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

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

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5
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99 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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60 views

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 ...
5
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1answer
156 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 ...
15
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1answer
331 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 ...
11
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2answers
639 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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105 views

$K$-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 ...
8
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1answer
169 views

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 ...
4
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1answer
143 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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126 views

K theory as derived global sectioins

I was watching the following lecture by P.Scholze, on $K$-theory of $p$-adic rings.. I am confused at around 13:40-14:00 of the lecture where he explains: A construction of $K$-theory of schemes $K$-...
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2answers
599 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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154 views

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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115 views

$1$-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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134 views

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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56 views

$k$-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-...
3
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48 views

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 ...
8
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1answer
200 views

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 ...
2
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1answer
274 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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1answer
463 views

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 ...
2
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1answer
302 views

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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157 views

“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 $U$ ...
4
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132 views

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
161 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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125 views

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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168 views

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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106 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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215 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....
8
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1answer
160 views

What is the inverse in K-theory represented by Clifford module extensions?

I am working on a model for topological KO-theory which is represented by explicit spaces of orthogonal Clifford module extensions. That is, assuming $M$ compact, $KO^{-n+1}(M) := [M,X_n]$ where the ...
4
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1answer
140 views

Higher homotopy groups of Joyal fibrant replacements of 2-coskeletal simplicial sets

Suppose $X$ is a 2-coskeletal simplicial set (meaning $X^{Δ^k}→X^{∂Δ^k}$ is an isomorphism for all $k≥3$). What is the easiest example of $X$ such that the Joyal fibrant replacement $Y$ of $X$ is not ...
3
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1answer
236 views

A class in the motivic cohomology group $H^{0,1}(\operatorname{Spec}k;\mathbb{Z}/p)$

In the following paper N. Yagita, Examples for the Mod p Motivic Cohomology of Classifying Spaces, on the first page, below display (1.1), it says "It is known that there is an element $\tau\in H^...
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95 views

Representation of $C^{*} (S_{\infty})$

I was wondering what is the group $C^{*}$-algebra of infinite symmetric group? Mainly, I was trying to calculate the k-theory of $C^{*}$-algebra of infinite symmetric group and I found K-Theory of $C^{...
4
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160 views

Cohomology classes coming from algebraic K-theory

Suppose $X$ is a smooth variety over $\mathbb{C}$. I believe there is a Chern class map $\operatorname{ch}:K_*(X) \otimes \mathbb{Q} \to H_{sing}^*(X(\mathbb{C}),\mathbb{Q})$ for algebraic $K$-theory. ...
3
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0answers
155 views

Loop spaces of motivic Eilenberg-Mac Lane spaces

Consider the unstable $\mathbb{A}^1$-homotopy category (say over $\mathbb{C}$). By the loop space $\Omega X$ of an object $X$, we mean the homotopy fiber of $pt\to X$. For an abelian group A and the ...
5
votes
1answer
112 views

On the width of the Catalan monoid and the rank of K-groups of the Furstenberg transformation group

The semigroup algebra of the Catalan monoid is isomorphic to the incidence algebra of $P_n$, where $P_n$ is the poset consisting of subsets of { 1,...,n } where for two subsets $X \leq Y$ if and only ...
2
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1answer
234 views

K-Theory of $C^{*}(X)$

I'm new to K-Theory for $C^{*}$-algebra and $C^{*}$-algebra of groups. If $X$ is the group of finite support bijections of natural numbers then what is the K-Theory of $C^{*}(X)$? I was planning to ...
32
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1answer
466 views

Equivalence of topological Hochschild homology and Mac Lane homology via an equivalence $QA\simeq HA \wedge_{\mathbb{S}} H\mathbb{Z}$

Mac Lane homology is a homology theory for (not necessarily commutative) rings. Given a ring $A$, Eilenberg and Mac Lane define its cubical construction $QA$ to be a certain connective chain complex, ...
0
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1answer
80 views

How to define an equivariant Kasparov's KK-theory map?

I'm looking for some references about how to construct an equivariant Kasparov's KK-theory map $$ \psi \ : \ KK^{G_{1}} ( A,B ) \to KK^{G_{2}} ( C,D ) $$, where, $ G_1 $ and $ G_2 $ are two distinct ...
3
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0answers
82 views

Reference request: $K$-theoretic wrong-way map for a boundary inclusion

Let $W$ be a compact manifold with boundary. Let $i:\partial W\hookrightarrow W$ be the natural inclusion. We have a long exact sequence in complex $K$-theory: $$\ldots\to K^*(\partial W)\xrightarrow{...
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49 views

A map from a $ G_1 $ - equivariant KK-theory of Kasparov, to a $ G_2 $ - equivariant KK-theory of Kasparov

Let $ G $ be a locally compact group. Let $ H $ and $ K $ be two normal subgroups of $ G $. In order to construct a map, $$ \psi \ : \ \ F(G/H,G/K) \to F(G/K,G/H) $$ where, $$ F(G/H,G/K) = KK^{G/H} ( ...
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129 views

Understanding the Exercise 9.9.5 of Weibel homological algebra

The exercise 9.9.5 of Weibel's homological algebra states that $\textbf{Exercises 9.9.5}$ If $Z$ is a nilpotent ideal of $R$ and $k$ is a field of $char(k) = 0$, show that $H_{dR}^{\ast}(R) \cong H_{...
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121 views

Riemannian version of topological $K$-theory

Let $X$ be a compact Hausdorff space.Put $Vec(X)$, the space of all real (or complex) vector bundles over $X$.We put also $Vec_g(X)$, the space of all Riemannian vector bundles over $X$, that is the ...
3
votes
2answers
287 views

Does the Gysin map in $K$-theory respect bordism?

Let $X_1$ and $X_2$ be two closed spin$^c$ manifolds that are bordant via a spin$^c$ manifold-with-boundary $W$. Let $Z$ be a closed spin$^c$ manifold with $\dim Z=\dim X_1$ mod $2$. Let $$f_1:X_1\to ...
10
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2answers
599 views

Good reference for topological Hochschild homology

I want to start reading topological Hochschild homology(THH) as well as topological cyclic homology (TC). I have read the Hochschild homology and cyclic homology from the book Cyclic homology by J. ...
5
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0answers
450 views

Does there exist a GRR-like generalization of the AS Index Theorem?

The Hirzebruch Riemann-Roch Theorem (HRR) expresses an analytic/algebraic invariant, namely the Euler-Poincaré characteristic of a vector bundle $V$ over a compact complex/algebraic manifold $X$, as ...
3
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1answer
162 views

The $\operatorname{spin}^c$ index for manifolds

$\DeclareMathOperator\spin{spin}\DeclareMathOperator\ch{ch}\DeclareMathOperator\ind{ind}$In the paper Čadek, Crabb, and Vanžura - Obstruction theory on 8-manifolds, the authors discussed the "$\...
4
votes
1answer
157 views

Compactly supported chern character

It is a standard result that for a CW complex $X$, the chern character $$\text{ch}: K^*(X)\otimes_{\mathbb{Z}} \mathbb{Q}\to H^*(X,\mathbb{Q})$$ induces an isomorphism. Suppose now that $X$ is an open ...
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0answers
95 views

Does the $K^1$-group of a complete flag variety vanish?

For $U(n)$ the Lie group of $n \times n$ unitary matrices, and $T^n$ its maximal torus subgroup, the homogeneous space $$ U(n)/T^n $$ is called the complete flag variety of order $n$. For the special ...
6
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2answers
204 views

If $A$ is a cofibrant commutative dg-algebra over a commutative ring of characteristic $0$, then its underlying chain complex is cofibrant

Let $R$ be a commutative ring with characteristic $0$, namely it contains the field of rational numbers. Higher Algebra Proposition 7.1.4.10 tells that the category of commutative $R$-dg-algebras $\...
5
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0answers
85 views

Rational functions with trivial Weil symbols at every point

Let $f, g$ be a pair of nonzero rational functions in $\mathbb{C}(t).$ For $\lambda\in \mathbb{C}$ let $a$ be multiplicity of $g(t)$ at $\lambda$ and $b$ - multiplicity of $f(t)$ at $\lambda.$ Weil ...
4
votes
1answer
143 views

Equivalence of families indexes of Fredholm operators

Let $F=F(H,H)$ be the space of bounded Fredholm operators in a Hilbert space $H$ with topology inherited from the norm operator topology, and let $X$ be a compact topological space. For a continuous ...

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