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

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$K$-groups and dual graphs of special fibers

Let $p$ be a prime number, let $E$ be an elliptic curve defined over $\mathbb{Q}_p$. Let $\mathcal{E}_p$ be the special fiber of the Néron model of $E$ over $\mathbb{Z}_p$ and let ...
4
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83 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 ...
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2answers
1k views

Who named it the Snake Lemma?

What is the history behind the colorful name of this result? Cartan-Eilenberg states it without any particular fanfare.
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493 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 ...
4
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1answer
253 views

K theory of a simplicial monoidal category, Cofinality theorem

Let $X=(d\mapsto X_d)$ be a simplicial symmetric monoidal category. We define the $K$-theory space of $X$ to be $K(X)=|d\mapsto K(X_d)|$, the geometric realisation of the simplicial space $d\mapsto ...
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2answers
338 views

When do the $\gamma$-filtration and codimension filtration of K-theory agree?

Let $X$ be a smooth quasiprojective algebraic variety over a field $k$. Then the $K$-groups $K_m(X)$ are defined, and there are two standard filtrations on them: the "codimension filtration" given by ...
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2answers
764 views

Who first noticed that the Hilbert symbol is a Steinberg symbol ?

Hilbert reformulated the quadratic reciprocity law of Gauß as a product formula $$ \prod_v(a,b)_v=1 $$ for the various local Hilbert symbols. For each place $v$ of $\bf Q$, the Hilbert symbol $(\ ,\ ...
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2answers
788 views

Finiteness of stable homotopy groups of spheres

Since the work of Serre in the early 50's on homotopy groups of spheres, it is known that the homotopy group $\pi_k(S^n)$ is finite, except when $k=n$ (in which case the group is $\mathbb{Z}$), or ...
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1answer
674 views

finite dimensional real division algebras

A celebrated theorem of Milnor and Kervaire asserts that any finite dimensional division algebra over the real numbers has dimension 1,2,4 or 8. This result is established using methods from algebraic ...
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1answer
382 views

second fundamental form

Hi all, Currently I'm reading a paper about the geometry of Grassmannians: www.omup.jp/modules/papers/riemann/04Nagatomo.pdf In there, the author regards the second fundamental form of the ...
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2answers
183 views

Steinberg Group as a Lattice in a lie group

Given an integral domain $R$, the Steinberg group $St_n(R)$ is the group given by generators $e_{pq}(\lambda) := \mathbf{1} + a_{pq}(\lambda)$, $p\neq q$, $1\leq p,q \leq n$ Subject to ...
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347 views

f.g. modules vs. f.g. projective modules

In algebraic K-theory one defines $K_0(R)$ as the result of application of the Grothendieck construction to the semigroup of isomorphism classes of left f.g. projective $R$-modules. But we can also ...
4
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1answer
251 views

Algebraic K-theory with commutative semirings?

My question is basically given in the title: Are there any references for a generalization of algebraic K-theory to the scenario where the domain of the functors consists of commutative semirings ...
5
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3answers
615 views

Is there a categorification of topological K-theory?

For a compact Hausdorff topological space $X$, its K-theory $K^0(X)$ is defined to be the Grothendieck group of the isomorphism classes of finite dimensional vector spaces on $X$. For example ...
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162 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 ...
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265 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 ...
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160 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: ...
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1answer
181 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). ...
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1answer
305 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
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1answer
536 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
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1answer
151 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 ...
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1answer
235 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 ...
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217 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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81 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 ...
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340 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 ...
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208 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 $A \subset \mathcal{L}(\mathcal{H}_1), B \subset \mathcal{L}(\mathcal{H}_2)$ ...
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117 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 ...
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3answers
354 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 ...
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183 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 ...
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1answer
480 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 ...
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4answers
509 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 ...
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200 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
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2answers
277 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
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1answer
354 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 ...
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284 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 ...
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154 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
200 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, ...
2
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1answer
267 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 ...
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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 ...
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297 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
899 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
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3answers
643 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 ...
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1answer
208 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 ...
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1answer
684 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 $H_*(G,F_p)=H_*(BG,F_p)$ ...
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2answers
684 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
584 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
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2answers
656 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 ...
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221 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 ...
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1answer
402 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 ...
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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 ...