All Questions
Tagged with algebraic-k-theory at.algebraic-topology
24 questions with no upvoted or accepted answers
12
votes
0
answers
551
views
Goodwillie's notes from MSRI Lecture Series
Does anyone know where I can find an electronic version of Goodwillie's (unpublished) notes from the MSRI Lecture Series in Spring, 1990? They're mentioned/cited as such in work of Dundas-McCarthy, ...
11
votes
0
answers
264
views
Direct proof of the equivalence of symmetric monoidal $K$-theory and exact sequence $K$-theory?
When all exact sequences split in $C$, we have $\Omega B C \simeq K(C):=\Omega Q(C)$. Heuristically, this is because the space of upper-triangular matrices is contractible. Can this be made precise? I ...
- 63.9k
11
votes
0
answers
340
views
$K$-theory spectrum of the category of finite groups
(I asked some people this question in person and got the answer "no", but wanted to see if the Internet had more to say)$
\newcommand{\FinGrp}{\mathbf{FinGrp}}
$
Way back in my first group theory ...
- 1,838
9
votes
0
answers
745
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When does algebraic K theory behave like a cohomology theory
Let $\mathbb{F}$ be a field. Let $K(\mathbb{F})$ be its algebraic (Quillen) $K$-theory spectrum. Let $X$ be a (nice, finite CW) topological space and let $\text{Rep}\Omega(X)$ be the DG category of (...
- 7,952
7
votes
0
answers
223
views
Does the Whitehead torsion of a homotopy equivalence depend on the CW structure?
In the (old) literature I've seen referenced the question of whether simple homotopy equivalence is a topological property, i.e. whether it depends only on the underlying space, rather than the ...
- 5,839
7
votes
0
answers
191
views
Torsion in Whitehead group
Let $\pi$ be a finite group of odd order. What do we know about the torsion subgroup of $Wh(\pi)$? I am especially interested in the $2$-primary part. Is it always trivial?
- 468
6
votes
0
answers
312
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 ...
- 5,575
5
votes
0
answers
107
views
Size of minimal generating set of ideal over Laurent polynomial ring
Recently in attacking a problem in algebraic topology relating to the construction of stably-free non-free modules over integral group rings I’ve noticed that it is often fairly easy to reduce to ...
- 187
5
votes
0
answers
162
views
Grothendieck group of coconnective dg-algebra
Is there an example of an $E_{2}$-coconnective differential graded algebra $A$ (with unit) such that $K_{0}(A)=0$ ?
- 169
5
votes
0
answers
311
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,...
- 6,028
5
votes
0
answers
363
views
making the group completion in homology sense unique via the plus construction
A paper by Mcduff and Segal justifies the following definition: A map of h-spaces $X \to Y$ is a group completion if the map is a localization on homology.
In the paper they prove that when $X$ is a ...
- 579
5
votes
0
answers
238
views
Tensor product of "difference bundles" ( Atiyah construction)
There is a well-known in index theory "difference bundle" construction of Atiyah( see for example the original paper "Clifford modules"). And also there is a corresponding formula for the tensor ...
- 51
5
votes
0
answers
178
views
Analytic refinement of generalized cohomology theories
Recently, U.Bunke and others developed in a number of papers (such as this, this or this) an approach to smooth extensions of cohomology theories based on stable homotopy theory. In this approach a ...
- 912
4
votes
0
answers
127
views
Almost acyclicity of the complex of configuration spaces of noncollinear points in projective plane over finite fields
Let $F$ be a finite field with many elements, say more than 7 for example, and $X$ be the corresponding projective plane $\mathbb{P}^2(F)$.
For a set of points in $X$, if any three of them are ...
- 441
4
votes
0
answers
226
views
How to to understand the homology groups $H_*(\Omega_0^\infty S^\infty)$?
The original statement of the Barratt--Priddy theorem says there is an isomorphism of homology groups
$$H_*(\Sigma_\infty)\cong H_*(\Omega_0^\infty S^\infty),$$
where $\Omega_0^\infty S^\infty$ is the ...
- 103
4
votes
0
answers
543
views
Comparing real topological K-theory and algebraic K-theory
Let $R$ be a commutative unital ring and let $i$ be a non-negative integer such that $K^i_{alg}(R)$ is finitely generated abelian group. Is it possible that there does not exist weak homotopy type of ...
- 257
3
votes
0
answers
180
views
Finite generation of algebraic $K$-theory with finite coefficients
Given a smooth connected complex quasi-projective variety $X$, is it possible that $K_i(X, \mathbb{Z}/l)$ to be infinitely generated for $i>0$? I think Quillen-Lichtenbaum implies that above the ...
- 5,901
3
votes
0
answers
260
views
Homotopy equivalence of $K$-theory and $G$-theory
Let $X$ be regular variety then it is known that $Q(Vect(X))\cong Q(\mathcal{M}_X)$. Where $Q$ is the Quillen's q-construction and $\mathcal{M}_X$ is the category of coherent sheaves on $X$. You can ...
- 5,901
2
votes
0
answers
205
views
What role does homotopy play in Karoubi's K-Theory?
In Karoubi's book K-Theory An Introduction, he defines the groups $K^{p,q}(\mathcal{C})$ for a pseudo-abelian Banach category as equivalence classes of triples $(E,F,\alpha)$, where $E,F \in \mathcal{...
- 233
2
votes
0
answers
123
views
Homotopy invariant $K$-theory spectrum version vs space version
Let $K^H$ be the homotopy $K$-theory spectrum. It is defined as a colimit of the form $|\mathbb{K}(X\times \Delta^{\bullet})|$. Here $\Delta^{\bullet}$ is the co-simplicial scheme defined by the ...
- 5,901
2
votes
0
answers
240
views
Algebraic K theory, Karoubi completion and splitting
Suppose $\mathcal{C}\subset \mathcal{C}'$ is a pair of pre-triangulated smooth DG categories over a characteristic-zero basefield (say, $\mathbb{C}$), such that $\mathcal{C}$ is faithfully embedded in ...
- 7,952
1
vote
0
answers
153
views
Stable homology of general linear groups
For what class of rings $R$, is the stable homology (with various choices of coefficients) of $GL_n(R)$ known? Borel computed it rationally for number rings, Quillen computed it for finite fields. Are ...
- 965
1
vote
0
answers
133
views
Contractibility of a $K_0^{\oplus}$ presheaf
Let's assume $X$ is a smooth projective variety over a field. Let $\Delta^{\bullet}$ be the cosimplicial scheme over the same field, where at level $n$ is just the $n$-th algebraic simplex. We can ...
- 5,901
0
votes
0
answers
307
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 ...
- 505