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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 …

It is well-known that topological K-theory is blessed with the Bott periodicity theorem, which specifies an isomorphism between $K^2(X)$ and $K^0(X)$ (where $K^n$ is defined from $K^0$ by taking suspe …

I have posed this question to some experts at my university who would probably know the answer if there were a complete one, so my expectations are limited. It's possible that the question deserves t …

Let $M$ be a compact manifold and let $f$ be a Morse function with exactly one critical point at each critical level. Then one can recover a CW-complex with the homotopy type of $M$ from just the cri …

Let $A$ be a C* algebra, $J$ an ideal, $\pi: A \to A/J$ the quotient map. Recall that the relative K theory group $K_0(A, A/J)$ consists of equivalence classes of triples $(p,q,x)$ where $p$ and $q$ …

Here I am referring to Kasparov's KK-theory, a bivariant functor on the category of separable C* algebras. It is well known that $KK(A, \mathbb{C})$ is K-homology and $KK(\mathbb{C}, B)$ is K-theory, …

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 …

Let $G$ be a discrete group and consider the reduced group C* algebra $C_r^\ast(G)$, viewed as an algebra of bounded operators on $\ell^2(G)$ by the regular representation. The canonical trace on $C_ …

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 …