All Questions
Tagged with at.algebraic-topology homological-algebra
388 questions
1
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0
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122
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Finitely presented homology group
Given a finitely presented $i$th singular homology group over $\mathbb Z$ of a topological space $X$. If one knows the family of $i$th singular homology groups of $X$ over all possible fields, can one ...
4
votes
1
answer
219
views
Is the normalized simplicial bar construction of an operad a cooperad?
Suppose that $\mathcal{P}$ is a connected, unital operad in $\mathbb{k}$-vector spaces (or complexes), i.e. $\mathcal{P}(1)=\mathbb{k}$ and the unit map for $\mathcal{P}$ is the identity. One may form ...
1
vote
0
answers
93
views
Spectral sequences associated to cohomologies of simplicial type and derived-functor type: Proof of convergence
Assume I have two cohomology theories $\mathrm{\tilde{H}^{*}}$ and $\mathrm{H^{*}}$, the latter being defined over a Grothendieck site $X$ as the derived functor of some left-exact covariant functor $\...
- 1,805
7
votes
1
answer
411
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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
7
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1
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812
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How to prove that topological Hochschild homology of a smooth proper stable k-linear infinity category is dualizable?
Let $k$ be a perfect field of characteristic $p$. I heard that the Topological Hochschild homology of a smooth proper stable infinity category (or dg-category) is dualizable as a THH(k)-module ...
3
votes
0
answers
107
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Inverse limit of chains of Eilenberg Mac Lane spaces
Let $... \to G_2 \to G_1$ an inverse system of abelian groups with inverse limit $G$, let $n \geq 2$ and $F$ a field. The induced inverse system $$... \to C_*(K(G_2,n);F) \to C_*(K(G_1,n);F) \ (*)$$
...
- 1,361
0
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0
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378
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Isomorphism of invariants and coinvariants over a field
Let $G$ be a finite group with normal subgroup $N$ acting on a vector space $V$ over a field $k$ in which the order of $N$ is invertible. Denote $H:=G/N$. The composite map $V^N \to V \to V_N$ and $\...
- 617
3
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1
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358
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Geometric interpretation of shuffle product
Let $A=k\mathbb \Pi$ be the group algebra of an abelian group $\Pi$ and let $B(A)=\bigoplus_{k=0}^\infty\,B^k(A)$ be the unnormalized bar complex of $A$ with generators $[a_0,\dots,a_k] \in B^k(A)=A^{\...
- 1,813
4
votes
1
answer
218
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Poincaré dual of the Alexander dual of the fundamental class of a knot is given by a Seifert surface
Let $K\subset S^3$ be an oriented knot and let $F:\overline{B^2}\times K\rightarrow S^3$ be a thickening with self linking number $0$. I will denote $F(B^2\times K)$ by $(B^2\times K)$ for simplicity. ...
7
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1
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353
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Does the category of cosheaves have enough projectives?
Given a general topological space $X$ does the category $\mathbf{coShv}(X,\mathbf{Mod}_R)$ have enough projectives ? I know that under some conditions this is true, for example if $X$ is a cell ...
- 173
6
votes
1
answer
316
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Groups with unusual cohomological dimension of direct product
$\DeclareMathOperator\cd{cd}$Are there any known examples of non-free groups with a property that $\cd(G)+1 = \cd(G \times G)$, or, less restrictive, $G, H$ with $\cd \neq 1, \infty$ such that $\cd(H)+...
- 4,600
8
votes
1
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414
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Augmentation ideal of a free group
If $F$ is a free group then it has cohomological dimension one, which implies that the augmentation ideal $IF=\operatorname{ker}(\epsilon:\mathbb{Z}G\to \mathbb{Z})$ of its group ring is a projective $...
- 35.9k
6
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1
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327
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Steenrod squares in terms of chain maps
$\DeclareMathOperator\Sq{Sq}$The Steenrod squares $\Sq^i: H^n({-};\mathbb{F}_2) \to
H^{n+i}({-};\mathbb{F}_2)$ are fundamental cohomological
operations. By the Yoneda lemma, they induce a map between ...
- 5,230
10
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1
answer
418
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Are all degree-1 cohomology operations Bocksteins?
I'm interested in cohomology operations (in ordinary cohomology)
$$H^i(-, G)\rightarrow H^{i+1}(-, H)\;,$$
that is, elements of
$$H^{i+1}(K(G, i), H)\;.$$
I know that $K(G, 1)=BG$, so for $i=1$, those ...
- 3,001
4
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0
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188
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Multi-variable cohomology operations
Intuitively, cohomology operations are ways to locally compute a cocycle $\alpha\in H^i(X, G)$ from any cocycle $\beta\in H^j(X, H)$. Formally, they are in one-to-one correspondence with homotopy ...
- 3,001
3
votes
1
answer
518
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
2
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0
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221
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Do the polyhedral homologies of a polyhedron coincide with the polyhedral homologies of its subdivision?
Definition. A convex polytope is a compact finite intersection of hyperplanes in $\mathbb{R}^n$
Definition. The polycomplex is the following data set:
a set of convex polytopes, closed under ...
- 6,962
7
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0
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439
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Transfer of E-infinity algebra structures
Skip to the bottom for my questions, first some discussion:
It is a celebrated theorem of Kadeišvili that $A_{\infty}$-algebra structures can be transferred along homotopy equivalences so that the ...
- 561
5
votes
0
answers
290
views
About the left adjoint of $f^*$
In lots of different cases (Verdier duality, Grothendieck duality, étale cohomology, ...) the very existence of a (right) adjoint to the sheaf functor $f_!$ gives useful information. (I'm going to ...
- 711
0
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0
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260
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Another definition of singular homology
The singular homology is defined via standard simplex. Now if I propose another definition of singular homology groups, based on arbitrary simplex, as follows:
Let $X$ be a topological space. A $n$-...
- 781
9
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2
answers
1k
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Hodge dual of de Rham cohomology and singular cohomology
We know that the de Rham cohomology is isomorphic to the singular cohomology, does the Hodge dual of differential forms induce a dual operation on de Rham cohomology, hence also on singular cohomology?...
- 10.5k
0
votes
1
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284
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Künneth formula and induced map in homologies
Let $X,Y,Z$ be smooth connected manifolds and $f \colon X \times Y \rightarrow Z$ a smooth map. Suppose that we have $H_{*}(X \times Y; \mathbb{Z})$ is isomorphic to $\bigoplus_{p+q=*}(H_{p}(X; \...
- 369
7
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0
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302
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The role of spectral Lie algebras and twisting for operads in spectra
In the theory of differential graded (co)operads, the notion of twisting is ubiquitous. The fundamental notion is the twisting map from a cooperad $C$ to an operad $P$. It is defined as a Maurer-...
- 5,839
7
votes
2
answers
321
views
Indexing categories of derivators
It is not clear to me the role of the domain and target in the definition of prederivators.
For instance, the classical references put the domain as $\mathit{Dia}$, others as $\mathit{Cat}$ itself.
...
- 894
4
votes
1
answer
592
views
Six functor formalism for quasi-coherent $D$-modules
Let $X$ be a smooth scheme over a field $k$ and let $\mathsf{D}_{\text{qc}}(\mathcal{D}_X)$ be the full subcategory of $\mathsf{D}(\mathcal{D}_X\mathsf{-Mod})$ composed of the complexes of $\mathcal{D}...
- 711
6
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1
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425
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Why does $p_*p^! A$ deserve to be called homology with coefficients in $A$?
Let $p:X\to S$ be the unique map from a (locally compact) topological space $X$ to a point. Since $\underline{\hom}(\underline{\mathbb{Z}},-)$ is the identity functor, we have that $\Gamma(X,-)=\hom(\...
- 711
1
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0
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222
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Cohomology spectral sequence of a CW complex filtered by its skeletons
Let $X$ be a CW complex. Suppose, $$\emptyset\subset X^0\subset X^1\subset \cdots \subset X^p\subset \cdots \subset X= \bigcup_{i=0}^{\infty} X^i,$$
is a filtration of $X$ by its skeletons $X^i$. Now ...
- 191
2
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0
answers
371
views
How to deduce Künneth from its relative version (in cohomology of sheaves)
Let $p:X\to S$ and $q:Y\to S$ be morphisms of "spaces" over $S$. We have an isomorphism
$$f_!(M\boxtimes N)=p_! M\otimes q_!N$$
in the derived category of "sheaves" over $S$, where ...
- 711
3
votes
0
answers
261
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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
13
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0
answers
864
views
A step in Toda's computation of a Cotor
I am trying to understand a proof from Toda's paper Cohomology of classifying spaces. The step I am stuck on is at page 96. Here is the setup.
We work with cohomology with $\mathbb{F}_2$ coefficients. ...
- 31
4
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0
answers
170
views
Relation between $Tor_{C^*(BG;K)}(K,K) $ and $K^G$?
Let $K$ be an algebraically closed field and $G$ a group.
Given a dg-algebra $A$, a left $A$-module $M$ and a right $A$-module $N$
let $Tor_A(M,N)$ denote the homology of the derived tensor product $M ...
- 1,361
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
3
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0
answers
133
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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
3
votes
0
answers
101
views
Geometric filtration for Eilenberg-Moore spectral sequence
I'm reading the paper by Eilenberg-Moore (https://link.springer.com/content/pdf/10.1007/BF02564371.pdf) about the Eilenberg-Moore spectral sequence.
In section 11, they introduce the notion of ...
- 449
5
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0
answers
148
views
Negative cyclic homology of the group algebra of discrete groups
I am looking for a reference for the calculation of the negative cyclic homology of the group algebra $\mathbb{K}[\Gamma]$ of a discrete group $\Gamma$ over a field $\mathbb{K}$ of characteristic 0. (...
- 173
4
votes
1
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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
4
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0
answers
158
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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
3
votes
0
answers
111
views
"Boundaries" in Free Simplicial Monoids
I suspect that this has been addressed somewhere already, but I cannot find anything. Let $\hat{\Delta}$ denote simplicial sets, and $Mon\hat{\Delta}$ simplicial monoids. There is a forgetful functor $...
- 713
5
votes
1
answer
187
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Homology of the free loop space of generalized flag varieties
Is it known whether for a generalized complex flag variety $X$ (that is, $G/P$ for a complex semisimple Lie group $G$ and a parabolic $P$), the homology of the free loop space $H_*(\Lambda X, \mathbb{...
- 1,677
7
votes
1
answer
397
views
A set theoretic question arising from trying to understand a sheaf cohomology question
I'm trying to understand the footnote to Example 5.3 in Wiegand - Sheaf cohomology of locally compact totally disconnected spaces which is about constructing a locally compact Hausdorff and totally ...
- 38.6k
7
votes
0
answers
270
views
Differentials in spectral sequences and Massey products
Consider a multiplicative spectral sequence such as the cohomological Serre spectral. It is known that differentials will satisfy a Leibniz rule. Is there a clean statement involving differentials and ...
- 965
4
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1
answer
497
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Twisted cochain as a model for universal cover
Let $X$ be a pointed connected cw-complex and $C_{\ast}(X)$ the singular chain complex associated to $X$.
Let denote $G=\pi_{1}(X)$ and $\tilde{X}$ the universal covering space for $X$.
As far as I ...
- 855
2
votes
1
answer
283
views
Cohomology of a simplicial abelian group $X_\bullet$, where $S_n$ acts on $X_n$
Let FinCar denote the category whose objects are the finite cardinal numbers $[n]=\{0,\dots, n\}$ and whose morphisms are all functions between them, and let $X$ be a a contravariant functor from ...
4
votes
0
answers
220
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Rational cohomology cohomology of $p$-adic analytic groups
It is a result of Lazard that given $G$ a compact $p$-adic analytic group then we have an isomorphism
\begin{equation} H^*(G; \mathbb{Q}_p) \cong H^*(T_eG; \mathbb{Q}_p) \end{equation}
where $T_eG$ is ...
- 767
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
0
votes
0
answers
278
views
Homology of a closed $3$-manifold with balls removed
This question has been posted on MSE with no answers.
Let $M^3$ be a closed, connected and orientable smooth $3$-manifold and let $\mathring{M}$ denote the manifold $M$ with $n$ disjoint open balls $...
- 2,095
5
votes
1
answer
416
views
triviality of homology with local coefficients
Let $X$ be a manifold or a CW-complex.
Let
$\pi: \tilde X\longrightarrow X$
be a covering map.
Let $\pi_1(X)$ be the fundamental group of $X$ and let $\rho: \pi_1(X)\longrightarrow O(n)$ be an ...
- 1,990
2
votes
1
answer
275
views
can the actions of fundamental groups annihilate homology?
Let $X$ be a path-connected manifold (or a CW complex).
Let $\pi_1(X)$ be the fundamental group of $X$.
Let $\pi: \tilde X\longrightarrow X$ be a covering map.
For each $m\geq 0$, let $C_m(\tilde X)$ ...
- 1,990
4
votes
1
answer
287
views
Conjugation action on relative homology
Let $G$ be a group and $K$ be a subgroup. Suppose $g \in G$ commutes with every element of $K$. Is it true that conjugation by $g$ will act trivially on $H_*(G,K)$?
- 965
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