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13
votes
0answers
67 views

To what extent can we characterise the image of the topological Chern character?

For a finite CW complex $X$, the Chern character gives an isomorphism of finite-dimensional vector spaces: $$ ch : K^*(X)\otimes \mathbb{Q} \to H^*(X, \mathbb{Q}). $$ The vector space $V = H^*(X, \...
6
votes
1answer
219 views

How to write K-theory Conner-Floyd Chern classes in terms of Adams operations?

From Adams, we know that the algebra of (unstable, degree-zero) cohomology operations $K^0(BU)$ can be written as formal infinite linear combinations of canonical generators $$\mu_n := \sum_{i=0}^{n}...
8
votes
0answers
342 views

Higher traces in Hochschild cohomology

Let $A$ be an associative algebra over a field $k$. Let $\rho:A \rightarrow \mathrm{End}(M)$ a left module, finite dimensional over $k$. Then the map $a \mapsto \mathrm{tr}_M \rho(a)$ is a well ...
5
votes
1answer
152 views

Generator of $K_0(C_0(\mathbb{C}))$

$\newcommand{\C}{\mathbb{C}}\newcommand{\Z}{\mathbb{Z}}$ I know from Bott-periodicity that $K_0(C_0(\mathbb{C}))\simeq \Z$, is there any easy way to compute an explicit generator of $K_0(C_0(\mathbb{C}...
7
votes
1answer
480 views

Entering to the K-Theory Realm

I am looking for a guidance in $K$-theory. My master thesis was in the field of Algebraic K-theory and its relation and interaction with the field of Algebraic Topology. I mainly had concentrated on ...
15
votes
1answer
345 views

Ring structure on K-theory of a quotient of the Fermat quintic

Let $Y$ be the Fermat quintic, i.e. $Y \subset \mathbb{C}P^4$ is defined by $$ \sum_{i=1}^5 z_i^5 = 0 $$ In Section 5.3 of this paper by Volker Braun the author computes the K-groups of a quotient $X =...
10
votes
2answers
173 views

Which $K$-groups $K(C^*_r(G))$ are computed?

We have the Pimsner-Voiculescu exact sequences and the Baum-Connes map for possible computation of the $K$-theory of the reduced group $C^*$-algebra $C^*_r(G)$ for a topological, locally compact, ...
5
votes
0answers
145 views

What can be said about the topological K-theory of non-singular varieties of small codimension in projective space?

Working over $\mathbb{C}$, the Barth-Larsen results tell us a lot about the ordinary cohomology of non-singular varieties of small codimension in projective space. For example if $X \subseteq \mathbb{...
2
votes
0answers
64 views

Computing the $K$-theory of the free inverse semigroup $C^*$-algebra

A typical example where I know the $K$-theory of the quotient, and wonder if this could help to compute the $K$-theory of the extension. (Or other way round: knowing one part out of three ones.) I ...
4
votes
1answer
173 views

What are some applications of virtual vector bundles?

K-theory gives a nice way to define vector bundles that don't actually exist. For example, given a singular variety $Y$ embedded into a smooth variety $X$ we can define the virtual normal bundle as $$ ...
3
votes
1answer
109 views

K-group properties of quasi-diagonal $C^*$-algebras

Let $A$ be a separable unital quasidiagonal $C^*$-algebra. What can be said about the $K$-theory of $A$, for example some properties? Especially, are there some criterions to decide whether or not $K_*...
4
votes
0answers
174 views

Basic Question: K-theory of a sphere bundle

Does anyone know how to compute the topological K-theory of the unit tangent bundle for a compact connected Riemann surface of genus greater than one? Thank you very much in advance.
3
votes
1answer
127 views

Invertibility of element in $K(X)$

If $\xi$ is a virtual bundle of virtual dimension $1$ in the ring $K(X)$, where $X$ is a compact Hausdorff topological space and $K$ stays for complex topological $K$-theory, then is $\xi$ invertible ...
10
votes
0answers
286 views

Reference for equivariant Atiyah-Jänich theorem

The equivariant Atiyah-Jänich theorem is an isomorphism $$ [X,F]_G \cong K_G^0(X), $$ where $G$ is a compact Lie group, $X$ is a compact $G$-manifold, $F$ is the space of Fredholm operators on a ...
45
votes
0answers
10k views

Atiyah's paper on complex structures on $S^6$

M. Atiyah has posted a preprint on arXiv on the non-existence of complex structure on the sphere $S^6$. https://arxiv.org/abs/1610.09366 It relies on the topological $K$-theory $KR$ and in ...