All Questions
Tagged with spectral-sequences kt.k-theory-and-homology
20 questions
56
votes
5
answers
9k
views
Why are spectral sequences so ubiquitous?
I sort of understand the definition of a spectral sequence and am aware that it is an indispensable tool in modern algebraic geometry and topology. But why is this the case, and what can one do with ...
- 25.6k
20
votes
1
answer
1k
views
Torsion in the Atiyah–Hirzebruch spectral sequence of a classifying space
Let $G$ be a compact, connected Lie group. There is an Atiyah–Hirzebruch spectral sequence
$$H^*(BG;K^*) \implies K^*(BG)$$
connecting $H^*BG$, which generally contains torsion, with $K^*BG \cong \...
- 2,995
16
votes
2
answers
2k
views
Torsion in K-theory versus torsion in cohomology
Inspired by this question, I wonder if anyone can provide an example of a finite CW complex X for which the order of the torsion subgroup of $H^{even} (X; \mathbb{Z}) = \bigoplus_{k=0}^\infty H^{2k} ...
- 8,803
10
votes
0
answers
325
views
Adams blue book lemma 17.14: computing a $\mathbb{F}_2$ basis for a filtration of $H\mathbb{Z}_*(bu \wedge bu)$
First off let me apologize for not being able to give all the context for this question. I'm learning how to do computations in stable homotopy theory and have been particularly spending a lot of time ...
- 101
10
votes
0
answers
352
views
Is there a spectral sequence of Atiyah's topological KR-theory that can be used to compute basic examples?
For Segal's complex $G$-equivariant $K$-theory, it is well-known that there is an Atiyah-Hirzebruch spectral sequence. If say $G$ a finite group and $X$ a finite CW-complex, the second page of this ...
- 357
8
votes
1
answer
475
views
Spin cobordism v.s. KO theory in low or in any dimensions
It seems that from this webpage, the spin cobordism is equivalent to KO theory in low dimension.
If we denote the $p$-torsion part (mean $\mathbb{Z}_{p^n}$ for some $n$) $$\Omega_d(BG)_p.$$
...
- 10.5k
7
votes
1
answer
845
views
Mayer Vietoris Spectral sequence for topological K theory
In Sheaf theory one can obtain the Mayer Vietoris spectral sequence for cohomology. For $\mathcal{U}$ an open cover of $X$ we get the convergence
$E_2^{pq} = \check H^p(\mathcal{U},H^q(-,F)) \...
- 659
7
votes
0
answers
541
views
Convergence of a spectral sequence of a double complex
In Weibel's book, a spectral sequence $E^r_{p,q}$ is said to weakly converge to a graded object $H_{\ast}$ if for every $n$ there exists a filtration $\dots \subset F_{r}H_{n} \subset F_{r-1}H_{\ast} \...
- 1,325
7
votes
0
answers
436
views
spectral sequence for a complex with two filtrations
Suppose $(C,d)$ is a chain complex: an abelian group with a map $d:C \to C$ such that $d^2 = 0$ (people like to assume $C$ is graded; if that helps - feel free to do so). A filtration is an ascending ...
- 353
6
votes
1
answer
564
views
a question about Bockstein spectral sequence
I find the following theorem for Bockstein spectral sequence at http://pages.vassar.edu/mccleary/files/2011/04/MC10.fin_.pdf, page 459:
Question. for a fixed $k$, if $\beta$ does not hit $H_k(X;\...
- 1,990
6
votes
1
answer
242
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
0
answers
163
views
Spectral Sequence for Twisted K-theory
Atiyah and Segal wrote in their Twisted K-theory that one can compute twisted K-theory using a spectral sequence similar to an Atiyah-Hirzebruch spectral sequence. They claimed that for any twisting $[...
- 61
5
votes
2
answers
2k
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 ...
- 16.9k
5
votes
0
answers
521
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
4
votes
1
answer
119
views
Kuenneth short exact sequence for K-homology
Atiyah proved a Kuenneth short exact sequence for K-theory. I need one for K-homology, but can not find any reference in the literature. Do you know one?
Using general spectra stuff, one gets a ...
- 2,998
3
votes
1
answer
470
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 ...
- 31
3
votes
0
answers
122
views
Twisted spin cobordism v.s. KO theory in low dimensions
Based on the background info and this this webpage, here is a more advanced problem:
Question: If we consider a different more subtle twisted structure, like
$${\Omega_d^{(\mathrm{spin} \times G)/N}},...
- 10.5k
2
votes
1
answer
186
views
Can I bound the degree of a contracting homotopy in an exact filtered complex?
Suppose that I have given you a bigraded vector space $V = \bigoplus_{i,j} V_{i,j}$. The first grading is a "homological" $\mathbb Z$-grading, and the second is an independent $\mathbb Z$-grading. ...
- 54.6k
2
votes
1
answer
185
views
How to compute the $G$-theory groups of $k[x,y]/(xy)$ for any field $k$
I am trying to compute the $G$-theory groups of the ring $k[x,y]/(xy)$ for any field $k$. What I have tried so far are two approaches.
Approach 1: Use the $G$-theory localization sequence for $k[x,y]/(...
- 639
0
votes
0
answers
80
views
Relation between Chow groups and K theory
I am reading about Chow groups and algebraic K-theory of schemes. I get to know that for smooth schemes the re is a strongly convergent spectral sequence
$$E_2^{p,q} = CH^{-q}(X,-p-q) \implies K_{-p-q}...
- 613