All Questions
Tagged with at.algebraic-topology sheaf-theory
111 questions
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
2
votes
0
answers
397
views
Terminology for "global sections" when sheaf is valued in general category
Let $\mathcal F$ be a sheaf (say on a topological space $X$) valued in some category $\mathcal C$.
What do we call $\mathcal F(X)$?
When $\mathcal C$ is some vaguely linear category (e.g. the ...
- 18.7k
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
9
votes
1
answer
413
views
Stable homotopy category of complexes of sheaves
Let $X$ be an Hausdorff space, which is locally compact and locally connected. Let $R$ be a commutative and unitary ring and let $D(X)$ be the derived category of unbounded complexes of sheaves of $R$-...
- 9,870
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
3
votes
1
answer
342
views
On the notion of conelike stratified (cs-) space
The notion of cs-stratification of a topological space is apparently due to Siebenmann, see also the paper by N. Habegger and L. Saper in the paper "Intersection cohomology of cs-spaces and Zeeman's ...
- 21.8k
5
votes
0
answers
189
views
Constructible sheaves on general stratified spaces
I am not an expert in the field, so my question might be rather standard. Let $X$ be a compact metric space. Assume that $X=\cup_{i=1}^NS_i$ is a finite disjoint union of locally closed topological ...
- 21.8k
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
11
votes
2
answers
413
views
Homotopy property of constructible sheaves on stratified spaces
Let $X$ be a stratified topological space (in my case $X$ is a compact space presented as a finite union of locally closed topological manifolds of finite dimension (strata) such that the closure of ...
- 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
8
votes
1
answer
529
views
Topology on the space of constructible sheaves
Let $X$ be a nice compact topological space with a fixed finite stratification by locally closed topological manifolds. At the beginning one may assume that $X$ is a complex algebraic manifold with ...
- 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
11
votes
3
answers
935
views
Examples of calculating perverse sheaves on algebraic varieties with easy stratification
I have been learning intersection homology and perverse sheaves in the following way. I started by reading the first $7$ chapters of Kirwan and Woolf's book
http://www.amazon.com/Introduction-...
- 123
11
votes
1
answer
335
views
Reference request: sheaves on the site of d-manifolds
I believe I know how to prove the following results. I also know to whom to cite fancy-shmancy results that have these as a very special case. My question is: what are the correct citations for ...
- 54.6k
10
votes
1
answer
612
views
How do the direct and inverse image sheaf functors interact with homotopy?
This is a crosspost of this MSE question.
The direct image sheaf functor $f_\ast$ and inverse image sheaf functor $f^\ast$ (here I mean the usual inverse image sheaf functor often denoted by $f^{-1}$)...
- 10.5k
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
1
vote
1
answer
356
views
Examples of nontrivial local systems in Decomposition Theorem
There is a proper map $f: X \rightarrow Y$ of projective varieties. The Decomposition Theorem of Beilinson–Bernstein–Deligne-Gabber states that
$$Rf∗IC_X \cong \oplus_a IC_{\bar{Y_a}}(L_a)[shifts]$...
- 1,719
8
votes
0
answers
337
views
What's the Hochschild homology of the category of constructible sheaves?
Let $X$ be a manifold. Does the Hochschild homology/cohomology of the category of constructible sheaves on $X$ have a more familiar name?
- 8,723
0
votes
1
answer
351
views
Birkhoff decomposition vanishing of the Chern numbers
Birkhoff decomposition vanishing of the Chern numbers of the holomorphic line bundles of the Birkhoff-Grothendieck decomposition, is some statement I read off in One of Connes papers. Without going ...
user39066
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
4
votes
1
answer
893
views
Holomorphic logarithmic De Rham complex
Let $X$ be a complex variety of dimension $n$ and $D$ a smooth hypersurface.
Let $\Omega_X(logD)^*$ be the holomorphic logarithmic De Rham complex: $\omega\in \Omega_X(logD)^k$ is a form of degree $k$...
13
votes
1
answer
1k
views
Sheaves on Contractible Analytic Spaces
Let $(X,\mathcal{O}_X)$ be a contractible complex analytic space. Suppose that $\mathcal{F}$ is a coherent sheaf of $\mathcal{O}_X$-modules. Can we invoke the fact that $X$ is contractible to conclude,...
- 4,920
17
votes
2
answers
1k
views
Is the site of (smooth) manifolds hypercomplete?
By site of manifolds Man, I mean the category of manifolds (maybe submanifolds to obtain a small category) with continuous maps between them. A Grothendieck topology is given by open covers. Actually, ...
4
votes
0
answers
166
views
Homotopy-theoretic measure of operations on sheaves failing to be sheaves
Here's something I've been wondering about for a few weeks:
Consider a topological space $X$ and a sheaf of rings $\mathscr O_X$ on $X$. Suppose $\mathscr{F}$ and $\mathscr{G}$ are $\mathscr O_X$ ...
user1437
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
14
votes
1
answer
2k
views
Hypercohomology of a complex via Cech cohomology
Let $X$ be a reasonable topological space. If $\mathcal{F}$ is a sheaf of abelian groups then Cech cohomology gives us a method to compute the cohomology groups $H^p(X, \mathcal{F})$ - the main input ...
- 141
16
votes
4
answers
2k
views
Coboundaries and Gluing in Cech Cohomology - Intuition?
I'm trying to develop an intuition for Cech cohomology geometrically, but am currently failing. A lot of people seem to say that the groups $H^n$ measure obstructions to gluing local sections to get ...
- 827
26
votes
1
answer
1k
views
Why are quasitopological spaces needed in sheaf theoretic approaches to the h-principle?
Recently I have been learning more about the h-principle and in particular the methods of "continuous sheaves". In many treatments of this I see people using "quasi-topological spaces" and I am trying ...
- 27.5k
7
votes
1
answer
467
views
Are constructible derived categories invariant up to weak homotopy equivalence?
Let $X$ and $Y$ be two topological spaces and $R$ be commutative ring. Let $D_c^b(X, R)$ and $D_c^b(Y,R)$ be their respective bounded derived categories of constructible sheaves of $R$-modules. I ...
- 121
2
votes
1
answer
812
views
Leray Spectral Sequence
Let $f:X\to Y$ be a smooth map between paracompact differential manifolds $X$ and $Y$.
Let $U$ be an open and dense subset of $Y$. For any $y\in U$, let $f^{-1}(y)=F$
be a generic fiber that is a ...
- 645
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
3
votes
1
answer
893
views
system of local coefficients on X, locally constant sheaves and orientation sheaves
Hi,
I try to understand the orientation sheaves. When searching it in the google, i meet new areas such as local coefficient system and locally constant sheaves. I realize that any system of local ...
- 225
1
vote
1
answer
1k
views
Spectral sequences in Hypercohomology of sheaves (For a complex of acyclic sheaves) - Follow-up to previous question
Alright, this is a follow-up to my previous question (Spectral sequences in Hypercohomology of sheaves), sorry I took so long to reply. Let $X$ be a topological space, let $F^\bullet$ be a cochain ...
- 360
8
votes
2
answers
993
views
Example of a Sheaf (on the site of smooth manifolds) with Nontrivial Cohomology on $\mathbb{R}^n$?
Does such a sheaf of abelian groups exist? If not, is there a reference or a proof? Does such a sheaf of non-abelian groups exist?
I realized recently that while I've taken it for granted that ...
- 1,138
2
votes
1
answer
1k
views
Spectral sequences in Hypercohomology of sheaves
Alright, here I go again, don't know if I'm missing something here but let $X$ be a topological space and let $F^{\bullet}$ be a cochain complex of sheaves, I want to compute the cohomology of this ...
- 360
1
vote
2
answers
1k
views
Hypercohomology of a complex of sheaves that might be acyclic (or might not)
Back again, check this out, let $X$ be a topological space and let $F^{\bullet}$ be a cochain complex of sheaves, I'm trying to compute the cohomology of the complex of global sections of the sheaves
...
- 360
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
2
votes
1
answer
2k
views
Orientation Sheaf and Double Cover
The orientation sheaf of an $n$-manifold $M$ is $\mathcal{O}_n=Sheaf(U\mapsto H_n(M,M-U;\mathbb{Z}))$, with stalks given by $(\mathcal{O}_n)_x = lim H_n(M,M-U)=H_n(M,M-x)=\mathbb{Z}$ (the limit is ...
- 17.5k
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
1
answer
1k
views
Does one need to sheafify when defining the inverse image of a sheaf with respect to an embedding?
This seems to be a rather simple (stupid?:)) question; yet I was not able to find an answer quickly.
For a morphism $f:X\to Y$ of schemes (or topological spaces) and an (etale or topological) sheaf $...
- 16.9k
4
votes
0
answers
382
views
Exercise concerning locally constant presheaves [closed]
Let $\mathscr{F}$ be a presheaf of abelian groups on some topological space $X$. We say that $\mathscr{F}$ is locally constant if there exists an open cover $\mathcal{U}$ of $X$ (i.e. $X=\bigcup_{U\in\...
- 4,902
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
7
votes
2
answers
7k
views
On a proof of the existence of tubular neighborhoods.
Studying analysis on manifolds, I have found, in the proof of the existence of tubular neighborhoods, a reference to theorem 3.1.2 in "Topologie algebrique et theorie des faisceaux" of Godement.
...
- 4,306
1
vote
2
answers
530
views
Why is this a local constant sheaf
If a group $G$ acts on a topological space $M$, and a representation of $G$ on a vector space $V$, why $M \times_G V$ is a local constant sheaf over $M/G$?
- 1,499
8
votes
2
answers
483
views
Swan-like theorem and covering spaces
Let $X$ be a finite CW complex. Swan's theorem provide an equivalence
$$
{\rm Vec}(X)\xrightarrow\sim{\rm ProjMod}(\mathop{\rm hom}\nolimits_{\rm Top}(X,\mathbb{R}))
$$
between the category of finite ...
- 2,782
10
votes
0
answers
484
views
Applications of sheaf theory to the computation of invariants of LS-category type
I would like to know if sheaf theory can be applied to a particular class of questions in topology.
The Schwarz genus (also known as sectional category) of a continuous map $p\colon\thinspace E\to ...
- 35.9k
3
votes
2
answers
2k
views
Relation between sheaf and group cohomology
Let $E=\mathbb{C}/L$ be an elliptic curve. Then $\mathbb{C}$ is contractible, and $L$ is the fundamental group of $E$. What's interesting is that we can find the cohomology of $E$, which is the same ...
- 15.4k