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by: Reyes Glover


Reyes Glover
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This 5 page Class Notes was uploaded by Reyes Glover on Sunday September 6, 2015. The Class Notes belongs to M 382C at University of Texas at Austin taught by Staff in Fall. Since its upload, it has received 33 views. For similar materials see /class/181483/m-382c-university-of-texas-at-austin in Mathematics (M) at University of Texas at Austin.

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Date Created: 09/06/15
CUP PRODUCTS AND INTERSECTION NUMBER DANIEL S FREED In lecture I sketched the proof of a theorem relating cup product and intersection number Via Poincare duality This note includes the details and more remarks We begin with a local model for intersections Let p q n be nonnegative integers and A standard af ne space with AP x1xp00 c A 1 Aq00mp1zn can These subspaces intersect transversely at the origin We use the standard orientations on AP Ag and A so that the local intersection number say as de ned in differential topology of AP and Ag at the origin is 1 Let B17 C N7 and Bq C Ag be the closed balls of radius 12 then B17 x Bq C A is compact and homeomorphic to the n ball B C A Throughout we work with oriented manifolds and Z coef cients There is an analogous discussion with unoriented manifolds and 222 coef cients Let M Pp Qq be compact oriented topological manifolds wherepq n Suppose f P a M and g Q a M are continuous then fJP C HpM and 9Q C HqM are homology classes rep resented by these manifolds Let ap C HqM and 0ch C HPM be the Poincare duals guaranteed by the Poincare duality theorem so that 2 De nition 3 We say 1 and g intersect transversely if fP gQ 1 zN C M is nite and about each m there is an open neighborhood Ui and a homeomorphism pi U gt A such that cplU AP cplU Ag f 1Ui and 971Ui are connected and the maps 4p 0 f filw and 4p 0 g lgiiw are orientation preserving homeomorphisms onto their images Remark 4 The word transverse is normally used in the smooth category I have co opted it here for an analogous concept in the category of topological manifolds For any i0 C gt M let i0 HM a HMlC denote the restriction where recall the notation HMlC HM M7 C for the homology localized at C De ne Ci 4p1Bp x B C U and de ne e i1 by the equation 5 WWQMMl en Date November 22 2008 2 D S FREED 6 i e HAMLpr x Ba is given by the standard orientation of A and the fundamental lemma which glues these together into consistent orientations localized on compact sets The number q is the local intersection number of f and g at m The theorem we prove in this note is the following1 N Theorem 7 04 v ap Z q i1 The right hand side is the global intersection number obtained by summing the local intersection numbers The left hand side is the intersection product of Gig and ap the theorem justi es the nomenclature intersection product Of course this term is used for any cohomology classes here we have cohomology classes Poincare dual to the images of manifolds We have not assumed that these are submanifolds but only that there is a nite number of intersection points and near these points the images of these manifolds look like submanifolds with a transverse intersection As a preliminary we give standard generators for certain homology groups in our local model A N7 X Ag Let AP C A be the p simplex with vertices 0717151717171 1917151717171 s 2711717171quot71 p71717117171 and Ag C A the q simplex with vertices 0717171717171 p171711717171 g p271717117171 pqn71717171711 The semicolon separates the rst p coordinates from the last q coordinates Let A C A be the simplex with vertices 0 n Lemma 10 The cohomology groups HAAHBZ X Ag HAAHAP X Bq and HnA pr X Bq are in nite cyclic with generators AP Ag and A The cohomology groups in other degrees uanish pqlt 1Our Sign conventions for cup and cap products are used of course Recall that a v 1039gt 71 a pa 0 74gt if a has degree p and 0 has degree 1 LECTURE 25 3 Proof For the last the pair ATLpr x B is homeomorphic to Anan An A 7 B which by excision has the same homology as APW QAPM Earlier in the course we showed that that this homology is only nonzero in degree n with generator AP For the rst two we use homotopy invariance to collapse the extra af ne space and then use excision again The key idea in the proof of Theorem 7 is to construct Poincare dual classes BPBQ which are localized in a neighborhood of the images fP 9Q It may be helpful to keep in mind a de Rham analog in our local model A N7 x Ag So if m1 m1 q are standard local coordinates then the local Poincare dual of N7 C A is pz1 1 Hap dz 1 A A gimp where p Ag a R is a function with compact supports such that A pmp1 zpq dz 1 A A dz 1 We can take the support of p as small as we like In particular this local model for the Poincare dual may be taken to be supported in N7 x Bq In the proof which follows we rst construct global compact sets KL C M whose interiors will contain the supports of 3p 3Q the latter are constructed by applying Poincare duality to those interiors which are noncompact manifolds Then the intersection product on the left hand side of Theorem 7 reduces to a sum of intersection products in the local model and we use the basic adjunction between cup and cap products to evaluate it One more preliminary which you can read about in Hatcher as recommended at the start of Homework 12 is the relative cap product If X is a space with open sets A B C X and pq n then the relative cap product is a map 11 AHPXAXHLXAUBgtHqXB Thus for closed sets K L C X we have a relative cap product 12 A HpXlK x HnXlK L gt HqXlL Proof of Theorem 7 First we construct2 compact sets KL C M such that intK D fP intL D 9Q K O L C UUi cplU O K AP x Bq and cplU O L B17 x Ag Let C be 1 the closed set which is the complement of U Ui in M Let A B be open sets which separate the L disjoint closed sets fP O C 9Q O C This means the respective closed sets are contained in A B respectively and A O B 0 These sets exist since M is a normal topological space as we assume it is Hausdorff and paracompact hence metrizable Then de ne KU 1APXB I o A 13 i LU4p71BPqu o B Next using the Poincare duality isomorphism DimltKgt H intK a HpintK we nd 3 E HgintK such that DimltKgtlt Pgt fat Since the support of BP is contained in K we can by Zln a smooth setting the analog of the interiors of K L are tubular neighborhoods of submanifolds 4 D S FREED excision regard 3p 6 HqMlK Let MK E Then by the naturality of the cap product we deduce 14 1 Pl 6P A MK Similarly we construct g 6 HPMlL with 15 9Ql 6Q A MK Again by naturality we deduce 04F il p 16 ac iZ Q where HqMlK a HqM and HPMlL a HPM Therefore 3Q v 3 E H MlK L and from 16 we have iijwQ v 3p 1Q v ap Let ji U gt M denote the inclusion and set 17 y 375p e HqUlU m K HqA lAP x 3 1 Similarly we de ne 39yg jf Q E HWAWBP x Ag Hence we compute ltO Q V 6m MD lt56 V 5 7iKmLlMlgt H M2 18 I 32 Vji Pv iUmeLMMl H 939 W5 v v910 ll MZ H In the last line we use the homeomorphisms 4p without indicating them in the notation Equa tion 18 is the promised reduction of the global intersection number to a sum of local intersection numbers which we now proceed to evaluate We use Lemma 10 and so the explicit cycle A to represent 1 in Note in the notation of that lemma that the front p face of A is AP and the back q face is Ag Recall the formula for the cup product which involves a sign and evaluation on the back face Thus using the symbol W to denote both a cohomology class and a cocycle representative we have m 9AM9AAWVeDWwampAAF To evaluate the next cap product we let A HP be the simplex with vertices in order p l 119 l 2 p q0 1 p see 8 and The front q face is 71 7 Ag and the back p face is AP We also have that A represents 711 1qu in homology Thus mwAwewmw wkewmmwMmea wmi LECTURE 25 5 Now7 by naturality we compute lt21 v A M mp A WWW imwp A m imLmPi M We use the cap product 12 in the second expression The third equality is 14 In the last equality we identify HpMlUi O L E HpUilUi O L E HPA lBI x Ag using excision and apl and we use the fact that Lpl o f f is orientation preserving Comparing 19 and 21 we nd 1Ui lt22 W M 4 Analogously to 21 we compute 23 m3 A U 256 A iui K LJ K iUmLMBQ A MK iUmthlQl Aq Thus comparing 20 and 23 we nd 24 507 Apgt 6i Finally7 we use 187 227 and 24 and the de nition of the cup product to evaluate 939 W8 v 797 Mgt MZ H ltO Q V 6m MD cg v 2 AM H MZ H eilt71gtmltvggt7 APgt Mil M G 3 H MZ H 6171 6139 1qu H MZ H H MZ H


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