A new proof of an index theorem of Freed and Melrose
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 of \(K_{\text{ev}}(P,\mathbb Z/k\mathbb Z)\). A \(\mathbb Z/k\mathbb Z\)-cycle on a smooth compact manifold \(P\) is a triple \(((M,N),(I,J,\gamma),f)\), (see page 701, the bottom). The equivalence relations among such cycles are bordisms and modifications. A key property used in the proof is the characterization of \(K_{\text{ev}}(point,\mathbb Z/k\mathbb Z)\): \begin{center} ``the elements of \(K_{\text{ev}}(point,\mathbb Z/k\mathbb Z)=\mathbb Z/k\mathbb Z\) are of the form \([(M,\emptyset),(E,\emptyset,\emptyset)]\)''. \end{center} Now the proof starts. The author uses the special-general strategy. The special case that \(Y=\emptyset\) follows the Atiyah-Singer theorem. Then the author shows the invariance of topological and analytic indices under bordisms and modifications. By the help of the key property we just mentioned, one can assume that \(Y=\emptyset\) and the general case follows the special case. At the end of the paper, the author presents an equivariant version of the main theorem for compact Lie groups. Also, the author proves an odd mod \(k\) index theorem through Toeplitz operator. This theorem is related to questions of \textit{R. J. Deeley} [J. \(K\)-Theory 12, No. 2, 235--256 (2013; Zbl 1297.19006)] and a Dai-Zhang conjecture [\textit{X. Dai} and \textit{W. Zhang}, J. Funct. Anal. 238, No. 1, 1--26 (2006; Zbl 1114.58011)].
scientific article; zbMATH DE number 7383949
scientific article