

3 of the product U R n to a topology on (f (π)) 1 (U ), and to glue these topologes to topologze f E. Note that as we vary (φ, U) the opens U do cover X and hence the open premages U do cover X. To glue, as usual we have to verfy two propertes: for any (φ 1, U 1 ) and (φ 2, U 2 ) such that U 1 U 2 s non-empty, we must prove () the overlap (f (π)) 1 (U 1 ) (f (π)) 1 (U 2 ) = (f (π)) 1 (U 1 U 2 ) s open n each of (f (π)) 1 (U 1 ) and (f (π)) 1 (U 2 ), and () ths overlap nherts the same subspace topology from both. (Note that f U 1 U 2 s empty then so s ts premage U 1 U 2 under f.) Snce U 1 meets U 2, the constant ranks for E on U 1 and U 2 are equal, say n. The open subset property () just says that φ carres the overlap to an open subset of U Rn, and ndeed t carres the overlap to the subset (U 1 U 2 ) Rn n U Rn that s obvously open. The agreement of subspace topologes n () s equvalent to the asserton that the transton mappng φ 2 (φ 1) 1 : (U 1 U 2) R n (U 1 U 2) R n s a homeomorphsm. Explctly, the map s gven by (u, v) (u, (φ 2 φ 1 1 ) f(u )(v)) where φ 2 φ 1 1 : (U 1 U 2 ) R n E U1 U 2 (U 1 U 2 ) R n s the transton somorphsm over U 1 U 2 for E. For u U ths latter map s gven on u-fbers by a matrx L(u) GL n (R) such that L : U GL n (R) s a C p mappng (ths encodes that φ 2 φ 1 1 s a C p mappng). Thus, (1) φ 2 (φ 1) 1 : (u, v) (u, ((L f)(u ))(v)). Snce L s a contnuous mappng and f s contnuous, so s L f. Thus, the formula for evaluaton of a matrx on a vector mples (va (1)) that the self-map φ 2 (φ 1 ) 1 of (U 1 U 2 ) Rn has contnuous component functons and so s contnuous; of course, the same argument apples to the nverse map (swap the roles of (φ 1, U 1 ) and (φ 2, U 2 )), so we get the homeomorphsm result. We have put a topology on f E such that for each local trvalzaton φ : E U U R n as above, wth n the constant rank for E over U, the subset (f (π)) 1 (U ) f E s open and the bjecton φ : (f (π)) 1 (U ) U R n over U nduced by the lnear fbral somorphsm (f E)(u ) = E(f(u )) φ f(u ) for all u U s a homeomorphsm. It follows that the projecton f (π) : f E X s contnuous snce for each of the (U, φ) s the part of f E over the open U X s homeomorphc to U R n (va φ ) n a manner that carres the restrcton of f (π) over to the contnuous standard projecton U R n U. Thus, we have gven f E a structure of topologcal vector bundle over X because the φ s are homeomorphsms that are lnear on fbers (.e., they provde local topologcal trvalzatons). Note that when U 1 U 2 s non-empty (so the constant ranks for E on U 1 and U 2 are equal, say n), the transton somorphsm φ 2 (φ 1 ) 1 s C p (snce by (1) t s gven n terms of the composte L f : U 1 U 2 GL n(r), wth f : U U a C p map and the matrx-valued functon L descrbng the transton mappng φ 2 φ 1 1 also C p on U 1 U 2 ). Hence, the hypotheses for glung C p -structures (as n Lemma 1.2 n the handout on the equvalence between bundles and O-modules) are satsfed for the topologcal space f E wth ts open cover by the (f (π)) 1 (U ) s and the homeomorphsms gven by the φ s. We thereby get a unque C p structure on f E wth respect to whch the maps φ : (f E) U U R n U are C p somorphsms for each C p trvalzaton φ : E U U R n U. It follows (by workng over the opens U n X ) that f (π) : f E X s a C p mappng and the lnear structures on the fbers (f E)(x ) make the φ s be C p trvalzatons that ensure f E s a C p vector bundle over X. Ths completes the constructon of the C p vector bundle structure on f (π) : f (E) X. R n 3

4 4 Remark 2.1. By constructon, f U X s any open subset (not necessarly wth E U trval) and U f 1 (U) s any open subset (not necessarly all of f 1 (U)), we have (f E) U = f U,U (E U) wth f U,U : U U the C p restrcton of f. In ths sense, the formaton of f E s local on X and X. Lemma 2.2. The set-theoretc mappng f : f E E over f : X X gven on fbers by the lnear dentty map (f E)(x ) = E(f(x )) E(f(x )) s a C p mappng. Proof. Ths problem s local over X and over X, so by Remark 2.1 we may assume that E s trval. Let φ : E X R n be a choce of trvalzaton. By the constructon of the C p -structure on f E, the bjecton φ : f E X R n gven on fbers over x X by (f E)(x ) = E(f(x )) φ f(x ) s a C p trvalzaton over X. Calculatng on fbers shows that the C p composte R n f E X R n f 1 X R n E over f : X X s exactly the map f, so ndeed f s C p. Example 2.3. Before we verfy that the par (f E X, f) satsfes the desred unversal mappng property, we gve two trval examples. If E X s a C p vector bundle and x X s a pont, for the ncluson mappng : {x} X the pullback E s the vector bundle over the 1-pont space {x} gven by the vector space E(x), wth ĩ : E E the ncluson onto the fber at x. If U X s an open subset and : U X s the ncluson map, then the vector bundle E U s the restrcton E U U (and ĩ : E E s the ncluson onto E U ). To gve nterestng examples wth mnmal mess, we verfy the unversal mappng property: Theorem 2.4. Let f : X X be a C p mappng. Let E X be a C p vector bundle, and let E X be a C p vector bundle. If T : E E s a C p bundle morphsm over f : X X then there s a unque C p bundle morphsm T : E f E over X such that f T = T. Set-theoretcally, snce f on x -fbers nduces the dentty map on E(f(x )) there s only one possblty for the fbral map T : E (x ) (f E)(x ) = E(f(x )), namely T x. Hence, the content of the theorem s that ths T s a C p mappng. Proof. The C p problem s local on X, so by Remark 2.1 we may work locally on X and X. Hence, the problem s reduced to the case when E and E are trval. Choose trvalzatons φ : E X R n and φ : E X R n over X and X respectvely. As we saw n the proof of Lemma 2.2, we get a C p trvalzaton f (φ) : f (E) X R n over X that s the map φ f(x ) : (f E)(x ) = E(f(x )) R n on x -fbers for each x X. The C p composte mappng X φ 1 n R E T φ E X R n s gven by (x, v ) (f(x ), (T (x ))(v )) for a lnear map T (x ) : R n R n, say gven by a matrx (a j (x )). The C p property of ths mappng s exactly the property that the matrx-valued mappng [T ] : X Mat n n (R) gven by x (a j (x )) s C p (.e., the matrx-entres a j : X R are C p functons). Indeed, suffcency follows from the C p property of f and the formula for evaluatng a matrx on a vector

5 n Eucldean space, and necessty follows from chasng (x, e j ) for standard bass vectors e j Rn. Defne the C p mappng T : E f E to be the composte E φ X R n X R n f (φ) 1 f E wth mddle map gven by (x, v ) (x, (a j (x ))(v )). Ths s a C p map (over X ) because each a j : X R s C p, and on fbers t gves the map E (x ) (f E)(x ) = E(f(x )) that s exactly T x. Hence, we have bult the desred C p mappng. An mportant feature of bundle pullback s that t s well-behaved wth respect to bundle morphsms over the ntal base space X: Corollary 2.5. Let f : X X be a C p mappng and let T : E 1 E 2 be a C p bundle morphsm over X. There s a unque map f (T ) : f (E 1 ) f (E 2 ) between C p bundles over X such that on fbers over each x X t s the R-lnear map (f (E 1 ))(x ) = E 1 (f(x )) T f(x ) E 2 (f(x )) = (f (E 2 ))(x ). (In partcular, f E 2 = E 1 and T s the dentty then f (T ) s the dentty.) The formaton of f (T ) behaves well wth respect to composton of C p bundle morphsms n the sense that f T : E 2 E 3 s another C p bundle map, then f ( T T ) = f ( T ) f (T ) as maps from f (E 1 ) to f (E 3 ) over X. Proof. The unqueness of f (T ) s clear on the set-theoretc level, and calculatng on fbers also verfes the compatblty wth composton n T. The only real problem s to prove that the map f (T ) defned set-theoretcally to be T f(x ) on x -fbers s n fact a mappng of C p bundles over X. But t s easy to drectly construct f (T ) as a C p bundle mappng by usng the unversal property of pullback bundles: consder the composte dagram of C p bundle maps 5 f E 1 f E 1 T E 2 X f X By the unversal property of the pullback bundle f E 2, ths composte map unquely factors through a C p vector bundle morphsm f E 1 f E 2 over X, and checkng on fbers shows that ths morphsm s f (T ). 3. Examples Let f : X X be a C p mappng between C p premanfolds wth corners, 0 p, and let π : E X be a C p vector bundle. Example 3.1. We dentfy C p -sectons s E(X) wth C p vector bundle morphsms [s] : X R E over X (gven by 1 s(x) on x-fbers). By the unversal property, the composte bundle morphsm X X R f 1 X R [s] E over f unquely factors through a bundle morphsm X R f E over X. Ths latter map corresponds to a C p -secton n (f E)(X ) that we denote f (s) and call the pullback secton. Concretely, on fbers we have (f (s))(x ) (f E)(x ) = E(f(x )) s s(f(x )). Ths fbral calculaton shows that the resultng map f : E(X) (f E)(X ) s R-lnear, and even lnear over the R-algebra map O(X) O (X ) gven by composton wth f. (That s, for h O(X),

6 6 f (h s) = (h f) f (s).) In the specal case E = X R we have f E = X R and va the equaltes E = O and f E = O the map f : O(X) = E(X) (f E)(X ) = O (X ) s h h f. In general, f : E(X) (f E)(X ) s nether njectve nor surjectve (even for E = X R, let alone for more nterestng examples such as E = T X). For any open U X we have (f E) f 1 U = f U (E U) wth f U : f 1 (U) U the C p restrcton of f (Remark 2.1), so we may apply the same constructon to f U nstead of f to defne f (s) (f E)(f 1 U) for any s E(U). Calculatng on fbers shows that the formaton of f (s) s local: for open U 0 U and s E(U), (f s) f 1 (U 0 ) = f (s U0 ) n (f E)(f 1 (U 0 )). Example 3.2. Fber calculatons show that n the setup of Example 3.1, f {s j } s a trvalzng frame for E U then {f (s j )} s a trvalzng frame for (f E)(f 1 (U)). That s, the vectors (f (s j ))(x ) = s j (f(x )) (f E)(x ) = E(f(x )) gve a bass for each x f 1 (U) when the vectors s j (x) E(x) gve a bass for each x U. The next two (rather lengthy) examples rest on the theory of the tangent bundle, to be developed n a few lectures. Postpone readng these examples untl that tme; skp ahead to Example 3.5. Example 3.3. Assume 1 p, and consder the total dervatve mappng df : T X T X of C p 1 bundles (over f : X X). By the unversal property of pullback, we arrve at a unque C p 1 bundle morphsm T X f (T X) over X whose restrcton to fbers over x X s the old tangent map df(x ) : T x (X ) (T X )(x ) (f (T X))(x ) = (T X)(f(x )) T f(x )(X). Hence, we may consder the C p 1 bundle morphsm T X f (T X) over X as merely another global repackagng of the collecton of tangent mappngs arsng from f : X X. In certan settngs (but not all!) t s more convenent to work wth ths map nstead of the C p 1 bundle morphsm df : T X T X over f : X X. As an mportant example, when f s an mmerson then the mappng T X f (T X) of bundles over X has fber map T x (X ) T f(x )(X) over x X, so ths bundle map over X encodes how the tangent spaces to X move nsde of the tangent spaces of X. Ths example wll be a partal motvaton for the noton of subbundle that we shall fnd to be very useful later on. We must warn the reader of a common source of confuson. Passng to X -sectons, df gves a mappng (T X )(X ) (f (T X))(X ) that assocates to any C p 1 vector feld v on X the X -secton f ( v ) of f (T X) whose value n each fber (f (T X))(x ) = T X(f(x )) = T f(x )(X) s df(x )( v (x )). Ths pullback secton f ( v ) s just a repackagng of the data of the df(x )( v (x )) s n the tangent spaces T f(x )(X) for varyng x X and t has nothng to do wth any global vector feld on X. More specfcally, although we have two maps (T X )(X ) (f (T X))(X ), (T X)(X) (f (T X))(X ) to the same target, these have nothng to do wth each other: the frst encodes the tangent mappng arsng from f (and so t s a specal constructon adapted to the fact that our vector bundles are tangent bundles) whereas the second s a general nonsense mappng that makes sense wth T X replaced by any C p 1 vector bundle on X (t does not encode any nformaton related to tangent maps, for example). We remnd the reader agan that C p 1 vector felds on X (resp. X ) do not gve rse to C p 1 vector felds on X (resp. X); the pullback bundle f (T X) on X s an abstract thng whose general X -sectons have no nterpretaton va vector felds on ether X or X (cf. the comments on f generally beng nether njectve nor surjectve n Example 3.1).

7 Example 3.4. Just as the tangent bundle globalzes pontwse tangent spaces, we can use the concept of pullback to globalze the pontwse lnear somorphsm (2) T (x1,x 2 )(X 1 X 2 ) T x1 (X 1 ) T x2 (X 2 ) for tangent spaces on products of C p premanfolds wth corners X 1 and X 2 (1 p ). Let π j : X 1 X 2 X j be the standard C p projecton map. We have C p 1 bundle dagrams T (X 1 X 2 ) dπj T (X j ) 7 X 1 X 2 π j X j and so by the unversal property of pullback we get C p 1 bundle morphsms T (π j ) : T (X 1 X 2 ) πj (T (X j)) over X 1 X 2. As s worked out n the handout on drect sums for vector bundles, these behave wth respect to bundle morphsms exactly as drect sums of vector spaces behave wth respect to lnear maps. Thus, we get a C p 1 bundle morphsm T (π 1 ) T (π 2 ) : T (X 1 X 2 ) π 1(T (X 1 )) π 2(T (X 2 )) over X 1 X 2. On fbers over (x 1, x 2 ) X 1 X 2 ths recovers the pontwse map dπ 1 (x 1, x 2 ) dπ 2 (x 1, x 2 ) : T (x1,x 2 )(X 1 X 2 ) T x1 (X 1 ) T x2 (X 2 ) that s exactly the standard somorphsm (2). Hence, T (π 1 ) T (π 2 ) s a C p 1 bundle morphsm that s an somorphsm on fbers, and so t s a C p 1 bundle somorphsm. Concretely, f U j X j s an open set over whch we have C p coordnates {x (j) n j }, then T (X j ) Uj has the trvalzng frame { x (j) 1,..., x(j) } 1 nj, and so ts pullback πj (T (X j)) U1 U 2 has the trvalzng frame gven by the sectons πj ( x (j) ) s for 1 n j (Example 3.2). By lookng n fbers over U 1 U 2 and usng Example 3.3 for π j, the nverse of the somorphsm T (π 1 ) T (π 2 ) on U 1 U 2 -sectons carres (π j ( x (j) ), 0) to (j) x, where the n π 1 + n 2 functons j (3) x (1) 1 π 1,..., x (1) n 1 π 1, x (2) 1 π 2,..., x (2) n 2 π 2 : U 1 U 2 R are the nduced C p coordnates on U 1 U 2. These n 1 + n 2 pullback sectons gve a trvalzng frame for π1 (T (X 1)) π2 (T (X 2)), and so va (2) we thereby recover the trvalzng frame for T (X 1 X 2 ) U1 U 2 gven by the partals wth respect to the C p coordnate system (3) on U 1 U 2. Example 3.5. Suppose that Γ s a group equpped wth rght C p -actons on X and on E that are free and properly dscontnuous, and assume that π s Γ-equvarant n the sense that π(v.γ) = π(v).γ for all v E and γ Γ. Assume also that the acton of γ Γ nduces the fbral bjecton E(x) E(x.γ) that s a lnear somorphsm for all x X. By the homework we know that the map of C p quotents π : E = E/Γ X/Γ = X s a C p vector bundle. There s a natural commutatve dagram E η E E π X ηx X where the horzontal maps are the natural local C p somorphsms onto the Γ-quotents. On fbers over x X and ts mage x = η X (x) the nduced map E(x) E(x) s clearly lnear, and even an somorphsm. In partcular, by the unversal property of pullback bundles we see that η E unquely π

8 8 factors through a bundle morphsm E ηx (E) over X, and ths latter morphsm must be an somorphsm snce t s an somorphsm on fbers over X (due to the fbral somorphsm property for η E ). To summarze: the pullback of E/Γ X/Γ by the quotent map X X/Γ recovers the orgnal bundle E X. An especally nterestng case s that of the Möbus strp wth nfnte heght. Let f : S 1 C be the quotent by the antpodal map and let M C be the Möbus strp wth nfnte heght, so M s the quotent of the trval bundle R S 1 S 1 va the nvoluton (t, θ) ( t, θ + π) that les over the antpodal map θ θ + π on S 1. We have seen that M s a C lne bundle over C that s not topologcally trval. However, ts pullback f (M ) S 1 s the trval lne bundle R S 1 S 1. Thus, a C bundle wth postve rank and no non-vanshng contnuous sectons may become a trval C bundle after a very mld pullback (such as the double coverng S 1 C). Another property of bundle pullback that s very useful n practce s that t s well-behaved wth respect to composton n the map along whch we are formng the pullback: Corollary 3.6. Let g : X X and f : X X be C p mappngs between C p premanfolds wth corners, 0 p, and let E be a C p vector bundle on X. There s a unque somorphsm of C p vector bundles c g,f,e : (f g) (E) g (f (E)) over X gven on fbers over each x X by the composte lnear somorphsm (4) ((f g) (E))(x ) E((f g)(x )) = E(f(g(x ))) (f (E))(g(x )) (g (f E))(x ). (As an example, for opens U X and s E(U) ths carres (f g) (s) to g (f (s)) on the level of sectons over (f g) 1 (U) = g 1 (f 1 (U)).) Moreover, these somorphsms are transtve n the sense that f h : X X s a thrd C p mappng then the two composte somorphsms (f g h) E h ((f g) E) h g f E, (f g h) E (g h) f E h g f E concde. That s, h (c g,f,e ) c h,f g,e = c h,g,f (E) c g h,f,e as somorphsms from (f g h) (E) to h (g (f (E))). Proof. The transtvty condton may be checked on fbers, where t s just the assocatvty of composton for maps of sets. Also, the unqueness of c g,f,e s mmedate because ts effect on fbers s specfed. The problem s therefore to prove that the set-theoretc map c g,f,e that s lnear on fbers over X s a C p mappng. Once agan, we use the unversal property of bundle pullback to recreate c g,f,e as a C p mappng: the composte bundle morphsm g (f E) f E E over f g : X X unquely factors through a bundle morphsm g (f E) (f g) (E) over X. Checkng on x -fbers gves the lnear somorphsm nverse to (4), so we have bult a C p vector bundle somorphsm whose C p nverse s c g,f,e. 4. Transton matrces Fx a postve nteger n. Perhaps the most concrete (and classcal) way to descrbe a rank-n C p vector bundle on X s through what are called transton matrces. Ths s a vector-bundle analogue of local C p charts on C p premanfolds wth corners; the purpose of the nterventon of matrces s to encode the lnear structure on the fbers of vector bundles over the base space X. Let π : E X be a rank-n vector bundle of class C p, 0 p, and let {U } be a trvalzng cover for E. Pck C p somorphsms of C p vector bundles φ : E U U R n over U. Over U j = U U j we have two trvalzng somorphsms va the restrctons of φ and φ j. That s, we have two C p bundle somorphsms φ : E Uj U j R n, φ j : E Uj U j R n.

9 We thereby get a transton somorphsm of C p bundles φ j = φ j φ 1 : U j R n E Uj U j R n. Ths nduces lnear fbral somorphsms T j (x) : R n R n over each x U j, so we have wth T j (x) GL n (R). φ j φ 1 : (x, v) (x, T j (x)(v)) Lemma 4.1. Fx and j. The map of sets T j : U j GL n (R) gven by x T j (x) s a C p mappng. Proof. Wrtng T j (x) = (a rs (x)) 1 r,s n for functons a rs : U j R, we have to prove that these functons are C p on U j. Consder the ncluson ι s : R R n onto the sth coordnate axs and the standard projecton π r : R n R onto the rth coordnate axs. The composte C p mappng U j R 1 ιs U j R n φ j U j R n 1 π r Uj R s exactly (x, c) (x, a rs (x)c), so usng the C p mappng x (x, 1) from U j to U j R we get that the map x (x, a rs (x)) from U j to U j R s a C p mappng. Composng wth the C p projecton U j R R gves that the functon a rs : U j R s C p. To summarze, usng the trvalzng open cover {U } and the choces of C p somorphsms of bundles φ : E U U R n we have bult a collecton of C p mappngs T j : U U j GL n (R) that we call the transton matrces for the trvalzaton of the E U s va the φ s. Note that ths has nothng to do wth local C p -charts on the base space X. These collectons of matrx-valued C p mappngs T j are not unrelated to each other: they satsfy the trple overlap condton T j T jk = T k as matrx-valued mappngs U U j U k GL n (R). Indeed, ths comes down to the elementary assocatvty calculaton (φ φ 1 j ) (φ j φ 1 k ) = φ φ 1 k on (U U j U k ) R n and the fact that matrx multplcaton encodes composton of lnear maps. The next result shows that ths procedure can be reversed: Theorem 4.2. Let {U } be an open coverng of X, and let T j : U U j GL n (R) be C p mappngs that satsfy the trple overlap condton T j (x) T jk (x) = T k (x) for all x U U j U k for all, j, k. There exsts a rank-n C p vector bundle π : E X wth trvalzatons φ : E U U R n satsfyng φ j φ 1 = T j on (U U j ) R n for all and j. Moreover, the data consstng of E and the φ s s unque n the followng sense: f π : E X wth trvalzatons φ : E U U R n s another such structure lkewse gvng rse to the T j s, then there s a unque C p bundle somorphsm f : E E over X such that φ f U = φ as Cp bundle somorphsms from E U to U R n for all. Before we prove the theorem, we make some remarks. A real nusance n ths theorem s that the trvalzng coverng {U } and the specfc trvalzng somorphsms φ play such a promnent role even though t s the vector bundle that s the prmary focus of nterest. A full treatment of the approach to vector bundles through the language of transton matrces requres the characterzaton of exactly whch changes n the U s and the φ s do not change (at least up to somorphsm) the bundle we construct from the data of the transton matrces. Moreover, to actually work wth ths 9

10 10 vewpont one has to certanly do more, such as express the noton of bundle morphsm (and many operatons wth vector bundles) n terms of transton matrces. Ths s a long story, and so we wll not delve nto t any further here. Our pont here s to smply record that the vewpont of transton matrces satsfyng a trple overlap condton s adequate to construct all vector bundles wth constant rank and t s very wdely used n practce and n mportant computatons wth vector bundles. Proof. Let us frst prove unqueness. Suppose that π : E X and π : E X wth respectve C p trvalzatons φ : E U U R n and φ : E U U R n over each U are both solutons to our exstence problem. We want to fnd the asserted unque C p bundle somorphsm f : E E over X satsfyng φ f U = φ over U for each. The restrcton f : E U E U of f over U has no choce but to be φ 1 φ, so each f s unquely determned and hence f s unquely determned. To actually buld f, we defne the bundle somorphsm f : E U E U over U to be φ 1 φ and we seek to glue the f s over X. Over U U j, we clam that the restrctons of f and f j to C p maps E U U j E U U j E do concde. Ths says that φ 1 φ = φ 1 j φ j as maps from E U U j to E U U j, or equvalently that φ (φ j ) 1 = φ φ 1 j as self-maps of (U U j ) R n. By hypothess on the data (E, {φ }) and (E, {φ }), both of these latter two self-maps are equal to (x, v) (x, (T j(x))(v)). Hence, we can ndeed (unquely) glue the C p bundle maps f over U to a C p mappng f : E E over X that s a bundle morphsm (lnearty may be checked on fbers over each x X, snce each x les n some U ). The same procedure apples to the nverses f 1 and so gves a bundle map f : E E of class C p that s an nverse to f (as may be checked on fbers over X). Havng settled the unqueness aspect of the problem, we now turn to exstence. There s a general procedure called glung of topologcal spaces that we have to use. In essence, what we want to do s to glue U R n to U j R n by pastng the open set (U U j ) R n U R n onto the open set (U U j ) R n U j R n va the fbrally lnear homeomorphsm (x, v) (x, (T j (x))(v)) over U U j for all and j. To make ths precse, we need to ntroduce a bg topologcal space wth an equvalence relaton. Let S = (U R n ) be the dsjont unon of the topologcal spaces U R n. We declare a subset of S to be open when t meets each U R n n an open set. Ths s obvously a topology on S. We defne an equvalence relaton on S as follows: for two ponts s = (u, v ) U R n and s = (u j, v j ) U j R n n S, we say s s f and only f u and u j are equal to a common pont x U U j and v j = (T j (x))(v ) n R n. Let us check that ths s an equvalence relaton on S. Certanly s s snce T (x) s the dentty matrx for all x U (ths follows from the trple overlap condton T (x) T (x) = T (x) and the nvertblty of T (x)), and lkewse f s s then s s because for all x U U j we have that T j (x) and T j (x) are nverse to each other (ther product s T (x), the dentty matrx). Fnally, suppose s = (u, v ), s = (u j, v j ), and s = (u k, v k ) satsfy s s and s s. The ponts u and u j concde n U U j and the ponts u j and u k concde n U j U k, so all three ponts are equal to a common pont x U U j U k. By hypothess v j = (T j (x))(v ) and v k = (T kj (x))(v j ) n R n, so the trple overlap condton gves v k = (T kj (x) T j (x))(v ) = (T k (x))(v ) n R n. Ths gves s s as desred. We therefore have an equvalence relaton on S. Let E denote the set of -equvalence classes n S. The projectons π : U R n U X have the property that f s = (u, v ) and s = (u j, v j ) are ponts n S wth s s then π (s) = π j (s ) n X. Hence, we get a well-defned map of sets π : E X that sends a -equvalence class to the

11 common pont π (s) X for any representatve pont s U R n S n the equvalence class for any. For x X, consder the fber π 1 (x) n E. We clam that ths has a natural structure of R-vector space. The representatves for the equvalence classes n π 1 (x) E are ponts of the form (x, v) U R n wth U contanng x. Snce two ponts (x, v), (x, v ) U R n are -equvalent f and only f v = v n R n (as T (x) s the dentty matrx), we conclude that for each U contanng x, every pont e π 1 (x) has a unque representatve of the form (x, v (e)) U R n wth v (e) R n. For any two ponts e, e π 1 (x) and c, c R, we wsh to defne ce + c e π 1 (x) to be the -equvalence n E class represented by (x, cv (e) + c v (e )) U R n S for any U contanng x. The crucal ssue s to show that ths defnton does not depend on the choce of such U. If U j also contans x then we have the relatons v j (e) = (T j (x))(v (e)) and v j (e ) = (T j (x))(v (e )) by the defnton of the equvalence relaton, so by R-lnearty of T j (x) : R n R n we get whence (T j (x))(cv (e) + c v (e )) = c (T j (x))(v (e)) + c (T j (x))(v (e )) = cv j (e) + c v j (e ), (x, cv (e) + c v (e )) (x, cv j (e) + c v j (e )) n S. Ths confrms that our proposed defnton of ce + c e π 1 (x) E s well-posed, and calculaton wth representatves (say usng a fxed U contanng x) shows that t defnes a structure of n-dmensonal R-vector space on π 1 (x). So far we have constructed a map of sets π : E X and we have put a structure of n-dmensonal R-vector space on the fbers of π. Consder the composte mappng of sets U R n S E. By the defnton of π (and of the equvalence relaton ), ths composte map s a bjecton from U R n onto π 1 (U ). Moreover, the resultng bjecton ψ : U R n π 1 (U ) carres the standard projecton U R n U over to the restrcton π : π 1 (U ) U of π : E X, and the nduced bjecton R n π 1 (x) of fbers over any x U s an R-lnear somorphsm. Ths lnearty s due to how the R-vector space structure on π 1 (x) was defned. We topologze E as follows: a subset Σ E s open f and only f ts premage n S = (U R n ) s open. Ths s clearly a topology on E. Lemma 4.3. The subsets π 1 (U ) E are open and the maps ψ are homemorphsms. Proof. The premage of π 1 (U ) n S meets each U j R n n the subset (U U j ) R n that s certanly open. Hence, all π 1 (U ) s are open n E. To prove that ψ s a homemorphsm, we have to prove that a subset Σ U R n s open f and only f ψ (Σ) π 1 (U ) s open, whch s to say that ψ (Σ) E has premage n S that meets each U j R n n an open subset. In vew of how and ψ are defned, the premage of ψ (Σ) n U j R n s the mage of Σ ((U U j ) R n ) under the mappng (U U j ) R n (U U j ) R n defned by (x, v) (x, (T j (x))(v)). Ths mappng s a homeomorphsm, and even a C p somorphsm, because t s obvously C p and usng T j (x) nstead of T j (x) gves a C p nverse. Hence, snce a subset of (U U j ) R n s open n ths product f and only f t s open n U j R n, our problem s equvalent to the obvous asserton that a subset Σ U R n s open f and only f Σ meets (U U j ) R n U R n n an open subset for every j (ncludng j =!). 11

12 12 We have gven E a topology such that the bjectons ψ : U R n π 1 (U ) over U are homeomorphsms that are R-lnear somorphsms on fbers over all ponts n U for all. In fact, E also has a structure of topologcal vector bundle (wth π as ts structure map to the base X). To see that π : E X s contnuous, we note that the U s are an open cover of X wth π 1 (U ) E an open set, so the π 1 (U ) s cover E and so by the local nature of contnuty t suffces to prove contnuty for the restrctons π : π 1 (U ) U of π. Snce the bjecton ψ : U R n π 1 (U ) s a homeomorphsm (wth the source gven ts product topology), t s equvalent to show that π ψ : U R n R n s contnuous. By the defnton of ψ and π, ths map s the standard projecton that s certanly contnuous. Wth π now shown to be contnuous, we lkewse see that the ψ s provde local topologcal trvalzatons, so π : E X s ndeed a topologcal vector bundle. The transton mappng ψj 1 ψ as a self-map of (U U j ) R n s exactly the map (x, v) (x, (T j (x))(v)). Hence, f we can promote E to a C p vector bundle over X such that the ψ s are C p trvalzatons then we wll have solved the exstence problem because the transton matrces lnkng these trvalzatons ψ and ψ j over U U j are gven by the map T j : U U j GL n (R) for all and j. (We take φ = ψ 1 n the statement of the exstence problem.) To gve the topologcal vector bundle E over X a structure of C p bundle makng the ψ s local C p trvalzatons, we frst note that (as has already been observed) the transton mappngs ψj 1 ψ are C p automorphsms of (U U j ) R n that are lnear on fbers, and more explctly are gven by the mappng (x, v) (x, (T j (x))(v)) wth T j : U U j GL n (R) a C p mappng (.e., ts matrx-entry functons are C p functons). The same apples to the nverse transton mappng, so we conclude that the transton mappngs are C p somorphsms. Hence, by the glung lemma for C p -structures (from Lemma 1.2 n the handout on equvalence between bundles and O-modules) there s a unque C p -structure on E wth respect to whch the fbrally lnear ψ s are C p somorphsms. Usng ths C p -structure, t remans to check that π : E X s a C p map. Ths goes exactly as n the proof that π s contnuous (workng over the U s and now usng that the ψ s are C p somorphsms and that each standard projecton U R n U s C p ).