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Though the Atiyah-Singer index theorem holds for pseudodifferential operators, all the applications of the index theorem I know of only need it for Dirac-type operators. I know that pseudodifferential …

I posted this question on Math.SE (https://math.stackexchange.com/questions/409444/), but got no answer. So I repost it here. Let M be a closed manifold. Then there is a cap product $K^\ast(M) \times …

Hi, I'm currently trying to understand the Atiyah-Singer index theorem and its proof as presented in the book "Spin Geometry" by Lawson and Michelsohn. I do not understand why the analytic index map …

This is a question I asked at Math.SE but got no answers: https://math.stackexchange.com/q/397164/7110/ Atiyah and Hirzebruch showed in their paper "Vector bundles and homogeneous spaces" that $\mathr …

The following is named the Künneth theorem for tensor products in the book by Blackadar on K-theory for operator algebras: If $A$ and $B$ are C*-algebras and $A$ is in the bootstrap class, then there …

Let $M$ be a compact spin$^c$ manifold, so that it has a fundamental class $[M] \in K_n(M)$. It is well-known that the cap product with $[M]$ induces Poincare duality isomorphisms $K^\ast(M) \cong K_\ …

Atiyah proved a Kuenneth short exact sequence for K-theory. I need one for K-homology, but can not find any reference in the literature. Do you know one? Using general spectra stuff, one gets a Kuenn …

Let $P\colon E \to F$ be an elliptic pseudodifferential operator over $M$. Assuming that $P$ defines a finitely summable Fredholm module, we may apply the Chern-Connes character to it to get a cyclic …

There is a homological Chern character $ch_\ast \colon K_\ast(X) \to H_\ast(X)$ for $X$ a smooth, compact manifold. I found only one definition of it (in the paper "K-Homology and Index Theory" by Ba …

Let $M$ be a closed, oriented manifold of dimension $n$. We know that the Chern character induces an isomorphism $K^\ast(M) \otimes \mathbb{Q} \cong H^\ast(M; \mathbb{Q})$ and now I was wondering how …