All Questions
33 questions
35
votes
4
answers
3k
views
References for sign conventions in homological algebra
There is no shortage of sign conventions in homological algebra. And once these conventions are set out, there is no shortage of diagrams where an obvious commutative diagram on the underlying ...
20
votes
5
answers
2k
views
Equivalence of ordered and unordered cech cohomology.
Given a topological space X and a finite cover X = $\cup X_i$, one can define Cech cohomology of a sheaf of abelian groups F with respect to the cover $\{X_i\}$ in two different ways:
(Ordered): ...
- 10.5k
15
votes
2
answers
2k
views
When is bar-cobar duality an equivalence?
Let $A$ be an augmented differential graded algebra over a field $k$. I will write $BA$ for its bar construction (whose homology is $Tor^A(k, k)$). This is a co-augmented differential graded ...
- 4,133
15
votes
2
answers
2k
views
Spectral Sequences reference
What is the best reference for spectral sequences for mathematicians who are not experts at the subject, but would just like to open a book and find the SS they need, without going in to deep.
I'm ...
- 1,589
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
2k
views
Cohomology ring of classifying space of spin group $\mathrm{BSpin}(n)$
$\DeclareMathOperator\BSpin{BSpin}\DeclareMathOperator\Pin{Pin}\DeclareMathOperator\BSO{BSO}\DeclareMathOperator\BO{BO}\DeclareMathOperator\BPin{BPin}$In the answer for question: Homology of ...
- 4,826
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
11
votes
0
answers
667
views
Pairing of cohomology and homology Künneth formulas
Let $k$ be a field, and let $X$ and $Y$ be CW-complexes of finite type (although the question makes sense for $k$ a ring and for more general chain complexes of finitely generated free abelian groups)....
- 35.9k
10
votes
3
answers
2k
views
Where can I find a proof of the de Rham-Weil theorem?
Where can I find a proof of the de Rham-Weil theorem?
Does anyone know?
- 360
10
votes
1
answer
657
views
Cap product on Leray-Serre spectral sequences
Let $\ p:X\to B\ $ be a fibration of spaces with fiber $F$. Then there are homological and cohomological Leray-Serre spectral sequences $E^{r}_{pq}$ and $E_r^{pq}$ that converge to $H_*(X)$ and $H^*(...
- 1,484
7
votes
1
answer
474
views
Explicit chain homotopy for the Alexander-Whitney, Eilenberg-Zilber pair
Let $A$ and $B$ be simplicial abelian groups, and let $N_\ast(-)$ denote the normalized chain complex functor. Let
$$AW_{A,B}\colon N_\ast(A\otimes B)
\longrightarrow N_\ast(A)\otimes N_\ast(B)$$
and
...
- 517
7
votes
1
answer
411
views
Under which conditions is the bar construction a conservative functor?
The bar construction is a functor $A\mapsto Bar(A)$ from the category of augmented differential graded algebras over a commutative ring $R$ to the category of chain complexes of $R$-modules. It sends ...
- 2,670
6
votes
0
answers
723
views
On the multiplicative structure in spectral sequences.
Let $f\colon X \rightarrow Y$ be a continuous map of sufficiently nice
topological spaces (say, smooth manifolds). Let ${\cal F}=(\dots\rightarrow
F_i \rightarrow F_{i+1}\to \dots)$ be a bounded ...
- 21.8k
5
votes
2
answers
683
views
Model categories and chain complexes
I'm fairly new to thinking about homological algebra and chain complexes in their own right, i.e outside of isolated examples such as for constructing simplicial homology, or for computing $Ext$ ...
- 168
5
votes
1
answer
425
views
Can we construct a filtered chain complex from a spectral sequence?
Suppose $\{(E_r,d_r)\}$ for $r>0$ forms a spectral sequence over a field $\mathbb{F}$, i.e. for any $r$, $(E_r,d_r)$ is a chain complex over $\mathbb{F}$ and $E_{r+1}=H(E_r,d_r)$. For simplicity, ...
- 673
5
votes
1
answer
323
views
Sullivan minimal model in the case of $H^1(V)\neq 0$
Is there a simple construction of a Sullivan minimal model $\Lambda U \rightarrow V$ in the case that $H^1(V)\neq 0$? Do you have a reference? I envisage a degree-wise construction as in the case of $...
- 466
4
votes
1
answer
230
views
Pontryagin product on the homology of cyclic groups
Consider the cyclic group $C_{p^N}$ of order $p^N$, and let $k$ be a field of characteristic $p$. I would like to know what the algebra structure on the homology $H_*(C_{p^N};k)$ induced by the ...
- 103
4
votes
1
answer
153
views
The homological negligibility of certain subsets in compact manifolds
Let $n\ge 3$ and $X$ be a compact connected $n$-manifold (without boundary).
I need a reference to the following facts (which I believe are true at least in dimension $n=3$):
Fact 1. For every ...
- 41.8k
4
votes
4
answers
284
views
Stratifications and Cohomology Computations
I am interested in references and suggestions concerning the use of stratifications in topology to inductively compute topological invariants. I would appreciate a fairly introductory reference on the ...
- 4,920
4
votes
0
answers
158
views
Postnikov square explicitly on a simplicial complex
$\DeclareMathOperator\Z{\mathbb{Z}}$
Following Wikipedia, a Postnikov square is a certain cohomology operation from a first cohomology group $H^1$ to a third cohomology group $H^3$, introduced by ...
- 10.5k
4
votes
0
answers
290
views
Generalized Postnikov square
Following Wikipedia (https://en.wikipedia.org/wiki/Postnikov_square), a Postnikov square is a certain cohomology operation from a first cohomology group $H^1$ to a third cohomology group $H^3$, ...
- 1,329
3
votes
1
answer
517
views
Künneth spectral sequence for cohomology of chain complexes of $R$-modules
Let $R$ be a unital ring. Let $\mathbf{A}_\bullet$ and $\mathbf{C}_\bullet$ be positive chain complexes of $R$-modules. If $\mathbf{A}_\bullet$ consists of flat $R$-modules then there is homology ...
- 855
3
votes
1
answer
149
views
Interesting examples of a 4-torsion X in a triangulated category such that $2 End(X/2X)\neq 0$?
It is well-known that for the sphere spectrum $S$ in the ('topological') stable homotopy category the object $S/2S$ i.e. the cone of $S\stackrel{\times 2}{\to}S$, is not $2$-torsion.
So I wonder ...
- 16.9k
3
votes
1
answer
575
views
What is the "higher version" of chain homotopy in singular homology?
In basic algebraic topology, we know the following well-known chain homotopy theorem:
Let $X$ be a topological space and $I=[0,1]$ be the unit interval. Let $S_*(X)$ and $S_*(X\times I)$ be the ...
- 9,019
3
votes
0
answers
133
views
Grothendieck spectral sequence (cohomology version) for posets with functor coefficient
In this paper, Quillen mentioned a spectral sequence as follows.
Let $f:X\to Y$ is a poset map and $\mathcal{F}:X\to Ab$, where $Ab$ is the category of abelian groups, a functor which is contravariant ...
- 139
3
votes
0
answers
148
views
An account of "Homologie nicht-additiver Funktoren. Anwendungen"'s results
Is there an account in English of results from "Homologie nicht-additiver Funktoren. Anwendungen" by Dold and Puppe? I am mostly interested in the spectral sequence of cross-effects which ...
- 1,374
3
votes
0
answers
248
views
Explicit computation of hyper Ext in terms of the homologies of the input chain complexes
This question was asked on math.stackexchange and didn't receive any traction, so I'm turning to the MO community. Hello!
Let $C_{\bullet}$ and $D_{\bullet}$ be chain complexes of $R$-modules for some ...
- 301
3
votes
0
answers
261
views
On the Hochschild cohomology of the minimal model of an $A_\infty$ algebra
Suppose $(A, (\mu_k))$ is a (curved) $A_\infty$ algebra, and let $(\tilde A, (\tilde\mu_k))$ be its minimal model. Now, we have two Hochschild cohomology rings $HH^*(A)$ and $HH^*(\tilde A)$. (It may ...
- 2,789
3
votes
0
answers
310
views
Functoriality of Leray homology spectral sequences of fibrations
Let $p\colon E\to B$ and $p'\colon E'\to B'$ be two fibrations. Assume for simplicity that $B,B'$ are simply connected. Let we have morphism of these fibrations, i.e. two continuous maps
$$f\colon E\...
- 21.8k
2
votes
0
answers
222
views
Reference request for a "truncated version" of the de Rham algebra
Let's start on the $n$-torus for sake of simplicity.$\newcommand{\T}{\mathbb T}$
If I understand the relevant definitions correctly, the usual de Rham algebra of smooth differential forms on $\T^n$ is ...
- 25.8k
1
vote
1
answer
361
views
Computing Ext groups in a functor stable $\infty$-category
Let $I$ be a small category and $\mathcal{D}=D^b_\infty(\mathbb{Z})$ the bounded derived $\infty$-category of abelian groups. Consider the $\infty$-category $\mathcal{C}:=\mathrm{Fun}(I,\mathcal{D})$. ...
- 617
1
vote
1
answer
634
views
Inception of modern view of Sheaf Cohomology in Mathematical Literature
From wikipedia entry on Sheaf Cohomology I have found the intriguing passage: 'The essential point is to fix a topological space X and think of cohomology as a functor from sheaves of abelian groups ...
- 37
1
vote
0
answers
178
views
Zeroth cohomology of tensor product of complexes concentrated in nonpositive degrees
This is probably an easy problem, but I can't find any reference.
Let $V$ and $W$ be cochain complexes over some commutative ring, and assume that they have both cohomologies concentrated in ...
- 2,079