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

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

722
questions

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### Spectral flow of Dirac operator twisted by instanton

Suppose $E$ is a $SU(2)$-bundle over a closed three manifold $M$ and $S$ is the spinor bundle over $M$. Also assume $D_{A(t)}:\Gamma(S\otimes_{\mathbb c} E)\to \Gamma(S\otimes_{\mathbb c} E)$ is a ...

**8**

votes

**0**answers

107 views

### Find a 4-manifold bounding a 3-torus with any abelian representation in SL_2

Fix $x,y,z\in \mathbb{C}^*$ and let $M=S^1\times S^1\times S^1$ with $\rho:\pi_1(M)\to \operatorname{SL}_2(\mathbb{C})$ mapping the three generators to diagonal matrices with entries $(x,x^{-1})$, $(y,...

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383 views

### Roadmap for Quillen 1

Question
Suppose you grasped and enjoyed reading Quillen's "Higher Algebraic K-theory I". Now, if you could go back in time to when you started studying algebraic topology and create a reading list / ...

**2**

votes

**1**answer

105 views

### For an additive category $\mathcal{A}$, how does one show $K_0(\mathcal{A})\cong K_0(\mathcal{K}^b(\mathcal{A}))$?

This is an exercise in §3.13 Beilinson's notes on homological algebra. He doesn't specify but I'm pretty sure $K_0(\mathcal{A})$ is defined as the free group on the isomorphism classes of $\mathcal{A}$...

**4**

votes

**1**answer

163 views

### Question on Cuntz' proof of Bott periodicity

I am reading the presentation of Cuntz' proof of Bott periodicity for $C^*$-algebras in Wegge-Olsen (Thm. 11.2.1). Here one considers the short exact sequence of $C^*$-algebras
$$0 \longrightarrow \...

**5**

votes

**0**answers

134 views

### Convergence of a spectral sequence of a double complex

In Weibel's book, a spectral sequence $E^r_{p,q}$ is said to weakly converge to a graded object $H_{\ast}$ if for every $n$ there exists a filtration $\dots \subset F_{r}H_{n} \subset F_{r-1}H_{\ast} \...

**3**

votes

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64 views

### Dixmier trace, Wodzicki residue and topological index

There are a well-known facts about Dixmier trace and Wodzicki residue. Let $P$ be an elliptic pseudodifferential operator of degree $−n$ on a compact Riemannian manifold $(M,g)$, than its Dixmier ...

**4**

votes

**1**answer

130 views

### What are the generator for $K_1(C(\mathbb{T})\otimes\mathbb{K})$?

I've been working computing several K-groups associated to some $C^*$-algebras involved with my master's thesis, however I've just got stucked finding some generators for $K_1(C(\mathbb{T})\otimes\...

**5**

votes

**1**answer

197 views

### Examples of noetherian local rings $R$ such that $K'_0(R)$ is not isomorphic to $\mathbb Z$

Does there exist a simple example of a commutative noetherian local ring $R$ such that $K'_0(R) = K_0(\mbox{Mod-}R)$ (by $\mbox{Mod-}R$ I mean the abelian category of finitely generated $R$-modules) ...

**11**

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195 views

### Criteria for a map of rings to induce an equivalence on K-theory?

Algebraic $K$-theory is Morita invariant, but surely it does not detect Morita equivalence. What are some examples of rings (or ring spectra) $R$ and $S$ that are not Morita equivalent, but ...

**1**

vote

**0**answers

55 views

### Functoriality of Hochschild cohomology for Drinfeld quotients

Let $C$ be a dg category and $C \to D$ a Drinfeld localization. Is there an induced pushforward map on $\operatorname{HH}^*(C) \to \operatorname{HH}^*(D)$, where $\operatorname{HH}^*$ denotes the ...

**11**

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336 views

### Looking for an invariant similar to algebraic K-theory

I'm wondering if there is an invariant, similar to algebraic K-theory, topological hochshild homologic, topological cyclic homology etc... that has the following properties:
a) It attach to each ...

**12**

votes

**0**answers

210 views

### Dennis trace map for stable $\infty$-category, naively

I'm trying to get more intiution about higher K-theory, Hochschild homology and the trace map between by thinking about these objects from an informal $\infty$-categorical perspective, instead of ...

**4**

votes

**1**answer

126 views

### Split cofibrations up to quasi-isomorphism

$R$ a ring $(1\neq 0)$, $\mathbf{Perf}(R)$ is the category of perfect complexes (of right $R$-modules).
Suppose that $A_{\bullet}\rightarrow B_{\bullet}\rightarrow B_{\bullet}/A_{\bullet}$ a short ...

**5**

votes

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128 views

### Descent properties for rational topological cyclic homology

Descent properties can be extremely useful for studying $\operatorname{TC}$ (topological cyclic homology), since it is a sheaf in many well behaved topologies.
I was wondering what is known about $\...

**2**

votes

**0**answers

85 views

### Left and right topological K-theory of Banach algebras

Let us consider the topological $K$-functor on the category of Banach algebras as described in page $18$ of "Introduction to the Baum–Connes conjecture" by Alain Valette.
The definition is based on ...

**5**

votes

**1**answer

175 views

### Exact subcategory with trivial Grothendieck group: what are the consequences and examples

Let $C$ be (a full) exact subcategory of the category of $R$-modules. We suppose that $C$ is essentially small. If the Grothendieck group $K_{0}(C)=0$, what can be said about the higher groups $K_{n}(...

**2**

votes

**3**answers

343 views

### Motivation for Karoubi envelope/ idempotent completion

This is the second part of my venture to become more comfortable with the concept of idempotent elements and idempotent splittings from category theoretical viewpoint. In the first part we considered ...

**15**

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699 views

### Application of higher categories in algebra

Higher topos and derived algebraic geometry are relatively new areas and probably fewer people are working on them compared to the majority of topologists or geometers. I believe higher categories ...

**5**

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**0**answers

320 views

### Modern context for hypercohomology spectra

In Thomason's paper Algebraic K-theory and étale cohomology, (Ann. ENS 1985, Numdam link) Thomason develops an elaborate theory of hypercohomology spectra, $\mathbb{H}(X,\mathcal{F})$ for presheafs of ...

**3**

votes

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128 views

### When is a virtual bundle an actual bundle?

As far as I understand, one can make a monoid from the space of vector bundles on a (compact) manifold $M$, with respect to the direct sum operation $\oplus$. In order to make this into a group, one ...

**3**

votes

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168 views

### Relations between rational algebraic K-theory and Chow groups

A consequence of Grothendieck's Riemann-Roch Theorem is the fact that the Chern character induces an isomorphism between algebraic
$ch: K_{0}(X) \otimes \mathbb{Q} \stackrel{\cong}{\rightarrow} C H^{*...

**23**

votes

**1**answer

547 views

### Vector bundles on $\mathbb{A}^n / G$

Let $G$ be a finite group acting linearly on $\mathbb{A}^n$. Do we expect algebraic vector bundles on $X := \mathbb{A}^n/G$ to be trivial? Here by the quotient I mean the singular scheme, not the ...

**4**

votes

**1**answer

187 views

### $KO_*$ groups of $\mathbb{R}P^\infty$, “Snaiths” theorem for $KO$

I posted this question some days ago at math.stackexchange, but didn't receive an answer.
I have two questions:
I wonder whether anyone has taken the time to compute $KO_*(\mathbb{R}P^\infty)$?
The ...

**3**

votes

**2**answers

200 views

### K-theory of free G-sets and the classifying space, and generalization [reference-request]

Let $G$ be a finite group, $\mathcal{G}^0$ be the category of finite free
$G$-sets and isomorphisms between them. Then $\mathcal{G}^0$ is a symmetric monoidal category with respect to the disjoint ...

**1**

vote

**1**answer

152 views

### Confusion regarding a definition of cycles

For a projective variety $X$ over $\mathbb{C}$, let us denote by $CH_k(X)$ the Chow group of $k$-cycles of $X$, modulo rational equivalence. Also, let $CH_k(X)_{hom}$ denote $k$-cycles modulo ...

**14**

votes

**0**answers

176 views

### Hauptvermutung for non-manifolds

The Hauptvermutung proposes the following: if two finite simplicial complexes are homeomorphic then they are PL-homeomorphic, meaning that they have a common refinement.
People are mostly interested ...

**7**

votes

**0**answers

168 views

### Recover the field from its Milnor K-groups

For every field $F$, consider $K_n^M(F)$ the $n$-th Milnor K-group of $F$ for each $n \in \Bbb N$, and form the Milnor K-ring $K^M(F)=\oplus_{n \geq 0}K^M_n(F)$. For instance, $K_1(F)=F^{\times}$.
...

**4**

votes

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98 views

### Alternative definitions of Weibel's homotopy K-theory

Consider a sort of $\mathbb{A}^1$-homotopy-stable algebraic $K$-theory for rings constructed as follows.
For $K_0$ we take a symmetrization subject to natural direct sum operation of $\mathbb{A}^1$-...

**1**

vote

**1**answer

114 views

### A kind of isomorphicity of vector bundles

Let $X$ be a connected topological space. Let $E$ be a $k$ dimensional sub vector bundle of the trivial vector bundle $X\times \mathbb{R}^n$. Then $E$ defines an idempotent with trace $k$ in $M_n(C(X))...

**4**

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242 views

### Other kinds of equivalence relations on the set of idempotents of a Banach or $C^*$-algebra or a ring (Can we get a new kind of K-theory?)

The standard equivalent relations on idempotents of a $C^*$ algebra or a Banach algebra are Murray von Neumann, similarity and homotopy equivalent. In this post we consider two other kinds of ...

**10**

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236 views

### Chromatic Homotopy Theory and Physics

Chromatic homotopy theory is a subfield of stable homotopy theory that studies complex-oriented cohomology theories from the "chromatic" point of view, which is based on Quillen's work relating ...

**9**

votes

**1**answer

254 views

### Computing K-theory for cellular vector bundles

One of the most computationally convenient properties of singular cohomology $X \mapsto H^\bullet(X;\mathbb{Z})$ is the fact that one can extract it from a good cover $\{U_i\}$ of $X$ via Cech ...

**7**

votes

**1**answer

327 views

### Cohomology theory with only one Adams operation?

Let $E$ be a multiplicative cohomology theory. Fix a prime p. Call a ring map $\psi^{p}:E\rightarrow E$ an Adams operation if it lifts the Frobenius map $E/p\rightarrow E/p$.
It is of course well-...

**3**

votes

**1**answer

149 views

### Bott periodicity homeomorphisms for spaces of Clifford extensions

I am trying to prove the following statement of real Bott periodicity, on the level of actual spaces of Clifford module extensions (i.e., not equivalence classes of modules).
Let $W = \mathbb{R}^{\...

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**0**answers

59 views

### Simpicial resolution in Cotangent Complex

I am reading Cotangent complex from Loday's book Cyclic Homology. My doubt is related to the simplicial resolution of $k$- algebra $A$ which is used in the definition of cotangent complex.. Let me ...

**11**

votes

**1**answer

1k views

### Why does K-theory need schemes to be Noetherian?

The definition of K-theory of a scheme $X$ is defined as
$G_i(X):=K_i(\mathrm{Coh}(X))$ or $K_i(X):=K_i(\mathrm{Vec}(X))$.
But usually the schemes are required to be (at least locally) Noetherian, and ...

**5**

votes

**0**answers

258 views

### Homotopy type of a $4$-manifold with finite fundamental group (paper by S. Bauer)

I'm studying Stefan Bauer's paper
The homotopy type of a 4-manifold with finite fundamental group. In: tom Dieck T. (eds) Algebraic Topology and Transformation Groups. Lecture Notes in Mathematics,...

**6**

votes

**1**answer

159 views

### Pinwheel Tilings and C* algebras, K-theory

I was reading that spaces of tilings can be related to C*-algebras and K-theory. Here is an example of the pinwheel tiling. [1]
They construct a space called $\mathcal{A}\mathbb{T}_{pin}$ and show ...

**4**

votes

**1**answer

219 views

### Is the motivic homotopy spectrum of Hermitian K-theory $\eta$-complete?

Theorem 1.2 of the paper "The motivic Hopf map solves the homotopy limit problem for K-theory" (see https://elibm.org/article/10011880) says that (under certain assumptions on the base field) the ...

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**0**answers

87 views

### Is the deformation along flow lines a simple homotopy equivalence?

Let $(M,g)$ be a compact, smooth $n$-manifold with boundary $\partial M$ and let $f: M \to [a,b]$ be a Morse function, whose critical points are interior and which satisfies $f^{-1}(b) = \partial M$.
...

**5**

votes

**1**answer

197 views

### When and why are Adams operations “non-negative”?

We can think of the unary operations in a lambda-ring as integer linear combinations of Young diagrams; for example the operation $\lambda^n$ corresponds to the Young diagram with $n$ rows and one ...

**11**

votes

**2**answers

317 views

### $d^3$ in the Atiyah-Hirzebruch spectral sequence for (twisted) $KO$

Cross posted from here after no responses and a bounty being placed on the question.
Let $h^n(-)$ be a generalised cohomology theory. For a space $X$ there is a spectral sequence known as the Atiyah-...

**3**

votes

**1**answer

340 views

### Chern character of coherent sheaf on singular variety

Does there exist a definition of Chern character (or Chern classes) for a coherent sheaf $\mathscr{F}$ on a singular variety $X$? In this case I might not be able to find a projective resolution for $\...

**5**

votes

**0**answers

77 views

### elementary matrices over a regular ring

Let $n\in\mathbb{N}$ with $n\geq 3$. Let $A$ be a regular ring and $\gamma\in GL_{n}(A)$. We suppose that $\gamma$ is homotopic to the identity, i.e. there exists $\alpha\in GL_{n}(A[X])$ such that $\...

**6**

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208 views

### Equivariant Venice Lemma

In the paper J. Simons and D. Sullivan. Structured vector bundles define differential K-theory, one of the key ideas is the so called Venice Lemma, which essentially can be stated as
Theorem: For ...

**4**

votes

**0**answers

190 views

### K-theoretic derivation of Bézout theorem

In the paper "$K$-Theory and Intersection Theory" by Henri Gillet in Handbook of K-theory, 2005 (link behind paywall at Springerlink), the author says:
"When the ground field $k = \mathbb C$, Bézout’...

**1**

vote

**1**answer

145 views

### $C^*$-algebras appearance in study of Lie groupoids and differentiable stacks

I am reading Differentiable stacks, gerbes, and twisted K-Theory by Ping Xu.
To talk about (twisted) K-theory of differentiable stacks, author introduced (page $41$) the set up of $C^*$-algebras. All ...

**5**

votes

**1**answer

286 views

### K-theory for a (geometric) stack

There is a notion of $K$-theory for a manifold $M$.
Is there a notion of $K$-theory for a stack $\mathcal{D}\rightarrow \text{Man}$ that is representable by a Lie groupoid $\mathcal{G}$; that is $...

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249 views

### Applications of Jordan-Holder theorem in an abelian category

The Jordan-Holder theorem says that any chain of subobjects of a finite length object can be refined to a composition series, and that any composition series has the same length.
This theorem holds ...