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

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25
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708 views

Functor that maps to both $KO^n$ and $KO^{-n}$

(my question is also meaningful for complex K-theory, but since Kn(X) is always isomorphic to K-n(X), it's less interesting) I start by recalling the analytic definition of KO-theory: The following ...
19
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480 views

What is to tmf as KR is to KO?

The $E_\infty$-ring spectrum $KU$ of complex K-theory carries a canonical involution induced from complex conjugation of complex vector bundles. The homotopy fixed points of this $\mathbb{Z}_2$-action ...
19
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878 views

derived category of equivariant coherent sheaves and fixed points

The K-group $K^T(X)$ of $T$(torus)-equivariant coherent sheaves on a variety $X$ is isomorphic to $K^T(X^T)$, that of the fixed point locus via the inclusion homomorphism, when we tensor the quotient ...
17
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416 views

Computational complexity of topological K-theory

I am a novice with K-theory trying to understand what is and what is not possible. Given a finite simplicial complex $X$, there of course elementary ways to quickly compute the cohomology of $X$ with ...
10
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547 views

Categorical Schur's Lemma

In attempt to prove (and compute) a formula for the dimensions of the HOMFLY homology of the (p,q)-torus knot one could try to follow original proof by Jones of a formula for HOMFLY polynomial of ...
9
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351 views

Rosenberg's proof of Bass-Heller-Swan

I'm reading the proof the Bass-Heller-Swan Theorem in Rosenberg's book Algebraic K-Theory and Applications (Theorem 3.2.22), which asserts $$K_1(R[t,t^{-1}]) \cong K_0(R) \oplus K_1(R) \oplus ...
9
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382 views

Two constructions for BU×Z

Consider the following two ways of getting the zeroth space in the $K$-theory spectrum $BU \times \mathbb{Z}$: 1) Take the groupoid of finite dimensional complex inner product spaces with isometries ...
8
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139 views

Do quotients of amenable groups C*-algebras satisfy the UCT?

Let G be a discrete amenable group. General Question: Let $J$ be an ideal of $C^*(G)$, the group C*-algebra of $G.$ Does $C^*(G)/J$ satisfy the universal coefficient theorem (UCT)? I am mainly ...
8
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172 views

Is the generation of rings by their units a question in K-theory?

Susan's question When can number rings be spanned (as $\mathbb{Z}$-modules) by units? smells like an algebraic K-theory question in disguise. I'll reformulate the question first: Given an integral ...
8
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203 views

(Reduced) cyclic homology of a free product of unital algebras

Shameless upfloat of 1-year old question - the motivation is that in general the corresponding Banach version is false, so I am trying to see where the proof breaks down, and what (if anything) can be ...
8
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241 views

H-space structure on the Calkin algebra

By the Atiyah-Jänich theorem the K-group $K^0(X)$ for a compact space $X$ may be represented as $[X, U(Q)]$, where $Q = B(H)/K(H)$ is the Calkin algebra and $H$ is a separable infinite dimensional ...
7
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142 views

Replacing commutative C*-algebras by simple ones

I am looking for functorial ways of replacing a commutative $C^*$-algebra $C$ by a simple one, say $A$ , such that the $K$-theory remains unchanged, i.e. $K_*(C) \cong K_*(A)$. I am particularly ...
7
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166 views

Is there an analog of Khovanov homology for edge deletion-contraction-extraction?

Motivated by Khovanov's categorification of the Jones polynomial, several authors have worked on the categorification of graph invariants. For the chromatic polynomial some references are: "A ...
7
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261 views

The residue class functor from a Frobenius category to its stable category induces a functor on cube-shaped diagrams - is it dense?

Let $\mathcal{E}$ be a Frobenius category, i.e. an exact category with sufficiently many bijective objects. (Such as e.g. the category of complexes over an additive category.) Let ...
7
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957 views

Regarding the Gerstenhaber bracket on Hochschild cohomology and Morita equivalence

Associated to any $A_\infty$ $k$-algebra $A$ the Hochschild cochain complex $CH^*(A)$ has the structure of a dg-Lie algebra and a dg-algebra which are compatible enough that the cohomology is a ...
6
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173 views

K-theory of topological groupoids

For a topological groupoid $G$, there are two kinds of ``topological'' $K$-homology theories, $K_{\ast}^{G}(\underline{EG})$ the $K$-homology of $G$ with $G$-compact support, and $K_{\ast}(BG)$ the ...
6
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529 views

Is there any “deep” relation between the localization theorem of equivariant cohomology and the localization theorem of equivariant K-theory

First let's consider equivariant cohomology: if a compact Lie group $G$ acts on a compact manifold $M$. We have the equivariant cohomology $ H_G(M)$ defined as the cohomology of the cochain complex ...
6
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229 views

What are the relations in the unbounded model of K-homology?

I have posed this question to some experts at my university who would probably know the answer if there were a complete one, so my expectations are limited. It's possible that the question deserves ...
6
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301 views

Higher K-theory of Orlik-Solomon algebras (and possible generalizations?)

This topic of this question is a bit outside my comfort zone, and I should say that my end goal is to really understand how much "graph theory" is captured by contraction-deletion relations. It seems ...
6
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281 views

Does every exact six-term sequence arise as the K-theory of a locally compact pair?

Consider six countable Abelian groups and six group homomorphims as in the following diagram G → H → I ↑ ↓ L ← K ← J Assume that the resulting sequence is exact ...
6
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263 views

homotopy domination that splits a non-split epimorphism and still wants to be a homotopy equivalence

Can a homotopy domination by a space supporting a free action of $G$ be promoted to a homotopy equivalence with such a space? As stated, this is not a serious question (multiply by an $EG$). But with ...
6
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214 views

What morphisms / Morita equivalences induce the 2-periodicity isomorphisms of KK-theory?

In Kasparov's paper, the canonical isomorphisms $KK_* \rightarrow KK_{*+2k}$ are defined rather implicitely (by tensoring and stabilization). Are there morphisms of $C^*$-algebras which induce them ...
5
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118 views

Reference for analyticity of $K$-theory

This is a follow-up to my last question, Homotopy excision for structured ring spectra -- reference?. The immediate reason why I care about Blakers-Massey theorems for ring spectra is to prove that ...
5
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147 views

What is Known about the $K$-Theory of Fukaya Categories?

Some Background: In Kontsevich and Soibelman's theory of motivic DT-invariants, one is interested in something like the ``number'' of objects in a 3-Calabi-Yau category $\mathcal{C}$ having a fixed ...
5
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115 views

Dense ideals in C*-algebras and K-theory

Let $A$ be a nonunital C*-algebra and let $I \subset A$ be a dense, $*$-closed, 2-sided ideal. I was under the impression that there existed some "obvious" argument proving that $I$ carries all the ...
5
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83 views

Comparison of K-groups of (affine) singular schemes with K'=G-groups

It is well known that Quillen K-theory coincides with $K'=G$-theory for regular schemes, and can be distinct from it for singular ones. Are there any methods for studying this distinctions? In ...
5
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154 views

Does the suspension isomorphism $K_1(A) \to K_0(SA)$ descend from a more refined invariant?

If $A$ is a C*-algebra, denote its minimal unitization by $\tilde A$ and its suspension by $SA$, thought of as all continuous $a:[0,1] \to A$ with $a(0)=a(1)=0$. The unitized suspension ...
5
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97 views

KK-witnesses of Gysin maps between differentiable stacks

In 1982 Alain Connes gave the construction of a KK-element $f! \in KK(C(X), C(Y))$ that "witnesses" the fiber integration/Gysin/Umkehr/wrong-way map on topological $K$-theory along a K-orientable map ...
5
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219 views

Roots of unity in algebraic K-theory

For any commutative ring $R$, the tensor product of (finitely generated, projective) $R$-modules equips the algebraic K-theory $K(R) = K_0(R)$ with the structure of a commutative ring with unit. For ...
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199 views

Are Tate twists of t-positive motives positive with respect to the Voevodsky's homotopy t-structure?

Let $X$ be a Voevodsky's motif (over a perfect field) that belongs to the positive part of the homotopy $t$-structure (i.e. its cohomology as an object of $D^-(ShSmCor)$ is zero in negative degrees). ...
5
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346 views

Does the Grothendieck ring inject to the Grothendieck group?

To a Noetherian scheme $X$, we can associate a Grothendieck ring of locally free coherent sheaves, and a Grothendieck group of coherent sheaves, with a natural map from the former to the latter. Is ...
5
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340 views

Attaching maps for Grassmann manifolds

The Grassmannian $G_n(\mathbb{R}^k)$ of n-planes in $\mathbb{R}^k$ has a CW-complex structure coming from the Schubert cell decomposition. What is known about the attaching maps in this CW-complex ...
4
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79 views

The 'most general' papers on rational Borel-Moore motivic homology and K'-theory?

There are two ways to define Borel-Moore motivic homology (of schemes) with rational coefficients: one should either consider certain complexes of algebraic cycles, or the $\gamma$-filtrations of ...
4
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113 views

Topology for bounded operators quotiented by Schatten ideal

I saw this particular question on stackexchange. Since there has been zero answers and since I've been interested in this question myself I want to ask it here. Given the $C^{\ast}$-algebra of bounded ...
4
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0answers
99 views

Comparison of products in Quillen and Waldhausen K-theory

I'm relatively new to algebraic K-theory and stumbled upon the following question. I would be very glad If someone could provide a reference to an answer or a short argument. We are given an exact ...
4
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116 views

Interpretation of the product $K(X)\otimes K^{-1}(X) \to K^{-1}(X)$

We can represent every element of the group $K^{-1}(X)=\tilde{K}(SX)$ by a isomorphism of trivial vector bundles $L:\, X\times \mathbb{C}^k \to X\times \mathbb{C}^k$ because $SX$ is the union of two ...
4
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84 views

Homology of special linear group over local field

I am trying to compute the group $H_1(Sl_2(\mathbb{Z}_2),M)$, where $\mathbb{Z}_2$ are $2$-adic integers and M is a module $\mathbb{Z}_2 \oplus \mathbb{Z}_2$. I suppose that the group acts on $M$ by ...
4
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130 views

Duality between K-theory and K-homology in the non-compact, spin$^c$ case

Let $M$ be a compact spin$^c$ manifold, so that it has a fundamental class $[M] \in K_n(M)$. It is well-known that the cap product with $[M]$ induces Poincare duality isomorphisms $K^\ast(M) \cong ...
4
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219 views

A canonical way to kill a subset of cohomology in a dg-algebra: via $A_\infty$-algebras? References?

Let $A$ be a differential graded algebra, $S\subset H^*(A)$. I would like to 'kill $S$ in a canonical way'. Is it possible to do it as follows: consider the $A_\infty$-algebra structure on ...
4
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0answers
84 views

Characterization of the sequences in the equivalence classe of the zero element in higher extension groups

Hello, I am looking for a characterization of the long exact sequences in the equivalence classe of the zero element (for the Baer sum) in $Ext^n(U,V)$ for $n>1$. If $n=1$, then these are the ...
4
votes
0answers
289 views

K-theory of compact Lie groups

The complex K-theory of a compact connected Lie group $ G $ is computed by Hodgkin in the case that $ G $ has torsion-free fundamental group. The result is that $ K^*(G) $ is an exterior algebra in ...
4
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121 views

What is the grading of x(x−1)R[x]? Loopspace for Karoubi-Villamayor K-theory.

I am reading the chapter on Karoubi-Villamayor K-theory in Weibel's K-book. In particular he defines $\Omega R=(x^2-x)R[x]$ for a ring. This will eventually lead to a model for the loopspace ...
4
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638 views

$Ext$ functor, filtered complexes and spectral sequences

Let $\mathcal{A}$ an abelian category. Take $M$ an object of $\mathcal{A}$, and $K_*$ a bounded complex in $\mathcal{A}$ equipped with a bounded increasing filtration $F$. By using homological and ...
4
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271 views

Q-construction and Gabriel-Zisman Localization

It might be a stupid question. When I took a look at the definition of Q-construction. It makes for an exact category $P$, one defines a new category $QP$ whose objects are the same as $P$ but ...
3
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152 views

A Question About the Elliott-Natsume-Nest Proof of Bott Periodicity

In Wegge-Olsen’s book K-Theory and C$ ^{*} $-Algebras, there is an outline of a proof of Bott Periodicity (the proof is due to George Elliott, Toshikazu Natsume and Ryszard Nest). The first step of ...
3
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98 views

Extending cohomology classes to compactifications of Kuga varieties

I am trying to understand the proof of lemma 3 in the paper "Algebraic cycles and the Hodge structure of a Kuga fiber variety" by B. Brent Gordon, available at ...
3
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101 views

How to deduce (8.1) in Lusztig's “Equivariant K-theory and representations of Hecke Algebras”

Let G be a connected complex algebraic group G and X=G/B, where B is the Borel subgroup. $ M=G\times C^{*}$, where $C^*$ acts on X trivially. Let $K_M(X)$ be the equivariant K-theory. Let $s\in S$ be ...
3
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168 views

The proof of the splitting principle in equivariant K-theory via flag manifolds

In Atiyah's famous paper "Bott periodicity and the index of elliptic operators" section 4, he proved the splitting principle for unitary groups (Propostion 4.9 in that paper), namely: Let $j: ...
3
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82 views

Torsion of Bass, Farrell and Waldhausen nil groups

Let H be an infinite virtually cyclic group. If H is orientable (resp. non-orientable) the Farrell nil groups $N_n(\mathbb{Z}H,\alpha) $ (resp. the Waldhausen nil groups ...
3
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156 views

Extensions of $Z/p$ by $Z/p$ and uniqueness of cokernels

I seem to run into something I cannot understand. Following Weibel (Homological Algebra), Ex. 3.4.1, p.76, it is claimed that if $p$ is prime, there are $p$ nonequivalent extensions of $Z/p$ by $Z/p$. ...