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

By Matthias Kreck (auth.)

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As the signature of the total space of a f i b r a t i o n over SI vanishes this image is contained in the kernel of ~ : % + 1 Hirzebruch-signature. We denote the kernel o f t and Winkelnkemper [ 5 ~ ( [34], b y % + 1. Neumann [ 3 4 ] have independently shown that the image of our homomorphism is equal to this kernel. 2 ) ~ where ~ i s the [51]): The image of . 28 Am~ ~"~m+l' [ M , f ] : > [ Mf ] is equal to ~m+l , the kernel of the signature homomorphism. 3: We wi'll give a new proof of this result later.

10 we can replace the statement about 84k by the s p l i t short exact sequence: 0 ) ~4k ",' W+(77 ;7"/ ) ~ ~')"4k ~ -~'4k+~ ~ e ~ 2' ~ O. 9). As a consequence of Theorem 2 we can determine odd dimensional diffeomorphism classes by numerical i n v a r i a n t s and we can do the same f o r even-dimensional diffeomorphisms up to t o r s i o n . 9: Two odd-dimensional diffeomorphisms are bordant i f and only i f a l l Stiefel-Whitney numbers of the underlying manifold and a l l characteristic numbers of the mapping torus are the same.

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. d. 4. Then the linking form L(~N F) on Tor H2k_I(QNF) vanishes in W(~/Z).