A new proof of an index theorem of Freed and Melrose
type: Item
instance of: scholarly article
title: A new proof of an index theorem of Freed and Melrose
zbMATH Open document ID: 1473.19009
author: Adnane Elmrabty
publication date: 2021-08-20T00:00:00Z
review text: This paper gives a new proof to the Freed-Melrose mod \(k\) index theorem [\textit{D. S. Freed} and \textit{R. B. Melrose}, Invent. Math. 107, No. 2, 283--299 (1992; Zbl 0760.58039)]: \[\text{Ind}^a(E,F,\beta)=\text{Ind}^t(E,F,\beta).\] The theorem is over \(\mathbb Z/k\mathbb Z\)-manifolds \((X,Y,\alpha)\), where \(X\) is a smooth compact manifold with boundary \(\partial X=\bigcup^k_{i=1}(\partial X)_i\), \(Y\) is a smooth closed manifold and \(\alpha\) is a sequence of diffeomorphisms \((\alpha)_i:(\partial X)_i\to Y\) for each \(i=1,\dots,k\). A complex vector bundle over such manifold is a triple \((E,F,\beta)\) such that \(E\), \(F\) are smooth complex vector bundles over \(X\) and \(Y\) with the isomorphism \(\beta:E|_{\partial X}\to\alpha^*F\). An illustrative example of such manifolds is, denoted by \((\mathbb S^{m,k},\mathbb S^{m-1},\pi_{m,k})\), formed by removing \(k\) interior parts of \(\mathbb B^{m}\) from \(S^m\), and the diffeomorphism is the identity map \((\pi_{m,k})_i:(\partial\mathbb S^{m,k})_i\to\mathbb S^{m-1}\) for \(i=1,\dots,k\). Fixing an embedding \(\iota\) of \((X,Y,\alpha)\) into the \(\mathbb Z/k\mathbb Z\)-sphere \((\mathbb S^{m,k},\mathbb S^{m-1},\pi_{m,k})\), one can define the induced vector bundle \(\iota_!(E,F,\beta)\) which gives an element in the \(K\)-group \(\tilde{K}(\mathbb S^{m,k},\mathbb S^{m-1})\). Hence the topological index of \((E,F,\beta)\) is defined as \[ \text{Ind}^t(E,F,\beta)=[\iota_1(E,F,\beta)]\in\tilde{K}(\mathbb S^{m,k},\mathbb S^{m-1})\cong \mathbb Z/k\mathbb Z. \] On the other hand, using the metrics on \((X,Y)\), the line bundle associated to the \(\text{Spin}^c\)-structure on \((X,Y)\), and fixing unitary connections, one can define the Dirac operator \(\mathcal D^E_X\) (see page 701, the top). Then the analytic index of \((E,F,\beta)\) is defined as \[ \text{Ind}^a(E,F,\beta)=index(\mathcal D^E_X,\alpha^*P_{\geq})\text{ mod }k\mathbb Z\in\mathbb Z/k\mathbb Z. \] Next, the author reviews the definition o…
reviewed by: Lin Shan
Mathematics Subject Classification ID: 19K56
Mathematics Subject Classification ID: 19K33
Mathematics Subject Classification ID: 58J20
zbMATH DE Number: 7383949
zbMATH Keywords: \(\mathbb{Z}/k\mathbb{Z}\)-manifolds
zbMATH Keywords: Dirac operators
zbMATH Keywords: direct images
zbMATH Keywords: geometric \(K\)-homology
zbMATH Keywords: index theorems
MaRDI profile type: Publication
full work available at URL: s12215-020-00524-3
OpenAlex ID: W3088586434
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DOI: 10.1007/S12215-020-00524-3
published in: Rendiconti del Circolo Matematico di Palermo
description: scientific article; zbMATH DE number 7383949