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Questions tagged [kt.k-theory-and-homology]

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

6
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1answer
122 views

$*$-algebras, completions, and $K$-theory

What is an example of a $*$-algebra $\cal{A}$, which admits two non-equivalent norms $\| \cdot \|_1$ and $\| \cdot \|_2$, with respect to which we can complete $\cal{A}$ to give two $C^*$-algebras $...
6
votes
0answers
100 views

Pin cobordism v.s. “KO” theory in low or in any dimensions

Fact: The spin cobordism is equivalent to "KO" theory in low dimension if we only consider the 2-torsion. This is related to a question and an answer supports the claim. Here we denote the $p$-...
7
votes
1answer
316 views

Reference request for K-Theory linearization

I posted this question on math.se here:https://math.stackexchange.com/q/2996787/482732, but I think it may be more appropriate here, sorry if I am wrong about that. In Waldhausen's paper Algebraic K ...
6
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0answers
124 views

Group $C^*$ vs group von-Neumann algebras

Let $\Gamma$ be a countable (discrete) group (in what follows, make additional assumptions as you wish). Let $C^*_r(\Gamma)$ and $W^*_r(\Gamma)$ be the reduced $C^*$-algebra respectively the reduced ...
6
votes
1answer
106 views

Example of nonvanishing Waldhausen Nil group

In a remarkable series of papers, both anticipating development in geometric topology and algebraic K-theory, specifically what we call now the Farrell-Jones conjecture, Waldhausen ...
19
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1answer
194 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, \...
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0answers
103 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
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1answer
506 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 ...
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152 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
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0answers
348 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-...
17
votes
1answer
301 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 ...
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80 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}},...
8
votes
1answer
260 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
2answers
106 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
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0answers
240 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
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0answers
81 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
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1answer
181 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
1answer
214 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 ...
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0answers
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
1answer
204 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 ...
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92 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
1answer
336 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 ...
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votes
0answers
86 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
1answer
154 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
1answer
225 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
0answers
59 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
1answer
523 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
1answer
276 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
1answer
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 $G$-...
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0answers
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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
1answer
239 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
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0answers
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
1answer
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
1answer
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 ...
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0answers
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
1answer
310 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
1answer
102 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 $K$-...
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145 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
1answer
253 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 ...
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1answer
382 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 ...
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0answers
238 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
1answer
347 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 =...
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0answers
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
2answers
245 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 ...
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votes
0answers
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
0answers
153 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
2answers
175 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, ...
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0answers
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 ...
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0answers
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
3answers
337 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–...