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I think the best answer you will find is the axiomatic characterization of the topological index in Index of Elliptic Operators I. One defines an index function to be a map $ind_X: K(TX) \to \mathb …

I'm assuming that by "torus" you mean the $1$-torus $T$, i.e. the circle; if you are interested in a higher dimensional torus, you can compute with K-theory products. Elements of $K_1(C(T))$ are repr …

For many basic examples, the usual tools of (co)homology theory work just fine (and very similarly) in K-theory and K-homology. Here's an example. How does one compute, say, the De Rham cohomology o …

This seems to follow from the fact that the Chern classes are integral, an observation which is more mysterious using some definitions than others. For instance, I know of no direct way to prove that …

In what follows, all tensor products are graded. The comments about the existence of canonical (up to homotopy) $\ast$-homomorphisms $\mathbb{C} \to K(H)$ and $K(H) \otimes K(H) \to K(H)$ right befor …

The main problem with dual algebras for non-separable C*-algebras is that they need not be functorial. Given representations $\rho_A \colon A \to \mathcal{B}(H_A)$ and $\rho_B \colon B \to \mathcal{B …

I can't give you an answer that is well-adapted to Lawson and Michelsohn's formulation of the pseudodifferential calculus. But here's how this sort of argument is supposed to go: two elliptic pseudod …

First, some comments. "A Dirac operator" is an operator naturally associated to a bundle which is a module over the Clifford algebra of the tangent bundle. "A twisted Dirac operator" (in the sense o …

I think the main topological significance of the element you identify is its close relationship with the Thom isomorphism $K(M) \cong K(T^*M)$. I would imagine that there is also some connection with …

1) The eta invariant itself depends on the metric, but the relative eta invariant is in many cases (see comments) a homotopy invariant. The relative eta invariant is defined to be the difference of t …

Index theory is fundamentally about a homomorphism $$K_n(M) \to \mathbb{Z}$$ from the top degree K-homology of $M$ (even dimensional) to the integers called the analytic index map. It is called this …

It's a bit easier to see this using a slightly non-standard definition of topological K-theory. Given a locally compact Hausdorff space $X$, let $\bf{E}$ be a complex of vector bundles, i.e. a sequenc …

The answer is basically "yes, because the definitions are rigged to make it so". The point is that you have to be careful both with C*-algebra K-theory in the non-unital case and with topological K-t …

KK-theory provides one way to eliminate the asymmetry in the definitions of K-theory and K-homology. A cycle in $KK(A, B)$ is a triple $(H, \rho, F)$ where $H$ is a (adjectives) Hilbert $B$-module, $ …

There are a number of ways to do this calculation, but at risk of shamelessly plugging my own work there is a nice way to see it using a Mayer-Vietoris principle. Decompose $\mathbb{R}$ as the union …