By Matthias Kreck (auth.)

ISBN-10: 3540133623

ISBN-13: 9783540133629

ISBN-10: 3540389121

ISBN-13: 9783540389125

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**Extra resources for Bordism of Diffeomorphisms and Related Topics**

**Example text**

By subtraction of handles with s i we obtain the admissible manifold N'F'. We denote the element in Trk(NF,Fo) represented by So(Sk ) by ~ and the element in II-k(N'F',Fo) represented by a f i b r e of the normal sphere bundle of Sl(S k) byp. 52 I I Then ITk(NF,Fo)/(O~) ~ 1Tk(N F ' ' F o ) / ( ~ ) ( [ 2 4 ] , p. 359). The behaviour of ~ can be controlled by the element represented by So(Sk ) in Trk(NF,FI) denoted by ~". For i f generator o f an i n f i n i t e ~" 6 Irk(NF,FI) is p r i m i t i v e , i .

The resulting manifold we denote by F ~. We denote the homotopy classes represented by the { 0 } x Sk by ~I . . . ~n and 6 1 . . . ~n" i -ITk(N F " I Fi) = llrk(NF'Fi) ~ ) ~ n ~ n and ~ I . . . ~-n span a d i r e c t summand in II-k(NF,FI) and vanish in ITk(NF,Fo) while ~ 1 . . ' ~ n span a direct summand in ITk(NF,Fo) and vanish in II'k(NF,F1). i Let W C'ITk(N F' ) be the subspace generated by ~ 1 + ~C-I. . . ~n + ~ ' ~ I +#1 . . . n-dimensional d ire ct summand in ~ k ( N ' F ' , F'i ) for i = 0 and 1 and for x,y E W the intersection number x o y vanishes.

5: Let NF be a n-dimensional admissible manifold with NF and F I-connected. Furthermore we suppose that there e x i s t s a k with I ~ k ~ ( n - 1 ) / 2 such that II"r(NF,Fi) = {0~ for 1~ ~ k and i = O. Let I : sk-lxD n-k > be a d i f f e r e n t i a b l e embedding and ~ ~ ~ k _ l ( F ) the element represented by l(sk-lx {0} ). 44 ! ) also hold i f n is even and k = n/2 -1. ) ~Tk_I(F ) ~ 1Tk_l(F ) / ( ~ ) and 1Tr(F ) : ll-r(F ) for r < k-1 where (~) denoted the subgroup generated by ~. The proof is standard and can be found for instance in [ 2 6 ] .

### Bordism of Diffeomorphisms and Related Topics by Matthias Kreck (auth.)

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