All Questions
Tagged with kt.k-theory-and-homology reference-request
105 questions
80
votes
2
answers
7k
views
Vladimir Voevodsky's works
Vladimir Voevodsky has made several contributions in abstract algebraic geometry, focused on the homotopy theory of schemes, algebraic K-theory, and interrelations between algebraic geometry, and ...
55
votes
8
answers
7k
views
Applications of Grothendieck-Riemann-Roch?
I am currently trying to learn a bit about Grothendieck-Riemann-Roch...
To try to get a better feeling for it, I am looking for examples of nice applications of GRR applied to a proper morphism $X \...
- 21k
36
votes
5
answers
6k
views
What is the equivariant cohomology of a group acting on itself by conjugation?
This question makes sense for any topological group $G$, but I'd particularly like to know the answer for $G$ a compact, connected Lie group.
$G$ acts on itself by conjugation. One has the equivariant ...
- 13.2k
23
votes
0
answers
647
views
Is this a model for $K$-theory of a triangulated category?
The recent question Complete the following sequence: point, triangle, octahedron, . . . in a dg-category reminded me of something I wanted to clarify long time ago; most likely this is now well known ...
- 18.4k
18
votes
5
answers
4k
views
"Must read" papers in algebraic K-theory?
I'm mainly interested (graduate student) in surgery theory and geometric topology.
If I have a chance to suggest "must read" papers in geometric topology for beginner,
I'm very glad to suggest "...
18
votes
1
answer
1k
views
Which motivic cohomology groups of complex numbers are non-torsion?
I would like to know which motivic cohomology groups of complex numbers are non-zero and ("better") non-torsion, i.e., for which $(i,j)$ the $i$th cohomology of the complex ${\mathbb{Q}}(j)$ over $\...
- 16.9k
18
votes
0
answers
734
views
How boundedly generated is $SL_3(\mathbb{Z})$?
The group $G = \mathrm{SL}_3(\mathbb{Z})$ is known to be boundedly generated, that is, there exists some $m \in \mathbb{N}$, and $g_1, \dots, g_m \in G$ such that we have the following equality of ...
- 11.3k
16
votes
3
answers
2k
views
Roadmap to a proof of the Atiyah-Singer Index Theorem which uses K-Theory
Lately my studies have been focusing on learning the machinery of K-Theory, and I thought that learning the Atiyah-Singer Index Theorem would be a good way to see K-Theory in action a bit and to learn ...
16
votes
2
answers
1k
views
Proof of Bott Periodicity in twisted K-theory
I have a question about the Proof of Bott Periodicity in twisted K-theory
by Atiyah and Segal in their paper Twisted K-theory.
Following their notation, to prove Bott periodicity in this context it ...
16
votes
1
answer
918
views
Explicit path in the unitary group of a $C^*$-algebra
For $G$ a discrete group, there is a canonical inclusion $g\mapsto u_g$ of $G$ into the unitary group of the reduced $C^*$-algebra $C^*_r(G)$. Denote by $[u_g]$ the class of $u_g$ in the (topological) ...
- 11.1k
14
votes
3
answers
3k
views
References for equivariant K-theory
I want a good introduction to localization in equivariant $K$-theory. This introduction can be simple in several ways:
I only care about torus actions.
I only care about $K^0$.
I only care about very ...
- 156k
14
votes
1
answer
1k
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 $\...
- 16.9k
13
votes
1
answer
700
views
Reference for the algebro-geometric proof of Matsumoto theorem
Matsumoto proved in his PhD thesis that if $F$ is a field then $$K_2(F)=(F^*\otimes F^*)/(x\otimes (1-x)).$$
The original Matsumoto proof as it is written in Milnor's book on algebraic K-theory looks ...
- 2,305
13
votes
1
answer
2k
views
Roadmap for Algebraic Geometry/Homotopy Theory/Algebraic $K$-Theory intersection
I’m afraid that this is quite a general question but I am hoping some experts can weigh in. I am a student generally interested in learning more about the intersection of algebraic geometry, algebraic ...
- 155
13
votes
1
answer
459
views
Algebraic K-theory of a ring
I started to learn some algebraic $K$-theory and its relation to geometric topology problems.
My question is: What is the list of rings such that all their algebraic $K$-theory groups are known?
I ...
- 443
12
votes
2
answers
341
views
Which $K$-groups $K(C^*_r(G))$ are computed?
We have the Pimsner-Voiculescu exact sequences and the Baum-Connes map
for possible computation of the $K$-theory of the reduced group $C^*$-algebra $C^*_r(G)$ for a topological, locally compact, ...
- 685
11
votes
1
answer
2k
views
A survey for various $K$-homology theories and their relationship
The ordinary Topological $K$ theory defined by Atiyah and Hirzebruch is a generalized cohomology theory (see wikipedia).There is the Bott spectrum associated to this generalized cohomology theory....
- 593
11
votes
1
answer
490
views
Reference to Chern classes in algebraic k theory
I am reading P. Schneider's paper, Introduction to the Beilinson conjectures. Section 4 in this paper is something very formal about Chern classes. Personally I find some terminologies in the paper a ...
- 2,971
10
votes
3
answers
725
views
Reduction mod $n$ of symplectic group
Let $g,n$ be positive integers, is there a reference that $\mathrm{Sp}(2g,\mathbb{Z})\to\mathrm{Sp}(2g,\mathbb{Z}/n\mathbb{Z})$ is surjection?
The only reference I could find is lemma 5.16 in Deligne–...
user39380
10
votes
2
answers
1k
views
Grothendieck group for projective space over the dual numbers
Fix a field $k$. For a singular variety $X$, I understand that the Grothendieck group $K^0(X)$ of vector bundles on $X$ is not necessarily isomorphic to the Grothendieck group $K_0(X)$ of coherent ...
- 2,955
9
votes
1
answer
837
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 ...
- 63.1k
9
votes
1
answer
932
views
The vanishing of non-connective K-theory in negative degrees
In the works of Cisinski, Tabuada, and Schlichting certain non-connective K-theory groups for a differential graded category $C$ are defined. As far as I understand, $K_i(C)$ is not necessarily zero ...
- 16.9k
9
votes
0
answers
1k
views
Some questions about Clausen's third IHES lecture on Efimov K-theory
I have some questions about the last theorem stated by Clausen at https://youtu.be/2xNG4rHUC6U?si=yw9eYiygLegH0nQK&t=4319. I'm not very familar with the definitions, so please correct me about any ...
- 2,356
9
votes
0
answers
371
views
Which of the physics dualities are closest in essence to the Spanier-Whitehead duality (with a subquestion)?
First of all, what I want to ask is slightly more elaborate than what stands in the title (hence the subquestion).
I am telling this since as it is, the title contains a meaningful question, but it ...
- 18.4k
8
votes
2
answers
437
views
Injectivity of the Baum-Connes assembly map for locally compact groups
Skandalis, Tu and Yu in "The coarse Baum-Connes conjecture and groupoids" proved that:
Let $\Gamma$ be a countable group with a proper left-invariant metric $d$. If $\Gamma$ admits a uniform ...
- 1,702
8
votes
1
answer
569
views
Continuity of l-adic cohomology: is the cohomology of the generic point isomorphic to the completion of the limit of cohomology of open subvarieties?
Let $X$ be a variety over an algebraically closed field $k$. Denote by $\eta$ its generic point; it is the inverse limit of the open subvarieties $X_i$ of $X$. It is well known that the etale ...
- 16.9k
8
votes
1
answer
943
views
Freedman's work on non-simply-connected 4-manifolds
In the late 1970's and in the 1980's, Michael Freedman showed a relationship between the topological surgery problem in 4-dimensions, the slice problem for links, and the classification of non-simply-...
- 22.1k
8
votes
0
answers
440
views
Poincaré duality for topological $K$-theory
Let $n$ be an even number. Let $X$ be a $n$-dimensional complex projective manifold with
$H^{2m+1}(X,\mathbb{Z})=0$, for all $0\leq m\leq n-1$.
$H^{2m}(X,\mathbb{Z})$ is a free $\mathbb{Z}$-module ...
user39380
8
votes
0
answers
411
views
Equivariant K-theory of projective representation on complex projective space
Let $G$ be a group and let $V$ be a complex projective representation of $G$, so that $G$ acts on the projectivization $\mathbb{P}(V)$. Is there any way to calculate the $G$-equivariant complex $K$-...
- 25.6k
8
votes
0
answers
256
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 ...
- 25.8k
7
votes
3
answers
2k
views
Complex structure on $S^4$
I have heard that there is a proof of non-existence of complex structure on the 4-sphere $S^{4}$ using only the topological K-theory (complex $KU$ and real $KO$). Moreover this argument can not be ...
- 451
7
votes
1
answer
448
views
Reference request for K-Theory linearization
I posted this question on math.se here:https://math.stackexchange.com/q/2996787/482732, but I think it may be more appropriate here, sorry if I am wrong about that.
In Waldhausen's paper Algebraic K ...
- 333
7
votes
1
answer
836
views
Chern Character Isomorphism for non-finite CW complexes, resp. for non-CW complexes
This is a question I asked at Math.SE but got no answers: https://math.stackexchange.com/q/397164/7110/
Atiyah and Hirzebruch showed in their paper "Vector bundles and homogeneous spaces" ...
- 2,998
7
votes
0
answers
116
views
A "lower-central" filtration of Steenrod algebra?
$\renewcommand{\Atwo}{\mathcal{A}_2}$ So, a lot of good work has been accomplished by filtering the Steenrod algebras $\mathcal{A}_p$ in powers of the Augmentation ideal; For reasons partly ...
- 731
6
votes
2
answers
684
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 ...
- 54.6k
6
votes
1
answer
674
views
Good references for K-theory of modular curves?
The title says it. I am looking for a good exposition on the K-theory of the curves $X_{i}(N)$, $Y_{i}(N)$, where $i\in\{0,1\}$.
I have some background in $K$-theory and also some background in ...
- 1,805
6
votes
1
answer
875
views
reference request for mod p and p-adic K-theory
Is there a good reference that explains mod p K-theory and p-adic or p-complete K- theory? All I know about K-theory is the topological K-theory of "vector bundles and k-theory" in Switzer's book (...
- 785
6
votes
1
answer
241
views
For which exact couples do associated spectral sequences degenerate at $E_1$?
It is well known that a bigraded exact couple of objects of an abelian category yields a spectral sequence (cf. https://ncatlab.org/nlab/show/exact+couple#SpectralSequencesFromExactCouples). My ...
- 16.9k
6
votes
1
answer
623
views
On various "extension closures" and "orthogonals" in triangulated categories
A vague form of my question is the following one: for a class of objects $D$ of a triangulated category $C$ we consider the class $E$ of objects that satisfy $Mor_{C}(d,e)=\{0\}\ \forall d\in D$; ...
- 16.9k
6
votes
0
answers
236
views
Fundamental class in equivariant K-theory
I'm looking for an accessible reference for the definition of the fundamental class in equivariant K-theory.
The setup I'm interested in is the following: suppose $V$ is a vector space equipped with ...
- 371
6
votes
0
answers
101
views
Squaring operation in KO theory
There's an operation $\Lambda^2: KO^0(X) \to KO^0(X)$ such that $\Lambda^2(x+y)-\Lambda^2(x)-\Lambda^2(y)=xy$, which comes from the antisymmetric power. Similarly, there's a $\Lambda^2: KO^4(X) \to KO^...
- 6,123
6
votes
0
answers
137
views
Comparison of K-groups of (affine) singular schemes with K'=G-groups
It is well known that Quillen K-theory coincides with $K'=G$-theory for regular schemes, and can be distinct from it for singular ones. Are there any methods for studying this distinctions? In ...
- 16.9k
5
votes
1
answer
374
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 (...
- 1,498
5
votes
2
answers
1k
views
Survey of Algebraic K-Theory Since 1980?
I just came across Charles Weibel's Development of Algebraic K-Theory until 1980, and found it really helpful. Is there been anything analogous which surveys the developments in the last 30 years? I'...
- 1,138
5
votes
1
answer
792
views
$K_0$ of integral group ring of cyclic group $\mathbb{Z}/p\mathbb{Z}$
Is there a table for the computation of $K_0(\mathbb{Z}[\mathbb{Z}/p\mathbb{Z}])$?
These groups are also known as ideal class group in number theory.In topology,they are the home of some important ...
- 593
5
votes
1
answer
223
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).
$...
- 9,019
5
votes
1
answer
141
views
Finding a proof within a paper: reduced $K$-theory of Higson compactification of $[0,\infty)$ is uncountable
Emerson and Meyer's Paper "Dualizing the Coarse Assembly Map" (2006) states the following Proposition (5.1):
Let $X = [0,\infty)$ be the ray with its Euclidean metric coarse structure.
Then the ...
- 1,903
5
votes
0
answers
520
views
Spectral sequences coming from filtrations (Postnikov systems) in triangulated categories: references and convergence
Let $c^i: M^{\ge i+1}\to M^{\ge i}$ for $i\in \mathbb{Z}$ be an sequence of morphisms in a triangulated category $C$ and assume that $M^{\ge i}$ are equipped with compatible morphisms into an object $...
- 16.9k
5
votes
0
answers
225
views
Can triangulated categories be "approximated by countable subcategories" (that are triangulated but not full!)?
For a given (finite) set of (objects and) morphisms $f_i$ in a triangulated category $C$ I am interested in a (non-full!) triangulated subcategory $C'\subset C$ of "small size" that would contain them....
- 16.9k
5
votes
0
answers
343
views
Duality between K-theory and K-homology in the non-compact, spin$^c$ case
Let $M$ be a compact spin$^c$ manifold, so that it has a fundamental class $[M] \in K_n(M)$. It is well-known that the cap product with $[M]$ induces Poincare duality isomorphisms $K^\ast(M) \cong K_\...
- 2,998