# Questions tagged [topological-k-theory]

Topological K-theory is a generalized cohomology theory, for which $K_0(X)$ is the Grothendieck group of isomorphism classes of vector bundles over topological space $X$. See also (algebraic-k-theory).

213 questions
Filter by
Sorted by
Tagged with
48 views

18 views

### Geometric Resolutions of C* -Algebras

I am reading "K-theory FOR OPERATOR ALGEBRAS" Bruce Blackadar, Proposition 23.5.1. Let B be a separable $C^{*}$-algebra. Then there is a separable commutative $C^{∗}$-algebra F, whose spectrum ...
138 views

### Why do we need the local triviality condition when working with vector bundles?

I'm currently reading through Adams' paper on the image of the J homomorphism, and wanted to brush up on vector bundles and K-theory before tackling this paper. The definition of (real) vector bundle ...
20 views

### On clutching functions

I'm reading Hatcher's "Vector bundles and K-Theory" (version 2.2, November 2017). In chapter 1, section 1.2, he describes how to construct vector bundles with base space a sphere. I can follow his ...
69 views

### How to get isomorphism $K(X)\cong \mathbb{Z}\times \underset{\to n}{\lim} {\rm Vect}_n(X)$

I'm reading Atiyah's K-theory, on page 44, the Lemma 2.1.1 claims that $$K(X)\cong \mathbb{Z}\times \underset{\to n}{\lim} {\rm Vect}_n(X)$$ I'm confused about how to get this isomorphism. Please ...
86 views

### Are these strengthenings of Serre-Swan and Gelfand-Naimark true?

Let $X$ be a compact Hausdorff space. The Serre-Swan theorem allows us to identify complex vector bundles with projective finitely generated modules over the ring $C(X;\mathbb{C})$ of complex-valued ...
60 views

### Mayer-Vietoris sequence in topological K-theory

In topological K-theory, we define functors $K^{-n}$ on the category of compact Hausdorff spaces. With this theory we have the Mayer-Vietoris exact sequence: if $X = A \cup B$, we have an exact ...
61 views

### Functoriality of twisted K-theory

In order to avoid the XY problem I will first state what I want, then what I think is the solution and how that failed until now. I'm trying to use May-Sigurdsson's definition of Twisted $K$-theory, ...
21 views

### the isomorphism of $D(B(H))$,where ,$H$ is an infinite dimensional inseparable Hilbert space.

We define $D( B(H))=\cup_n P(M_n( B(H)))/\sim$, where $P(M_n(B(H))$ is the set of projections in $M_n(B(H))$,$\sim$ is the equivalence relation as follows:suppose $p$ is a projection in $P(M_n(B(H)))$...