All Questions
Tagged with at.algebraic-topology homological-algebra
388 questions
0
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248
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A possible generalization of "Group Cohomolgy"
The group cohomology of a group $G$ is defined as the derived functor associated to the following left exact functor:
$$FIX: \mathcal{M_G} \to \mathcal{Ab}$$
where $FIX$ is the functor from the ...
- 356
6
votes
1
answer
980
views
Group (Co)Homology of Symmetric Group
The question concerns the group homology or group cohomology of symmetric groups.
The entries in groupprops.subwiki.org and in this MO post show the results for the symmetric group S$_4$.
groupprops....
- 10.5k
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
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
5
votes
0
answers
220
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Non existence of commutative singular cochains vs quasi iso between cochains on BT and its cohomology
I believe that there isn't a commutative model for the DGA of cochains on a space, because of cohomology operations. This question has some nice explanations of why this is so. For example, there is a ...
- 561
5
votes
0
answers
358
views
Long exact sequence of Quillen derived functors
Let $F:\mathcal A\to \mathcal B$ be an additive right exact functor between abelian categories $\mathcal A$ and $\mathcal B$. Then for a short exact sequence
$$
0\to A\to B\to C\to 0
$$
in $\mathcal A$...
- 385
0
votes
0
answers
92
views
$Q(f+g)_*=Q(f_*+g_*)$ (The maps induced by the sum is the sum of induced maps modulo decomposables [Reference request]
Let $X, Y$, let's say, homotopy commutative $H$-spaces, $f,g$ maps from $X$ to $Y$. (Actually we only need $Y$ to be homotopy commutative $H$_space,
but the statement is easier if we also suppose $X$ ...
- 3,051
5
votes
0
answers
113
views
How "commutative" are Cech cochains of a sheaf of commutative (dg) algebras?
Let $X$ be a topological space and $\mathcal{F}$ be a sheaf of commutative dg-algebras over $X$. Let $\mathfrak{U}$ be a fixed open covering of $X$ and $C^\bullet(\mathfrak{U},\mathcal{F})$ be the ...
31
votes
1
answer
3k
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What was the error in the proof of Roos' theorem?
Background: In 1961, Roos (who, sadly, apparently passed away just last month) purported to prove [1] that in an abelian category with exact countable products (AB4${}^\ast_\omega$), limits of inverse ...
- 63.9k
10
votes
1
answer
920
views
Is the homotopy category of an abelian model category abelian?
A model structure on an abelian category $A$ is called an abelian model structure if the cofibrations are precisely the monomorphisms with cofibrant cokernel, and if the fibrations are precisely the ...
- 30.3k
9
votes
0
answers
439
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(Torsion in) homology of free nilpotent groups
It is known that for free $k$-step nilpotent group on $r$ generators $N(r, k)$ its integral homology is torsion-free in degrees $\leq 3$ (obvious for 1 and 2, Igusa&Orr computations for 3). ...
- 4,600
6
votes
0
answers
236
views
Explicit formula for higher order Bockstein
The formula for the Bockstein $\beta:H_n(X;\mathbb{Z}/p\mathbb{Z})\to H_{n-1}(X;\mathbb{Z}/p\mathbb{Z})$ is $$\beta[c\otimes 1]=[\frac{1}{p}\partial c\otimes 1]$$
(McCleary page 456)
How about for ...
- 1,229
5
votes
0
answers
543
views
Strong Convergence vs Conditional Convergence for Spectral Sequences (Is there a simple explanation?)
I am curious if there is a relatively simple explanation of what is the difference between strong convergence and conditional convergence for Spectral Sequences?
(Hopefully a simpler explanation than ...
- 1,229
9
votes
1
answer
777
views
Intuition for torsion of a chain complex and application to lens spaces
I have read a bit about the torsion of an acyclic complex. One of my concrete hopes was that I could understand why $L(7,1)$ and $L(7,2)$ are not homeomorphic - I am under the impression that ...
- 5,349
4
votes
1
answer
195
views
Can Bockstein Spectral Sequence detect multiple summands of the same power, in homology?
I understand that the differential $d^k$ of the Bockstein S.S. (mod p) is nonzero iff the homology $H_*(X;\mathbb{Z})$ has summand of the form $\mathbb{Z}/p^k$.
How about for multiple summands in the ...
- 1,229
7
votes
3
answers
314
views
Eilenberg-Zilber-type theorem for good fiber products?
My question is:
If $p\colon X \to B$, $q\colon Y \to B$ are proper submersions, is there a characterization of $H_*(X \times_B Y)$ in terms of $H_*(X)$, $H_*(Y)$, $H_*(B)$ that is simpler than the ...
- 1,589
12
votes
1
answer
1k
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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
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votes
0
answers
97
views
First Betti number of a Reeb graph is not greater than that of the space?
(I have asked this question at math stackexchange, it was upvoted but got no answers; maybe you can help.)
It is well-known that $\beta_1(R(f))\le\beta_1(X)$, where $\beta_1$ is the first Betti ...
4
votes
0
answers
101
views
What is the correct generalization of "sigma-free" to props?
This is a question about props, a generalization of operads (used to model operations with several inputs and several outputs).
By forgetting the composition structure of an operad one obtains a so ...
- 517
7
votes
1
answer
197
views
Homology of a limit of semidirect products
Suppose I have two families of groups $A_k$ and $B_k$ indexed by the natural numbers and suppose $B_k$ acts on $A_k$. Suppose there are groups homomorphisms $A_{k+1} \rtimes B_{k+1} \to A_k \rtimes ...
- 111
93
votes
3
answers
11k
views
What is homology anyway?
Disclaimer: I don't feel qualified to ask this question and yet it's been troubling me for some time now and I lost my patience and decided to ask to get some kind of answer. If there are any stupid ...
- 7,789
10
votes
0
answers
414
views
Segal-Freed-Hopkins-Teleman = Atiyah-Hirzebruch/Leray-Serre?
Freed-Hopkins-Teleman (section 3.7) generalise Segal's (Proposition 5.3) spectral sequence for equivariant K-theory to more general local quotient groupoids (that is, topological groupoids locally ...
- 35.5k
4
votes
0
answers
234
views
Is there an analogue to the koszul complex for constructible sheaves?
Given a variety $X$ and a complete-intersection morphism
$$
Y \to X
$$
is there an analogue of the Koszul complex for $\mathcal{O}_Y \in \textbf{Coh}(X)$ in the setting of constructible sheaves? ...
- 1,716
8
votes
1
answer
363
views
Adams spectral sequence and short exact sequences. Some clarifications
as the title suggests I'm looking for some clarifications in the computations of the ext charts of some $A(1)$-modules arising as extensions of other modules. In particular, I've the following example ...
- 503
1
vote
1
answer
117
views
Is singular barycentric subdivision injective?
This question has been asked on mathstackexchange without any answers.
Let us note $\Delta_p(X)$ the $p$-singular chains on a topological space $X$. We have a well-known barycentric subdivision
$$b:...
- 1,017
3
votes
0
answers
310
views
Functoriality of Leray homology spectral sequences of fibrations
Let $p\colon E\to B$ and $p'\colon E'\to B'$ be two fibrations. Assume for simplicity that $B,B'$ are simply connected. Let we have morphism of these fibrations, i.e. two continuous maps
$$f\colon E\...
- 21.8k
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
$$
H_n(K)=\mathbb{Z}^{\oplus i}\oplus (\oplus _{p \...
- 1,990
4
votes
1
answer
417
views
Second differential in long exact sequence for Cech cohomology for nonabelian groups
I do not really think it fits MO, but I posted it in MathStackExchange with little success, so...
Assume that we have a short exact sequence of, say, Lie groups $1\rightarrow A\rightarrow B\...
- 49
4
votes
1
answer
573
views
Homotopy colimit of a simplicial DGA
It seems to be well-known that the homotopy colimit of a simplicial chain complex (unbounded) can be computed by taking the totalization of the associated (half-plane) double complex. The totalization ...
- 403
2
votes
0
answers
108
views
Relating inflation maps from spectral sequences in lower and higher dimensions
The spectral sequence has some nice property.
Consider $ N \to G \overset{R}{\to} Q $ and $G/N=Q$. There is a spectral sequence $\{E^{p,q}_n, d_n\}$ with: (i) The differential is defined as a map $...
- 10.5k
0
votes
0
answers
308
views
Generalising the Mayer-Vietoris principle
My understanding of the general Mayer-Vietoris principle is as follows. We want to compute the cohomology of some sheaf $\mathscr{F}$. We start by taking a resolution $$\mathscr{F}_0 \rightarrow \...
user81684
5
votes
2
answers
651
views
Inflate a finite-group cocycle into coboundary in non-Abelian groups
Edit: In case that there is no solution for the original question, I modify to enrich the question.
We like to ask a possible specific inflation a $H^3(Q, \mathbb{R} /\mathbb{Z})$ cocycle with a ...
- 755
0
votes
1
answer
143
views
Trivialize a cup-product 2-cocycle of $G$ in a larger group $J$
I like to ask a simple question: how to trivialize a cup-product 2-cocycle of $G$ into a 2-coboundary of $J$ in a larger group $J$.
Let us take a nontrivial 2-cocycle $\omega_3^G(g_a, g_b) \in H^2(G,\...
- 755
10
votes
0
answers
268
views
Isomorphisms between minimal $A_\infty$-algebras having identical $k$-truncations
Let $A_m =(A,0,m_2,m_3,\dots)$ and $A_n=(A,0,n_2,n_3,\dots)$ be two $A_\infty$-structures on a vector space $A$. Assume that
i) $A_m$ and $A_n$ are isomorphic, and
ii) $A_m$ and $A_n$ have the same ...
- 523
2
votes
1
answer
264
views
Trivialize a cup-product 3-cocycle of $G$ in a larger group $J$
Inspired by this question, let us take a nontrivial 3-cocycle $\omega_3^G(g_a, g_b, g_c) \in H^3(G,\mathbb{R}/\mathbb{Z})$ in the cohomology group of $G$ with $U(1)=\mathbb{R}/\mathbb{Z}$ coefficient. ...
- 755
5
votes
1
answer
782
views
Resolutions by free Differential Graded Algebras
I am struggling to find references for explicit computations of things like symmetric algebras and resolutions in the dg context. (any pointers in this direction would be highly appreciated!) I have ...
- 1,889
28
votes
1
answer
3k
views
Two points of view about Borel-moore homology
They are several ways to define the Borel-Moore homology on a locally compact space $X$.
The first one is by analogy with the singular homology but instead of using finite chains, we use locally ...
- 1,017
5
votes
1
answer
184
views
Symmetric multiplication on homotopy quotient of topological space
According to http://ocw.mit.edu/courses/mathematics/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/lecture-notes/lecture2.pdf. , if $X$ is any topological space, then its ...
- 2,809
8
votes
0
answers
736
views
Homology of inverse limits over inverse systems more complicated than towers
Most textbooks discussions of homology of inverse limits of chain complexes consider only “towers,” i.e. inverse systems indexed by the natural numbers. I’d like to find a reference that explains what ...
- 5,509
8
votes
1
answer
480
views
Can the Hochschild cochain complex be given the structure of a "homotopy BV algebra"?
In a 1993 letter, Deligne posed the following (paraphrased from a paper of Gerstenhaber and Voronov's):
Conjecture (Deligne). The Hochschild cochain complex $CC^*(A)$ of an associative algebra ...
- 1,589
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
7
votes
0
answers
192
views
mod $p$ homology module of unordered configuration spaces of the projective plane
Let $M$ be a manifold and $k$ be a positive integer. Let $F(M,k)$ be the $k$-th ordered configuration space over $M$, consisting of all ordered $k$-tuples of distinct points in $M$. Let $\Sigma_k$ be ...
- 2,223
1
vote
1
answer
520
views
Free Symmetric Operads and $\mathbb{S}$-modules
In the definition of operads, if we restrict our attention to $\mathbb{S}$-modules where the action by the symmetric groups is free, then the free operads
have still an underlying free $\mathbb{S}$-...
- 133
9
votes
1
answer
804
views
Known results in the Cohomology of finite groups
I am learning to compute cohomology of finite groups and came across this survey article http://www.ams.org/notices/199707/adem.pdf "Recent Developments in the cohomology of finite groups" by ...
- 529
25
votes
0
answers
922
views
Does the Tate construction (defined with direct sums) have a derived interpretation?
Any abelian group M with an action of a finite group $G$ has a Tate cohomology object $\hat H(G;M)$ in the derived category of chain complexes. There are several ways to define this. One is as the ...
- 52.6k
6
votes
0
answers
542
views
N-periodic derived categories
I have some seemingly basic questions about $N$-periodic derived categories to which I have not found answers in any of the usual places.
Let $R$ be a ring, and let $D(R)_{\mathbb Z/N\mathbb Z}$ ...
- 18.7k
21
votes
1
answer
2k
views
A spectral sequence for computing cohomology of a space from that of its strata
Let $X$ be a smooth complex variety (not necessarily compact) and let $D$ be a normal crossings divisors with components $D_1$, $D_2$, ..., $D_N$. For a set of indices $I$, let $D_I = \bigcap_{i \in I}...
- 156k
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
7
votes
0
answers
668
views
Is there an obstruction which classifies "quasi-isomorphism but not chain equivalence"?
Fix a ring $R$ and let $C_\bullet$, $D_\bullet$ be (possibly unbounded) chain complexes of $R$-modules. Assume that $f_\bullet:C_\bullet \to D_\bullet$ is a quasi-isomorphism: that is to say, $f$ is a ...
- 15.5k
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