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**13**

votes

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**2**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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 ...