Cobordism invariance of the family index

✍ Scribed by Catarina C. Carvalho

• Publisher: John Wiley and Sons
• Year: 2011
• Tongue: English
• Weight: 206 KB
• Volume: 285
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.200810132

✦ Synopsis

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 \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$M\hookrightarrow \mbox{$\cal M$}\rightarrow B$\end{document}, and define a notion of cobordant families using K^1^‐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 \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$B\times \mathbb {R}^n$\end{document}.