**13**

votes

**2**answers

1k 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 ...

**12**

votes

**1**answer

849 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 ...

**4**

votes

**1**answer

444 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 ...

**4**

votes

**2**answers

198 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 ...

**9**

votes

**2**answers

378 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**

votes

**1**answer

282 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 ...

**6**

votes

**3**answers

727 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 ...

**7**

votes

**0**answers

193 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 ...

**4**

votes

**0**answers

334 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 ...

**3**

votes

**0**answers

187 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

**1**answer

191 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

**1**answer

419 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

**1**answer

593 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

**1**answer

161 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

**1**answer

244 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**

votes

**0**answers

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**

votes

**0**answers

97 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**

votes

**0**answers

405 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**

votes

**0**answers

260 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**

votes

**0**answers

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**

votes

**3**answers

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**

votes

**0**answers

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**

votes

**1**answer

567 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

**4**answers

552 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

**0**answers

229 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

**2**answers

297 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

**1**answer

399 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 ...

**7**

votes

**2**answers

303 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**

votes

**0**answers

173 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**

votes

**1**answer

216 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

**1**answer

334 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**

votes

**1**answer

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 ...

**6**

votes

**2**answers

350 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**

votes

**2**answers

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

**4**answers

779 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

**1**answer

268 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**

votes

**1**answer

773 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**

votes

**3**answers

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**

votes

**1**answer

685 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

**2**answers

736 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**

votes

**0**answers

254 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

**1**answer

506 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**

votes

**0**answers

192 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

**2**answers

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**

votes

**0**answers

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

**1**answer

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**

votes

**0**answers

397 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

**0**answers

340 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 ...

**11**

votes

**3**answers

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

**2**answers

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 ...