All Questions
16 questions
23
votes
3
answers
6k
views
Does homology detect chain homotopy equivalence?
Is the following true: If two chain complexes of free abelian groups have isomorphic homology modules then they are chain homotopy equivalent.
- 425
23
votes
3
answers
3k
views
Homology theory constructed in a homotopy-invariant way
Singular homology sends homotopic morphisms on equal morphisms and weakly equivalent spaces on isomorphic objects. So singular homology is in fact defined on the homotopy category of topological ...
- 3,033
13
votes
0
answers
680
views
Singular chains generated by manifolds with corners --- does it really work?
Usually we define singular homology via the complex of singular chains:
$$C_\ast(X)=\bigoplus_{n\geq 0}\mathbb Z\langle\sigma:\Delta^n\to X\rangle$$
where the right hand side denotes the free abelian ...
- 18.7k
12
votes
1
answer
1k
views
Getting the most general form of Mayer-Vietoris from the Eilenberg-Steenrod axioms
I asked this question a while ago on MSE, got no answer, put a bounty on it, still got no answer, was advised to ask here instead, hesitated, forgot about the question for a while and now remembered ...
- 9,650
10
votes
1
answer
855
views
finite complex with non-finitely generated homology with local coefficients
I am looking for an explicit example, if one exists, of a (pointed) finite connected CW-complex $X$ such that some homology group with local coefficients $H_n(X,{\mathbb Z}[\pi_1 X])$ is not a ...
- 6,210
8
votes
2
answers
1k
views
Differentials in the Lyndon-Hochschild spectral sequence
The Lyndon-Hochschild(-Serre) spectral sequence applies to group extensions in a manner analogous to the Serre-Leray spectral sequence applied to a fibration.
Does anyone know of a good description (...
- 1,422
8
votes
1
answer
1k
views
Geometric intuition behind this chain homotopy
My question has to do with the chain homotopy that appears in Lee's Introduction to Topological Manifols and Rotman's Introduction to Algebraic Topology proofs that the inclusion
$$C_\bullet^\mathcal{...
- 516
7
votes
4
answers
685
views
Realizing complexes with bases as cellular complexes
This is a question a friend of mine asked me some time ago. I suspect the answer is "no" but can't prove it.
Every free complex of abelian groups is isomorphic to the reduced cellular complex of some ...
- 23.5k
7
votes
0
answers
434
views
spectral sequence for a complex with two filtrations
Suppose $(C,d)$ is a chain complex: an abelian group with a map $d:C \to C$ such that $d^2 = 0$ (people like to assume $C$ is graded; if that helps - feel free to do so). A filtration is an ascending ...
- 353
6
votes
1
answer
563
views
a question about Bockstein spectral sequence
I find the following theorem for Bockstein spectral sequence at http://pages.vassar.edu/mccleary/files/2011/04/MC10.fin_.pdf, page 459:
Question. for a fixed $k$, if $\beta$ does not hit $H_k(X;\...
- 1,990
6
votes
0
answers
136
views
torsion part of homology of simplicial complexes [duplicate]
Let $n$ be a fixed positive integer and let $K$ be a simplicial complex with $N$ vertices. Suppose the $n$-th integral homology group of $K$ is
$$
H_n(K)=\mathbb{Z}^{\oplus i}\oplus (\oplus _{p \...
- 1,990
4
votes
1
answer
479
views
Cellular homology of the universal cover
Let $X$ be a connected pointed CW complex. Let $\tilde{X}$ be its universal covering space and $G=\pi_{1}(X)$.
Lets denote $(C^{Cell}_{\ast}(\tilde{X}),d)$ the cellular chain complex associated to $\...
- 855
3
votes
0
answers
133
views
Milnor exact sequence for homology of hopf algebras
Let $K$ be a field and $\mathrm{Hopf}^K_{E_\infty}$ the $\infty$-category of
homotopy-coherent hopf algebras over $K$ that are coherently commutative and cocommutative.
Precisely, $\mathrm{Hopf}^K_{E_\...
- 1,361
2
votes
1
answer
216
views
Compute the singular homology group modulo barycentric subdivision
Let $X$ be a topological space, and let $C(X)$ denote its singular chain complex with boundary operator $\partial$ and $n$-th chain group $C_n$. We know there exists a barycentric subdivision operator ...
- 807
2
votes
0
answers
194
views
A covariant functor on a given abelian category and comparison of homology in target and source
The definition of cohomology of a complex is based on the following:
We have a complex (of appropriate objects) $$0\leftarrow C_0\leftarrow C_1\leftarrow C_2\ldots \leftarrow C_n\ldots$$
Then for an ...
- 356
2
votes
0
answers
168
views
Singular homology: Lifting simplices gives map in homology
Let $X$ be a space, $k=k_1+\dotsb+k_r$ and let $G:=\mathfrak{S}_{k_1}\times\dotsb\times \mathfrak{S}_{k_r}$ act freely on the right on $X$. Fix a commutative ring $R$ and another space $Y$.
Then the ...
- 1,623