Algebraic and topological K-theory, relations with topology, commutative algebra, and operator algebras

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73 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, \...

**7**

votes

**0**answers

80 views

### Torsion in Atiyah Singer index formula

In the papers of Atiyah and Singer, they first show their index theorem and then derive some index formulas.
For the Fredohlm index living in the integers, they use the fact that on spheres the Chern ...

**12**

votes

**1**answer

468 views

### Reference for the algebro-geometric proof of Matsumoto theorem

Matsumoto proved in his PhD thesis that if $F$ is a field then $$K_2(F)=(F^*\otimes F^*)/(x\otimes (1-x)).$$
The original Matsumoto proof as it is written in Milnor's book on algebraic K-theory looks ...

**7**

votes

**0**answers

130 views

### Does Deligne's exceptional series lead to an “exceptional K-theory”?

To a certain extent, Deligne's exceptional series $A_1 \subset A_2 \subset G_2 \subset D_4 \subset F_4 \subset E_6 \subset E_7 \subset E_8$ plays a role analogous to the classical series $A_n \subset ...

**31**

votes

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291 views

### Chern character of a Representation

Let $G$ be a finite group. Under the identification of the representation ring $R_{\mathbb{C}}(G)$ with the equivariant K-theory $KU^0_G(\ast)$ of the point, followed by Atiyah-Segal completion-...

**16**

votes

**1**answer

274 views

### Milnor Conjecture on Lie groups for Morava K-theory

A conjecture by Milnor state that if $G$ is a Lie group, then the map $B(G^{disc})\to BG$ sending the classifying space of $G$ endowed with the discrete topology to the classifying space of the ...

**3**

votes

**0**answers

74 views

### Twisted spin cobordism v.s. KO theory in low dimensions

Based on the background info and this this webpage, here is a more advanced problem:
Question: If we consider a different more subtle twisted structure, like
$${\Omega_d^{(\mathrm{spin} \times G)/N}},...

**5**

votes

**0**answers

114 views

### Spin cobordism v.s. KO theory in low or in any dimensions

It seems that from this webpage, the spin cobordism is equivalent to KO theory in low dimension.
If we denote the $p$-torsion part (mean $\mathbb{Z}_{p^n}$ for some $n$) $$\Omega_d(BG)_p.$$
...

**1**

vote

**2**answers

104 views

### Why is the flat cotorsion pair actually a cotorsion pair?

I asked this question some while ago on Stack Exchange but didn't get an answer (link), so I am trying it here as well.
Fix a ringed space $(X,\mathcal{O})$ and denote by $\mathcal{F}$ the class of ...

**8**

votes

**0**answers

200 views

### Is there a citeable source for generators and relations of simplicial sets?

Simplicial sets can be specified using generators and relations,
in complete analogy with groups, rings, etc.
More precisely, a system of generators of relations for a simplicial set
consists of a ...

**3**

votes

**0**answers

74 views

### About mod 2 Index of Dirac Operators in 3D on Non-Orientable Manifold

I was reading Witten's paper "Fermion Path Integrals and Topological Phases" (https://arxiv.org/abs/1508.04715). He claimed that
Indeed, on
an orientable 3-manifold, the eigenvalues of the Dirac ...

**5**

votes

**1**answer

177 views

### Are these two constructions of $K_0(A)$ isomorphic?

The following question is extracted from this question on MSE, which got no answer so far, probably because it was a bit hidden by another question which a posteriori was totally obvious.
Let $A$ be ...

**3**

votes

**1**answer

212 views

### Is Quillen's bracket a “universal enveloping” something?

$\newcommand{\G}{\mathcal{G}}$
In K-theory, there is a construction due to Quillen as follows. Let $(\G, \oplus, 0)$ be a monoidal groupoid. Then the bracket $\langle \G, \G \rangle$, sometimes also ...

**1**

vote

**0**answers

33 views

### Functoriality in the group $G$ of the domain of the Baum-Connes map

Lück claims in his preliminary book, that the left hand side of the Baum-Connes map is functorial in the group $G$. For the right hand side $K(A \rtimes G)$ this is clear for the full crossed product, ...

**3**

votes

**1**answer

200 views

### Slice theorem for proper groupoids

Let $G$ be a locally compact Hausdorff (second countable) groupoid with Hausdorff (second countable) unit space $X$.
Assume $G$ is étale, i.e., the source and range maps of $G$ are local ...

**5**

votes

**0**answers

90 views

### Spectral Sequence for Twisted K-theory

Atiyah and Segal wrote in their Twisted K-theory that one can compute twisted K-theory using a spectral sequence similar to an Atiyah-Hirzebruch spectral sequence. They claimed that for any twisting $[...

**9**

votes

**1**answer

327 views

### Functoriality for wrong way maps

In the K-theory formulation of the index theorem one defines the topological index in terms of the so called wrong way maps. Those maps are defined for embeddings of compact manifolds $i:X \to Y$: see ...

**7**

votes

**0**answers

85 views

### Counting symmetric convex bodies with no nonzero lattice point in the interior

In order to estimate the size of the torsion in the algebraic $K$-groups of $\mathbb Q$ one needs to understand the homology of $\mathrm{GL}_n(\mathbb Z)$, or alternatively, the homology of the space ...

**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}...

**3**

votes

**1**answer

218 views

### Chern classes of generators of $K(S^{2n})$

Calculate the Chern classes $$ c_n \in H^{2n}(S^{2n})$$ for the generator of the group $$ K(S^{2n})$$ where $S^{2n} $ - sphere of dimension $ 2n $, $ K(S^{2n})$ - group from K-theory.
I found the ...

**2**

votes

**0**answers

57 views

### How to compute k-homology group of 2-sphere?

By M-V sequence we all know that $K_0(S^2)$ is isomorphic to $\mathbb{Z}\oplus\mathbb{Z}$. I wonder the explicit computation. What are the two generators of $K_0(S^2)$? And how do they pair with ...

**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 ...

**11**

votes

**1**answer

273 views

### Hilbert 90 for higher K-groups

For a field $F$, Let $K_n(F)$ be the Quillen's $n$-th K-group of $F$.Then $K_0(F)\cong \mathbb{Z}$, $K_1(F)\cong F^\times$.
For a finite Galois extension $L/K$, $K_n(L)$ are Galois modules.
Then $\...

**8**

votes

**1**answer

226 views

### Equivariant bundles invisible in K-theory and Borel cohomology

For a given topological group $G$ there are natural transformations $$K^* \leftarrow K^*_G \overset a\to H^{**}(EG \times_G -;\mathbb Q)$$ from equivariant K-theory, the first forgetting the $G$-...

**5**

votes

**0**answers

103 views

### When is K0 of a C* algebra finitely generated?

Are there workable conditions that imply that $K_0(A)$ is finitely generated, for a noncommutative unital C* algebra $A$. My actual question is in fact much more specific than this:
Is it possible ...

**9**

votes

**1**answer

234 views

### Positive cones in K-groups

Let $X$ be a topological space or a scheme, and let $K^0(X)$ be $K$-group of vector bundles of $X$. One may ask when an element $x$ of $K^0(X)$ is represented by an actual vector bundle, and not just ...

**31**

votes

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2k views

### Homology of $\mathrm{PGL}_2(F)$

Update: As mentioned below, the answer to the original question is a strong No. However, the case of $\pi_4$ remains, and actually I think that this one would follow from Suslin's conjecture on ...

**3**

votes

**1**answer

1k views

### Who was the first to capitalize Real?

For example in Atiyah's $KR$-theory there is the notion of a Real vector bundle in contrast to complex or real vector bundles. I am also familiar with the notion of a Real $C^*$-algebra and there are ...

**4**

votes

**1**answer

170 views

### Absoluteness of motivic cohomology and restriction of scalars

Maybe this is a question to naive for the MO community! For a projective smooth variety $X$ defined over a field $F$, and for simplicity let's assume $F$ is a number field. One way to define motivic ...

**2**

votes

**0**answers

94 views

### Is the Milnor boundary map, a natural transformation?

Consider the Milnor $K_n$-functors for discrete valuiation fields. For any discrete valuation field $F$ we can associate an abelian group $K_n(F)$ and the construction is given thanks a universal ...

**6**

votes

**1**answer

294 views

### Group (Co)Homology of Symmetric Group

The question concerns the group homology or group cohomology of symmetric groups.
The entries in groupprops.subwiki.org and in this MO post show the results for the symmetric group S$_4$.
groupprops....

**6**

votes

**1**answer

100 views

### Periodicity isomorphism in KR theory

I am currently working my way through Atiyah's paper "On K-Theory and Reality" and can't get my head around a remark stated in the paper.
In the third section $KR$-Theory is related to regular $K$-...

**10**

votes

**0**answers

139 views

### Is there a spectral sequence of Atiyah's topological KR-theory that can be used to compute basic examples?

For Segal's complex $G$-equivariant $K$-theory, it is well-known that there is an Atiyah-Hirzebruch spectral sequence. If say $G$ a finite group and $X$ a finite CW-complex, the second page of this ...

**3**

votes

**1**answer

252 views

### What do Atiyah and Segal mean by $K_G^*(X)$?

I'm reading Atiyah and Segal's article "Equivariant K-theory and Completion" and need a little help understanding the notation they use. At various points in the paper they talk about objects of the ...

**11**

votes

**1**answer

376 views

### Is K theory ever trivial because of the ring, and not because of the kinds of modules we look at?

Let $\mathcal C$ be some Waldhausen category; we know that the K-theory $K(\mathcal C)$ might be trivial if $\mathcal C$ contains objects that are too ``big'' in some sense---for instance, via the ...

**11**

votes

**0**answers

234 views

### Direct proof of the equivalence of symmetric monoidal $K$-theory and exact sequence $K$-theory?

When all exact sequences split in $C$, we have $\Omega B C \simeq K(C):=\Omega Q(C)$. Heuristically, this is because the space of upper-triangular matrices is contractible. Can this be made precise? I ...

**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 =...

**0**

votes

**0**answers

65 views

### A refined comparison of isomorphic vector bundles on a compact space

Let $X,Y$ be $2$ topological spaces such that the compact open topology on $Homeo(X)$ makes it as a topological group where $Homeo(X)$ is the group of homeomorphisms of $X$.
We define an ...

**2**

votes

**2**answers

238 views

### Finitely generated $K_0$ of $C^*$-algebras

Let $A$ and $B$ be $C^*$-algebras. Let $A_0$ be a dense *-subalgebra of $A$, and let $B_0$ a dense *-subalgebra of $B$. Assume that $A_0$ is isomorphic to $B_0$.
Finally, assume that $K_0(A)$ is ...

**3**

votes

**0**answers

175 views

### Vector fields on quasi-spheres

In 1962, Adams proved that there do not exist $\rho(n)$ linearly independent vector fields on the sphere $S^{n-1}$, where $\rho(n)$ is the Hurwitz-Radon number. I wonder if this is still true in the ...

**5**

votes

**0**answers

151 views

### spectral sequence for a complex with two filtrations

Suppose $(C,d)$ is a chain complex: an abelian group with a map $d:C \to C$ such that $d^2 = 0$ (people like to assume $C$ is graded; if that helps - feel free to do so). A filtration is an ascending ...

**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, ...

**4**

votes

**0**answers

190 views

### $\mathbf{A}^1$- contractibility

Suppose $U$ is an $\mathbf{A}^1$-contractible smooth scheme over a field $k$, that is, it is isomorphic to a point in the $\mathbf{A}^1$-homotopy category of smooth schemes over $k$.
Does motivic ...

**0**

votes

**0**answers

45 views

### Higher order algebraic and topological k group [duplicate]

Is the higher order algebraic k group has some direct analogy with the corresponding topological k group ? since the 0th,1th algebraic k group are direct algebraic version of topological k group,but ...

**7**

votes

**3**answers

327 views

### Reduction mod $n$ of symplectic group

Let $g,n$ be positive integers, is there a reference that $\mathrm{Sp}(2g,\mathbb{Z})\to\mathrm{Sp}(2g,\mathbb{Z}/n\mathbb{Z})$ is surjection?
The only reference I could find is lemma 5.16 in Deligne–...

**1**

vote

**1**answer

225 views

### $SO(6) \to SU(2) \times SU(2) \times U(1)$ branching rules

What do these branching rules mean?
\begin{eqnarray*} SO(6)_E &\to& SU(2)_\ell \times SU(2)_r \times U(1)_\Sigma
\end{eqnarray*}
I am taking these examples from a paper of Gukov (on p.51) ...

**2**

votes

**0**answers

185 views

### Open problems in the theory of manifolds relating to construction [closed]

A while ago I stumbled across a paper of Thurston: Some Simple Examples of Symplectic Manifolds, where Thurston constructs closed symplectic manifolds with no Kaehler structure. My question is: What ...

**2**

votes

**1**answer

183 views

### Does $A\oplus M_n(R)\cong B\oplus M_n(R)$ imply $A\cong B$? $R$ Dedekind domain

Let $R$ be a Dedekind domain and $A, B$ be finitely generated projective $M_n(R)$-modules. Is it true that
$A\oplus M_n(R)\cong B\oplus M_n(R)\:\:\Rightarrow\:\:A\cong B$?
Here, the isomorphism is ...

**5**

votes

**0**answers

186 views

### KK-theoretical proof of Atiyah-Singer index theorem

Does anyone know of any detailed proof of the Atiyah-Singer Index Theorem using KK-theory/ Kasparov products? References to any papers and textbooks are greatly appreciated. Thanks!

**4**

votes

**1**answer

113 views

### Finding a proof within a paper: reduced $K$-theory of Higson compactification of $[0,\infty)$ is uncountable

Emerson and Meyer's Paper "Dualizing the Coarse Assembly Map" (2006) states the following Proposition (5.1):
Let $X = [0,\infty)$ be the ray with its Euclidean metric coarse structure.
Then the ...