All Questions
25 questions
6
votes
0
answers
223
views
Under what generality are the compactly supported singular and sheaf cohomologies equal?
Edit: I have since resolved my question.
If X is locally compact Hausdorff in addition to being cohomologically locally contractible with coefficients in $A$ - eg it is a manifold or an open subset of ...
- 1,021
3
votes
1
answer
240
views
Cohomology of the complement of a subvariety
Let $X$ be a complex manifold, $Y\subset X$ a subvariety, and $U:=X\setminus Y$ of codimension $d$. It is well known that the restriction map
$$
H^i(X,\mathbb Q)\to H^i(U,\mathbb Q)
$$
is an ...
- 178
2
votes
0
answers
148
views
Equivalence of cohomology with compact support
Let $𝑋$ be a connected CW complex, $𝜌:\pi_1(𝑋)→\mathrm{Aut}(G)$ a representation, and $G_\rho$ the associated sheaf. It follows from here, that the following two cohomologies are isomorphic.
(1)The ...
- 241
6
votes
1
answer
326
views
Spectral sequence generalizing Čech cohomology
Let $X$ be a 'nice' topological space. Let $\left(U_i\right)_{i\in I}$ be a finite open covering of $X$. Let $\mathcal{F}$ be a sheaf of abelian groups.
For a subset $A\subset I$ denote $$U_A:=\cap_{...
- 21.8k
11
votes
1
answer
406
views
Resolutions of unbounded complexes: Condition ($\ast$) in Spaltenstein's paper
In the paper "Resolutions of unbounded complexes" (Compositio Math., vol. 65, no. 2, pp. 121-154) N. Spaltenstein generalizes the 6 functor formalism to unbounded complexes of sheaves over ...
- 23.5k
6
votes
1
answer
327
views
Topology on cohomology of a sheaf of topological groups
Let $X$ be a topological space and $\mathcal{F}$ be a sheaf of commutative topological groups on $X$. I am interested in the following question:
Is there a natural way to introduce topology on $H^i(X,...
- 1,170
2
votes
1
answer
501
views
Higher direct image with compact support of a constant sheaf
Let $f: X \to Y$ be a locally trivial fibration between locally compact spaces with fiber $F$. It is well known that for a constant sheaf $A_X$ on $X$, the higher direct images $R^n f_* A_X$ are ...
3
votes
1
answer
225
views
Subspace inclusion with non-vanishing higher direct images
I'm looking for concrete topological intuition for the derived pushforward.
Let $f:X\to Y$ be a continuous map. The derived pushforward $\mathbf Rf_\ast$ takes a sheaf $F$ to the sheafification of ...
- 10.5k
2
votes
1
answer
600
views
Pushforward in Compactly Supported Cohomology
Suppose $X,Y$ are locally compact Hausdorff spaces and $f:X\to Y$ is a topological submersion of relative dimension $n$. By this we mean that for all points $x\in X$, there exists an open neighborhood ...
- 1,761
2
votes
2
answers
219
views
Continuous map with homeomorphic fibers whose associated $H^{k}_c$ sheaf is not a local system?
Let $ f: X \to Y$ be a continuous map between connected manifolds s.t. for all $y \in Y$ the fiber $f^{-1}(y)$ is homeomorphic to some fixed connected manifold $Z$.
Let $k$ be a ring and for every $...
- 7,789
9
votes
0
answers
570
views
In terms of sheaf cohomology, what does Bott & Tu's relative de Rham cohomology $H^\bullet(f)$ compute for $f: S \to M$ a smooth map?
Given a map $f: S \to M$ of smooth manifolds, Bott & Tu define on page 78 a complex by $\Omega^q(f)=\Omega^q(M) \oplus \Omega^{q-1}(S)$ and $d(\omega, \theta)=(d\omega, f^*\omega - d\theta)$ where ...
- 6,025
3
votes
0
answers
163
views
Question about the precise statement of Leray spectral sequences and a simple example
On Bott's paper "Homogeneous vector bundles" there is the following statement of Leray spectral sequence:
Let $X$ and $Y$ be paracompact and locally compact spaces and $f : Y \to X$ be a proper map....
1
vote
0
answers
205
views
Sheaves and isomorphisms with chain complex of singular chains (Sheaf Theory, Bredon)
Let $\Delta_{\ast}(X,A)$ (resp. $\Delta_{\ast}^c(X,A)$) be the chain complex of locally finite (resp. finite) singular chains of $X$ modulo those chains in $A$.
How to show that the homomorphism of ...
- 11
8
votes
1
answer
1k
views
Relative version of de Rham cohomology with local coefficients
Given a vector bundle $E \to M$ with connection $\nabla$, we get a twisted de Rham sequence using the exterior covariant derivative:
$$\mathcal{E} \xrightarrow{d^\nabla=\nabla} \Omega^1_M \otimes_{\...
- 6,025
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
4
votes
2
answers
315
views
Equivalence of different cohomology groups
Let $X$ be a topological space (may be assumed to be locally compact). Let $A$ be either a field or $\mathbb{Z}$. One can consider various cohomology groups:
(1) singular cohomology $H_{sing}^*(X,A)$;...
- 21.8k
4
votes
1
answer
1k
views
On push-forward of the constant sheaf for fibrations
Let $f\colon E\to B$ be a fiber bundle with a connected fiber $F$, $f$ is proper. Let $\underline{\mathbb{C}}_E$ be the constant sheaf on $E$. Let $f_*(\underline{\mathbb{C}}_E)$ denote its direct ...
- 21.8k
5
votes
0
answers
377
views
Push forward of the constant sheaf for a Serre's fibration
Let $f\colon X\to Y$ be a proper continuous map of topological spaces which is a Serre's fibration. $X$ and $Y$ may be assumed to be locally compact, $Y$ is connected topological manifold of finite ...
- 21.8k
5
votes
2
answers
331
views
Sheaf cohomology on non paracompact topological spaces
I have some confusion on the subject of sheaf cohomology on non-paracompact topological spaces, i hope you can help me.
My reference is Godement's book "Topologie algebrique et theorie dex faisceaux"....
- 609
4
votes
1
answer
601
views
Explicit examples presheaves associated to higher direct images which fail to be sheaves
So I would like to have a few simple examples where the presheaf associated to higher direct image of sheaf fails to be sheaf. So I'm looking for two (natural and simple) topological spaces $X$ and $Y$...
- 7,609
4
votes
1
answer
478
views
Euler Characteristic of Coverings via Sheaf Theory
Let $X$ be a nice space (compact $CW$-complex or triangulated space, compact manifold, whatever works),
$f:Y\to X$ be a finite covering of degree $n$, and $\chi(X)$ be the euler characteristic.
By the ...
- 2,554
10
votes
1
answer
2k
views
When does the sheaf cohomology of a topological space vanish?
The question is in the title. A more precise formulation is:
Let $X$ be a topological space. When does $H^i(X,F) = 0$ for all $i > 0$ and all abelian sheaves $F$ on $X$?
The obvious example is a ...
- 281
1
vote
0
answers
101
views
How can one compute the cohomology of $i'^*C$, for $i':\mathbb{A}^{N-1}\setminus \{0\}\to \mathbb{A}^{N}\setminus \{0\} $?
For an (etale or 'topological', constructible bounded) complex of sheaves $C$ on $X'=\mathbb{A}^{N}\setminus \{0\} $, $i'$ being the embedding $\mathbb{A}^{N-1}\setminus \{0\}\to \mathbb{A}^{N}\...
- 16.9k
5
votes
1
answer
468
views
Derived Equivalence of Sheaves and Homotopy
This question loosely elaborates on an earlier question. It is pretty silly, but I'd like to hear some authoritative answers.
Recall that if $f:S^{\bullet}\to T^{\bullet}$ is a quasi-isomorphism of ...
- 2,684
8
votes
0
answers
370
views
Dualizing complex of the product of two locally compact spaces
Hello!
In the setting of locally compact Hausdorff spaces, I would like to understand the relation between the exterior product ${\mathbb D}_X\boxtimes{\mathbb D}_Y$ of the dualizing complexes of two ...
- 2,756