All Questions
Tagged with spectral-sequences homology
14 questions
17
votes
2
answers
2k
views
Are the homology and cohomology Serre spectral sequences dual to each other?
If we use homology and cohomology over a field $k$, if a space has homology and cohomology groups of finite type in each degree, then $H_\ast(X;k)$ is dual to $H^\ast(X;k)$ using the universal ...
- 8,167
9
votes
1
answer
399
views
Third bordism group of BG, where G is an arbitrary compact Lie group.
Is anything known about $\Omega_3(BG)$, where $G$ is an arbitrary compact Lie group; i.e., is it possible to describe the structure of $\Omega_3(BG)$ for any compact Lie group? I know that $H_3(BG)$ ...
- 1,709
8
votes
2
answers
492
views
Conditions under which the preimage of a submanifold in nontrivial in homology
Let $\pi: M^{n+k} \to N^n$ be a fibre bundle with fibre $F$ between compact smooth manifolds. What are “mild” sufficient conditions on the topology of $M$, $N$ and $F$ so that given a closed $p$-...
- 2,095
7
votes
2
answers
1k
views
Proof of the ''trangression theorem''
Here is what I would call the transgression theorem. Let $X$ be a pointed space and $\Omega X$ its loop space. There are two maps $H_{p}(\Omega X) \to H_{p+1}(X)$ which should be the same. I am ...
- 20.9k
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
2
answers
408
views
Homology spectral sequence for function space
The question is in the title. Suppose that $X$ and $Y$ are two pointed connected CW-complexes. I was wondering if there exists a spectral sequence computing the homology of the function space $$H_{\...
- 101
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
1k
views
Unbounded complexes, resolutions and computation of derived functors
Hey guys, let $F: \mathcal{A} \rightarrow \mathcal{B}$ be a left exact functor between abelian categories with enough injectives, let $K \in Kom(\mathcal{A})$ be an unbounded complex, I've heard that ...
- 456
4
votes
1
answer
233
views
Mayer–Vietoris sequence for coproduct of Hopf algebras
Is there a Mayer–Vietoris-type sequence for the homology of a coproduct of two Hopf algebras over an ideal? The definition of the coproduct can be found in Agore - Categorical Constructions for Hopf ...
- 1,374
3
votes
1
answer
938
views
Computing the homology groups of spaces in a fibration
Let $F\rightarrow X\rightarrow B$ be a fibration. If we know very well the spaces $F$ and $B$ and wish to compute the homology of $X$. One possible tool is the Serre Spectral Sequence. However, it ...
- 1,108
2
votes
1
answer
201
views
Identifying $d_1$ in the Atiyah-Hirzebruch-Serre spectral sequence
In A Primer on Spectral Sequences (also later published in More Concise Algebraic Topology), J. Peter May describes the Serre Spectral Sequence for any homology theory. To recap, suppose $p\colon E\...
- 508
2
votes
0
answers
108
views
Computation of mod p homology of $MSU$
I am trying to proof Novikov theorem
\begin{equation}
MSU_*\otimes \mathbb Z[\frac 1 2] \cong \mathbb Z[\frac 1 2][y_2, y_4, \ldots],\quad \deg y_i = 2i.
\end{equation}
This can be proved by using ...
user131113
1
vote
0
answers
91
views
non zero differential in a spectral sequence
This is the situation:
Let $A = R_* \otimes C_*$ be an $R$-module where $C_*$ is a finitely generated graded ($*\geq 0$) vector space over a field $F$ which is also bounded above, and $R$ is a ...
- 81
1
vote
0
answers
112
views
Reference request for Leibniz rule and spectral sequences
Suppose $A_*$,$B_*$, and $C_*$ are chain complexes equipped with filtrations and a map $m:A_* \otimes B_* \to C_*$ respecting these filtrations. I am looking for a reference for the fact that the map $...
- 41