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

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3
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185 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: ...
4
votes
1answer
190 views

What is the role of $\sum (-1)^p[\wedge^pT^*M]$ in the K-theory $K(M)$

I apologize for the vague title. Let $M$ be a compact smooth manifold, then we have $T^*M$ and hence $\wedge^pT^*M$ as vector bundles on $M$. There for we have $$ \sum (-1)^p[\wedge^pT^*M] \in K(M). ...
4
votes
1answer
408 views

K-theory, monoidal vs. exact

My question is somewhat related to this one. However I think it adds something new to the table so I decided to post it sperately. There is a construction of K-theory for symmetric monoidal ...
21
votes
1answer
588 views

Comparison map between de Rham cohomology of analytic and formal neighborhoods of singularities

Suppose that $X$ is a complex algebraic (or complex analytic) variety, and $x \in X$ is a singular point. I am interested in two types of local differential forms at $x$: analytic and formal. First, ...
2
votes
1answer
160 views

Does a dense submodule of a free module always contain a basis?

Let $R$ be a completed normed ring, eg Banach algebra. Suppose that $F$ is a free $R$-module of infinite rank with a norm defined by the square root of sum of all norms of its components. If $F'$ is a ...
6
votes
1answer
243 views

Projective modules over non-rational group rings

Let $G$ be a finite group. We know that the $K$-group $K_0(QG)$ of the rational group ring $QG$ is a free abelian group generated by the irreducible representations of $G$ over $Q$. Now let $R$ be a ...
5
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224 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 ...
3
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94 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 ...
9
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395 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 ...
2
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257 views

K-Thy mapping cone in pullback diagram

Let $X, Y$ be connected, compact Hausdorff spaces and $\mathcal{H}_1, \mathcal{H}_2$ Hilbert spaces with $A \subset \mathcal{L}(\mathcal{H}_1), B \subset \mathcal{L}(\mathcal{H}_2)$ ...
4
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122 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 ...
0
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3answers
367 views

The definition of the $K$-theory groups $K_{0}$ and $K_{1}$.

I have read few textbooks and papers about the $K$-theory groups $K_{0}$ and $K_{1}$ of (reduced) $C^{*}$-algebra and most of them didn't give a clear simple way to define these groups. Just ...
8
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215 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 ...
10
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1answer
564 views

K-Theory space of finite abelian groups

Consider the abelian category $\mathsf{finAb}$ of finite abelian groups. It is easy to prove that there is an isomorphism $\mathrm{ord} : K_0(\mathsf{finAb}) \to \mathbb{Q}^+$. Can you give a ...
4
votes
4answers
547 views

Characteristic classes detecting nontrivial fiberwise homotopy of sphere bundles

I am looking for characteristic classes of vector bundles (either real or complex) with values in generalized multiplicative cohomology theories such that: i) they vanish if the bundle of unit ...
2
votes
0answers
222 views

k-theory of $\mathbb{Z}$

I have a doubt. Borel computed the rank of the higher algebraic k-theory of $\mathbb{Z}$: $rank(K_n)(\mathbb{Z})= 1$ if $n\equiv1 mod4$, otherwise this rank is equal to 0. On the other hand Bjorn ...
5
votes
2answers
296 views

Induction theorems for finite-dimensional complex representations of infinite groups

Let $G$ be a group, usually infinite. I am interested in finite-dimensional complex unitary representations of $G$, i.e. group homomorphisms $G \rightarrow U_n(\mathbb{C})$. The category of these ...
13
votes
1answer
398 views

Beilinson's formula for the product of two modular curves

In his cellebrated 1984 paper "Higher regulators and values of L-functions", Beilinson proved (among many other exciting things) that the value at the non-critical point $s=2$ of the Rankin L-function ...
6
votes
2answers
302 views

Trivial cobordism group in dimensions 1, 3, 7 related to H-space structures on the spheres in these dimensions?

Is there a connection between the existence of H-space structures on $S^1$, $S^3$ and $S^7$ and the fact that every (closed) 1-manifold, 3-manifold and oriented 7-manifold is a boundary, or is this a ...
3
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0answers
171 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 $Z/p$. ...
6
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1answer
214 views

Significance of the vanishing of $K_{-1}(A)$

In M. Schlichting's paper, he defines the negative $K$-theory for derived categories. In this he states that for $\mathcal{A}$ an idempotent complete (see below) triangulated category, ...
3
votes
1answer
327 views

Does bundle with torsion Chern classes admit flat connection?

I want to know something about torsion in topological k-theory. So, consider complex bundle with chern classes lying in torsion part of integer homologies and my question is : does it admit a flat ...
49
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1answer
3k views

Derived Functors Versus Spectral Sequences

Let $A{\buildrel F\over\rightarrow}B{\buildrel G\over\rightarrow}C$ be additive functors between abelian categories. Hartshorne, in Proposition 5.4 of Residues and Duality, constructs the obvious ...
5
votes
2answers
343 views

whitehead group of product of groups

I am wondering is there a formula for the whitehead group of product of groups. In other words, if we know the whitehead group of two groups, are we able to calculate the whitehead group of their ...
13
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2answers
1k views

Can anyone explain to me what is an assembly map?

Or can you give me a good place to read about things related to assembly map, besides wikipedia? I am specially interested in the case of algebraic K-theory. Would appreciated if you could provide ...
12
votes
4answers
769 views

Injective dimension of graded-injective modules

In "Existence theorems..." Van den Bergh proposes the following "pleasant excercise in homological algebra": Let $A$ be a connected graded noetherian $k$-algebra (that is, $\mathbb N$-graded with ...
7
votes
1answer
263 views

Triviality of direct multiples of flat complex vector bundles

Atiyah Patodi and Singer [Spectral asymmetry and Riemannian geometry III] write that if $E$ is a complex flat bundle (non holomorphic, just smooth and complex) on a compact manifold $X$ (more ...
10
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1answer
769 views

Friedlander Conjecture

I just read on the ALGTOP discussion list that Morel has announced a proof of the Friedlander conjecture. Question: Are there other applications besides the Milnor conjecture $H_*(G,F_p)=H_*(BG,F_p)$ ...
17
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3answers
1k views

Brauer Groups and K-Theory

Is there some a priori reason why we should expect the Brauer group of real [complex] super vector spaces to be closely related to periodicity in real [complex] K-theory? By "a priori" I mean a proof ...
11
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1answer
676 views

Motivic cohomology vs. K-theory for singular varieties

As far as I understand, for a smooth variety $X$ its motivic cohomology could be described as the corresponding piece of the $\gamma$-filtration of (Quillen's) $K^*(X)$; this is completely true for ...
4
votes
2answers
730 views

On two spectral sequences for the cohomology of a double complex

For a (bounded) double complex (of abelian groups or vector spaces) one can consider two spectral sequences that converge to the cohomology of the totalization: one can first compute either the ...
6
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0answers
251 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 ...
10
votes
1answer
501 views

Homology of infinite loop spaces $QX$

Let $X$ be a simply connected space. By $Q$ I denote $\Omega^{\infty}\Sigma^{\infty}$. Then $QX$ is an infinite loop space and the homology $H(QX)$ in $\mathbb{F}_p$ is a Hopf algebra over the ...
0
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0answers
191 views

Topological K-theory of Bohr compactification of real numbers

I am interested in the K-theory of the Bohr compactification $\mathbb{R}_B$ of the real numbers. Do we have $K_0(C(\mathbb{R}_B))$ isomorphic to $K_1(C(\mathbb{R}_B))$ ? More generally, what do we ...
10
votes
2answers
1k views

Atiyah-Patodi-Singer Eta invariant and Chern-Simons form

I am trying to understand the Atiyah-Patodi-Singer index theorem in the case of Dirac operators in four dimensions. I have three questions about the eta invariant: 1) Is eta a topological invariant ...
0
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0answers
271 views

A modified version of K-theory for manifolds ?

If $X$ is a compact smooth manifold, $K^{0}(X)$ can be defined as the algebraic $K_{0}$-group of $C^{\infty}(X)$. In order to do that we use the following equivalence relation: we say that two ...
16
votes
1answer
2k views

Voevodsky's counterexample to the existence of a motivic t-structure

I have been trying to unravel some of the known relationships between various ideas on mixed motives. I find the litterature quite hard to follow -"from experts, for experts". Voevodsky in ...
9
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0answers
396 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 ...
3
votes
0answers
337 views

K-theory of differential graded modules over differential graded algebras

Suppose you have a smooth vector bundle $E$ over a smooth manifold $X$. If you consider the algebra $ \Omega^\ast (E)$ of differential forms on $E$, it will be homotopy equivalent to the algebra of ...
10
votes
3answers
1k views

State of the art for Gersten's conjecture for K-theory?

Does anyone know (of a reference to) under what restrictions on the regular scheme $X$ it is known that we have an exact sequence $$0 \to \mathcal{K}_n(X) \to \bigoplus_{x \in X^{(0)}} K_n(k(x)) \to ...
16
votes
2answers
1k views

Algebraic K-theory of the group ring of the fundamental group

I know of two places where $K_{*}(\mathbb{Z}\pi_{1}(X))$ (the algebraic $K$-theory of the group ring of the fundamental group) makes an appearance in algebraic topology. The first is the Wall ...
2
votes
1answer
350 views

Spectral sequence for H-space bundles

Let $F \rightarrow E \rightarrow B$ be a fibre bundle such that $B$ is a smooth and compact manifold and $F$ obtains an associative H-space structure. Explicitly, it is not a principal bundle. One ...
4
votes
1answer
224 views

Index vs. equivariant index (and then taking invariant part)?

Let $C$ be a smooth projective curve and let $C^{(n)}$ be its $n$th symmetric power. Let $E$ be a $S_n$-equivariant vector bundle over the Cartesian power $C^n$. Suppose that $E$ descends to a vector ...
6
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2answers
418 views

Somewhat general question that includes: “Do quasi-isomorphic cdgas have quasi-isomorphic spaces of derivations?”

Question: Given two quasi-isomorphic dg commutative algebras (over a field of characteristic zero, if you like), to what extent do their various homological geometric data agree? Example: Given a dg ...
11
votes
2answers
767 views

Twists of K-theory and tmf

I read in a paper by Christopher Douglas that third cohomology twists of $K$-theory may be interpreted as TMF-classes via a map $K(\mathbb{Z},3) \to TMF$, which is related to String orientations. How ...
3
votes
2answers
280 views

Extensions which define the same element of $\text{Ext}^n(M,N)$ are in fact equivalent

It is well known (and wouldn't be so-named unless it were) that: If $\xi$, $\eta$ are $n$-fold extensions of $N$ by $M$ (modules over a ring $R$) which yield the same element of $\text{Ext}^n(M,N)$, ...
0
votes
1answer
263 views

Can injective resolutions be 'enlarged' (or shrunk) to admit only injective maps from extensions?

Let $M$ and $N$ be $R$-modules for some ring $R$. There is a standard result involving the computation of $\text{Ext}^n(M,N)$, using projective resolutions, which says that you can always choose a ...
6
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0answers
313 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 ...
2
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1answer
311 views

A question about a proof in Rosenberg's algebraic K-theory book

In corollary 4.3.13 of Rosenberg's book Algebraic K-theory and its applications, it's proved that $K_2(F)=0$ if $F$ is a finite field. The last sentence in the proof says that the central extension ...
2
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218 views

The restriction of the Gersten resolution (for etale cohomology) onto a closed subvariety.

There is a very important result of Bloch and Ogus: for any smooth variety $X$ and fixed $r\in \mathbb{Z}$, $r\ge 0$, $l$ is prime to the residue field characteristic, the Zariski sheafification of ...