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

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

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Subalgebras of $B(H)$ consisting of all operators leaving a given finite dimensional space invariant

Let $H$ be an infinite dimensional separable Hilbert space. Let $V$ be a finite dimensional subspace of $H$. Put $$A=\{T\in B(H)\mid T(V)\subseteq V\}.$$ So $A$ is a Banach algebra. Can we equip $A$ ...
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construction of $K_0$-group and Karoubian completion

Let $A$ be a ring. The $K_0$ group of $A$ can be defined in most old fashioned way as the Grothendieck group of the set of isomorphism classes of its finitely generated projective $R$ modules, ...
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164 views

Sheaf of chain complexs glued by chain homotopy equivalences

Let $(X,\mathcal O_X)$ be a locally ringed space with an open covering $\mathscr U$. Suppose: For any $U\in\mathscr U$, we have a chain complex $(C_U, d_U)$ such that $C_U$ is an $\mathcal O_X(U)$-...
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Complex structure on $S^4$

I have heard that there is a proof of non-existence of complex structure on the 4-sphere $S^{4}$ using only the topological K-theory (complex $KU$ and real $KO$). Moreover this argument can not be ...
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Question on the classification of Cuntz algebras via their extension groups and via their K-theory

I've recently been reading Kenneth Davidson's book on C*-algebras by example. One thing that particularly interested me was the classification of the Cuntz algebras by looking at the extensions of the ...
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About the algebraic structure of the $G$-equivariant $KK$-theory

Let $ G $ be a second countable locally compact group. Let $ A $ and $ B $ be two $G$-$C^*$-algebras. Let $ KK^G (A, B) $ be the $G$-equivariant $KK$-theory of the pair $ (A, B) $. Could you tell me ...
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1answer
357 views

L-theory of additive category

Reading some articles in the field, I found the following statement: Proposition: Let $\mathcal{B}$ be an additive category and $\mathcal{A}$ a full additive subcategory of $\mathcal{B}$. If $\mathcal{...
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1answer
231 views

Algebraic K-theory and intersection theory (Bloch's formula)

It seems to be a well known fact that algebraic K-theory can be used to understand intersection theory, at least for varieties (or stacks!) over a field. A first glimpse of this result seems to be ...
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Elliptic operators with with same index but non homotopic symbols

Let $\mathcal{D}:\Gamma(E)\to \Gamma(F)$ be an elliptic operator of order $k$. Where $E,F$ are $\mathbb{C}$-vector bundles over $X$, a compact smooth manifold. In Atiyah-Singer "the index of ...
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1answer
222 views

Group ring with infinite stable rank

In searching for a counterexample in homological stability, I came across the following question: Is there a known example of a finitely presented group $G$, so that the group ring $\mathbb{Z}[G]$ ...
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1answer
194 views

Algebraic K-theory of a category containing all perfect complexes

Let $R$ be a ring and let $\mathcal{C}$ be the category of perfect $R$-complexes. Suppose that $$S=\bigoplus_{i=1}^{\infty}R$$ Let us define $\mathcal{D}$ the smallest thick category generated by $S$. ...
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224 views

Stable Adams operations

I have come across a paper by Adams, Harris and Switzer on the Hopf algebra of cooperations of real and complex K-theory. The Adams operations are stable in the $p$-local setting, however I have not ...
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496 views

Subspaces of $ A_{n}(\mathbb {Q})$ in which all nonzero matrices are invertible

Let $A_{n}(\mathbb{Q}) $ denote the $n$ times $n$ skew symmetric matrices over the rational number field. Let $N$ be a subspace of $A_{n}(\mathbb{Q}) $. If all the non-zero matrices in $N$ are ...
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Central division algebras over $ \mathbb{Q} $

Quaternions over $ \mathbb{Q} $ are an example of a Central Division algebra over $\mathbb{Q} $ for which the basis elements $\{ i,j,ij \} $ other than $1$ are represented by skew-symmetric matrices ...
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1answer
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Problems concerning subspaces of $M_{n}(\mathbb{Q}) $

Let $M_{n}(\mathbb{Q}) $ denote the $n$ times $n$ matrices over the rational number field. $N$ be a subspace of $M_{n}(\mathbb{Q}) $.Then if all the non-zero matrices in $N$ are invertible, what is ...
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Irreducible representations of the symmetric group on homology of simplicial complex

I am following Wall's paper A note on symmetry of singularities and I have some questions regarding representation theory and the homology of some objects: Consider an action of $\Sigma_k$ on a finite ...
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421 views

What is a most natural categorification of a vector space?

Few days ago I became excited when I learned from an answer to Examples of simple vertex operator algebras (VOAs) that The irreducible modules of the rank $d$ free boson are naturally parametrized by ...
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Is there algebraic $K$-theory of a group independent of the base ring?

Given a ring R and a group $G$, I can consider the group ring $R[G]$ and then take the algebraic $K$-theory $K(R[G])$. This the $K$-theory of the category $\operatorname{Rep}_R(G)$. As a variant, one ...
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567 views

Algebraic K-theory "with proper support"

I would like to know what is the "correct" algebraic $K$-theory "with proper support". I suppose that the answer should be found in the condensed world, which is mostly inspired ...
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Adams blue book lemma 17.14: computing a $\mathbb{F}_2$ basis for a filtration of $H\mathbb{Z}_*(bu \wedge bu)$

First off let me apologize for not being able to give all the context for this question. I'm learning how to do computations in stable homotopy theory and have been particularly spending a lot of time ...
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About the representation ring of a compact group

A question stuck in my mind when I was reading the paper "The representation ring of a compact Lie group" by Segal. He says on page one that I confine myself to the case of a compact Lie ...
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1answer
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What is known about the discrete group cohomology $H^2(\mathrm{SL}_2(\mathbb C), \mathbb C^\times)$?

The cohomology ring of $\mathrm{SL}_2(\mathbb C)$ as a topological group is straightforward (it's generated by a Chern class), but what is known in the discrete case? I'm particularly interested in $H^...
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1answer
150 views

K-theory of a coconnective dga

I have seen somewhere that if a differential graded algebra $A$ is connective (homologically graded), then the Grothendieck group $K_{0}(A)=K_{0}(H_{0}(A))$. Suppose that $A$ is a differential graded ...
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What is the multiplicative structure of K-theory $\Omega$-spectrum $KU$?

The K-theory $\Omega$-prespectrum $KU$ has spaces $KU_{2i}=BU\times \mathbb{Z}$ and $KU_{2i+1}=U$, according to Bott periodicity we have a $\mathbb{Z}_2$-graded cohomology theory $\tilde{K}^*(X)$, ...
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118 views

Which t-structure extend from subcategories of compact objects uniquely?

Let $T$ be a compactly generated triangulated category, that is, $T$ is closed with respect to small coproducts and equals its own smallest triangulated subcategory closed with respect to coproducts ...
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Special case of Elliott's Theorem

Let $A$ and $B$ be unital $AF$-algebra. By Elliott's theorem we know that if there an order isomorphism $\psi: K_0(A) \rightarrow K_0(B)$ with $\psi([1_{A}]) = [1_{B}]$, then there exists an ...
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1answer
181 views

Integral homology of braid groups as a ring

Let $Br_k$ denote the braid group on $k$ strands. In Corollary A.4 of "Homology of Iterated Loop Spaces" (Page 348), Cohen-Lada-May compute $H_i(Br_k;\mathbb Z)$ as an abelian group for each ...
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410 views

Diffeomorphism groups of h-cobordant manifolds

Do we have specific examples of h-cobordant smooth manifolds $M$ and $M'$ such that $\operatorname{BDiff}(M) \not \simeq \operatorname{BDiff}(M')$? Perhaps something can be said in terms of K-theory ...
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665 views

Solving polynomial equations in spectra?

Let $M$ be the mod-$p$ Moore spectrum where $p \geq 3$ is a (power of) a prime. Then $M$ satisfies the "polynomial equation" $M \wedge M \cong M \oplus \Sigma M$. Is this a general ...
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$K$-theory with respect to two different choices of quasi-isomorphisms

This question is related to another question asked here. Let's assume we have an exact category $C$ that consists of specific vector bundles on a variety. Furthermore assume $C$ is idempotent complete ...
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Stable rank one and corners of $C^\ast$-algebras

Thanks to a result of Herman and Vaserstein in [3], Rieffel's notion of stable rank [4] coincides with the Bass stable rank [1] for every $C^\ast$-algebra $A$: we denote it by $\mathrm{sr}(A)$ and we ...
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1answer
167 views

Locally trivializing a G vector bundle?

In §1.6 of Atiyah's K-theory, he defines the notion of a $G$-(vector)-bundle, which is a sort of "equivariant vector bundle" with respect to a finite group action. More specifically, let $G$ ...
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778 views

Magic behind idempotent-complete categories a.k.a. why (sometimes) be Karoubian is sexier than be Abelian

It is well know that Karoubian categories (also called idempotent-complete categories) are living between additive and Abelian categories. While one of the most famous advantages to work with ...
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Which derived categories of coherent sheaves are equivalent (or "$t$-related") to derived categories of rings?

As far as I understand, it was Beilinson who proved that the bounded derived category of coherent sheaves $D^b(\mathbb{P}^n)$ is equivalent to the bounded derived category of a certain (non-...
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Definition of odd topological K-theory using circles

I wanted to check whether the following characterization of odd complex topological $K$-theory is correct (reposted from Math.SE). Let $X$ be a compact Hausdorff space. Then $K^{-1}(X)$ can be defined ...
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$1$-periodic mod-$2$ K-theory

Complex $K$-theory mod $2$ is $2$-periodic, $K/2_* = \mathbf{F}_2[u,u^{-1}]$. Is there an extension $K/2 \to K'$ of ring spectra such that $K'_*=\mathbb{F}_2[q,q^{-1}]$ with $|q|=1$ and such that the ...
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How much vanishing of odd K-groups implies the vanishing of odd equivariant K-groups?

The main quetion is For a compact Lie group $G$, and a $G$-space $X$ with $K^1(X)=0$. How much can we say about the vanishing of $K_G^1(X)$? Moreover, how much $K^0_G(X)=K^0(X)\times R(G)$? Here $...
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$k$-invariants of $KO$ and $ko$ and differentials in the AHSS spectral sequence

Let $KO$ and $ko$ denote real $K$-theory and connective real $K$-theory. It appears to be a well done result that the $k$-invariants can be used to determine the early differentials in the Atiyah-...
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Relation of $KR(X)$ and $K(Y)$ for $X\to Y$ a $C_2$ principal bundle

It is an important property of usual equivariant $K$-theory that $K_G(X)\cong K(X/G)$ whenever $G$ acts freely on $X$. What can be said about $KR(X)$ when the action of $C_2$ on $X$ is free? In the ...
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K-theory on finite-dimensional (possibly not finite) CW complexes

I am trying to understand why (at least my most elementary understanding of) topological K-theory breaks down for non-compact things (which I have seen asserted in various places). In particular, as ...
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1answer
290 views

Calculating topological $K(X)$ for complex projective manifolds

In the introduction to the book Vector bundles and K-theory http://pi.math.cornell.edu/~hatcher/VBKT/VBpage.html two approaches to classification of (topological) vector bundles are discussed - the ...
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On the definition of A-theory

Waldhausen's A-theory is a version of algebraic K-theory of spaces. Concretely, for a (pointed) space $X$, he considers the 'Waldhausen category' $\mathcal R_f(X)$ of finite retractive CW-complexes ...
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K/G-theory of affine bundles

Setting: $f : C \to D$ is a morphism of Artin stacks over $X$ which is a torsor for a vector bundle $T \to X$: étale-locally in $X$, we have $C \simeq D \times_X T$. I want to conclude that $f^*: G(D)...
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"Somewhat connected" spaces or algebras

Before we state our question, we give a motivational simple example: Put $X$ for disjoint union of two circles. However $X$ is not a connected space but it has an open dense subset $U$ such that $U$ ...
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Solution without using any k-theory tools

Let $A$ be the UHF-algebra of type $2^{\infty}$. Suppose that $p$ and $q$ are two projections in $A$ and $\tau(p) = \tau(q)$, where $\tau$ is the unique normalized trace. Then there is a partail ...
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1answer
168 views

Computation of the groups $K(BU \times \mathbb{Z})$ and $H^*(BU \times \mathbb{Z})$

Let $U$ denote the limiting group of the chain $U(1) \to U(2) \to U(3) \to \cdots$ I wish to compute the group $K^{-1}\mathbb{C}/\mathbb{Z}(BU \times \mathbb{Z})$. For this, we have the long exact ...
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A possible kind of $K$ theory via comparison of sphere bundles associated to given vector bundles

Let $E$ be a vector bundle on a topological space $X$.Thanks to Allen Hatcher's book "Vector Bundles and K theory", the construction of sphere bundle $S(E)$ can be done without any inner ...
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Geometric foundation of the Grothendieck polynomials

Grothendieck polynomials were firstly defined in Alain Lascoux and Marcel-Paul Sch¨utzenberger. Structure de Hopf de l’anneau de cohomologie et de l’anneau de Grothendieck d’une vari´et´e de drapeaux....
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110 views

Tensor product and cohomology of dg categories

Let $\mathcal{C}$ and $\mathcal{D}$ be dg categories over a field $k$ of characteristic zero. Then one can form their tensor product $\mathcal{C} \otimes \mathcal{D}$: the objects of the tensor ...
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241 views

Proving faithful flatness of a K-theoretic map without the moduli stack of formal groups

I'm in the process of writing an expository paper on complex K-theory and Snaith's theorem; the proof of Snaith's theorem that I'm following along (located at http://math.uchicago.edu/~amathew/snaith....

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