Cobordism invariance of the family index

2008/05/28 by Catarina Carvalho, Carvalho, Catarina
Mathematics · #14D99 (Secondary) #19K56 (Primary) #58J20 #FOS: Mathematics #K-Theory and Homology (math.KT) #math.KT #msc:14D99 #msc:19K56 #msc:58J20

paper · pdf · doi:10.48550/arxiv.0805.4412

20 pages, minor changes and corrections, additions in Section 4 - Corollary 4.6 is new plus discussion on Dirac operators

arxiv created 2009/11/06 · arxiv updated 2009/12/01

Abstract

We give a K-theory proof of the invariance under cobordism of the family index. We consider elliptic pseudodifferential families on a continuous fibre bundle with smooth fibres over a compact base space B, and define a notion of cobordant families using K¹-groups on fibrations with boundary. We show that the index of two such families is the same using properties of the push-forward map in K-theory to reduce it to families on B x Rⁿ.

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