All Questions
Tagged with at.algebraic-topology sheaf-theory
111 questions
98
votes
10
answers
14k
views
equivalence of Grothendieck-style versus Cech-style sheaf cohomology
Given a topological space $X$, we can define the sheaf cohomology of $X$ in
I. the Grothendieck style (as the right derived functor of the global sections functor $\Gamma(X,-)$)
or
II. the Čech ...
- 1,871
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
33
votes
4
answers
6k
views
What (if anything) happened to Intersection Homology?
In the early 1990's, Gil Kalai introduced me to a very interesting generalization of homology theory called intersection homology, which existed for like 10 years back then I believe. Defined ...
- 6,734
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
26
votes
2
answers
2k
views
Loop Spaces as Generalized Smooth spaces or as Infinite dimensional Manifolds?
There are two ways to define smooth mapping spaces and I want to know how they compare.
Let's take the concrete special case of free loops spaces. I think this is the most studied example so will ...
- 27.5k
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
23
votes
4
answers
5k
views
De Rham decomposition theorem, generalisations and good references
De Rham decomposition theorem states that every simply-connected Riemannian manifold $M$
that admits complementary sub-bundles $T'(M)$ and $T''(M)$ of its tangent bundle parallel with respect to the ...
- 28.9k
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
18
votes
4
answers
1k
views
Cohomology of a sheaf of functions locally constant along a foliation
Take a smooth manifold $M^n$ with a smooth foliation $F$. Consider the sheaf $\cal F$ of $C^{\infty}$ functions on $M^n$, locally constant along the foliation $F$. What is known about Chech cohomology ...
- 28.9k
18
votes
0
answers
462
views
Is there a model category describing shape theory?
Is there a nice model structure on some category of topological spaces compatible with shape theory? In particular, weak equivalences should induce isomorphisms on sheaf cohomology.
As an example, ...
- 6,813
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, ...
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
15
votes
1
answer
2k
views
How to motivate constructible sheaves
I'm writing some notes for some students which just finished a first course in scheme theory. There I would like to talk about constructible sheaves, but I found it hard to give a compelling ...
- 721
14
votes
3
answers
2k
views
Recommendations for getting into sheaves with emphasis on differential geometry and algebraic topology
I want to study the theory of sheaves from a categorical point of view with an emphasis on applications in algebraic topology and differential geometry and I'm looking for a good introductory book to ...
- 141
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
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
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
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
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
11
votes
1
answer
866
views
Serre spectral sequence for de Rham cohomology
Suppose we a given a fibration of manifolds $p\colon E\to M$ with a path connected fiber $F$ and simply connected $M$, then we have the Serre spectral sequence with
$$
E_2^{p,q} = H^p(M,\underline{H^...
- 2,305
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
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
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
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
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
9
votes
2
answers
1k
views
Is there a description of sheaf cohomology in algebraic-topological terms?
Is there a description of of sheaf cohomology for the sheaf of sections of a continuous function in terms of common constructions in Algebraic Topology?
In more detail: Any sheaf on a space X can be ...
- 5,309
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
9
votes
0
answers
308
views
Refinement of hypercovers by ordinary covers
I am asking for references and discussions of statements of the form
Every bounded hypercover can be refined by an ordinary cover
By "bounded" I mean "finite height". E.g., are ...
- 4,519
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
8
votes
2
answers
4k
views
Sheaf cohomology question
For a topological space $X$ and a sheaf of abelian groups $F$ on it, sheaf cohomology $H^n(X,F)$ is
defined.
Singular cohomology of $X$ can be expressed as sheaf cohomology if $X$ is locally ...
- 221
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
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
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}))
$$
between the category of finite ...
- 2,782
8
votes
1
answer
1k
views
Relative version of de Rham cohomology with local coefficients
Given a vector bundle $E \to M$ with connection $\nabla$, we get a twisted de Rham sequence using the exterior covariant derivative:
$$\mathcal{E} \xrightarrow{d^\nabla=\nabla} \Omega^1_M \otimes_{\...
- 6,025
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
8
votes
0
answers
681
views
Stalks of limit sheaves
Let $\{\mathcal{F}_i\}_{i\in \mathbb{N}}$ be an inverse system of sheaves of abelian groups on a space $X$. Then for any $x\in X$ we have a natural map
$$\left(\lim_i \mathcal{F}_i\right)_x\rightarrow ...
- 4,418
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
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
7
votes
2
answers
3k
views
Grothendieck spectral sequence and Mayer-Vietoris sequence
Suppose $U'\cup U''=X$ is an open cover $U$ of a topological space $X$ and $F$ is a sheaf on $X$ with values in abelian groups. There is a special instance of the Grothendieck spectral sequence ...
- 319
7
votes
1
answer
490
views
Equivariant perverse sheaves and orbit stratification
Let $X$ be a complex algebraic variety with an action of a connected algebraic group $G$.
The forgetful functor from the category of $G$-equivariant perverse sheaves on $X$ to the category of perverse ...
- 3,446
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
7
votes
2
answers
614
views
Explicit description of exponentials of étalé spaces
It is well known that the category $\mathit{Sh}(X)$ of sheaves of sets on a topological space $ X $ is a topos.
On the other hand, there exists a natural equivalence of categories between $\mathit{Sh}(...
- 751
7
votes
1
answer
353
views
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 ...
- 213
7
votes
0
answers
362
views
What is a morphism of ∞-sites?
Recall that a morphism of sites
is a covering-flat functor
that preserves covering families.
Morphisms of sites can be identified with those
geometric morphisms of induced toposes
for which the ...
- 37.8k
6
votes
1
answer
911
views
Putting sheaves to work for algebraic topology?
This is cross-posted from math.se after receiving points and no answers. I apologise if this question is too basic for MathOverflow.
I'm refreshing my memory of ...
- 1,221
6
votes
2
answers
566
views
commutativity of restriction and Gysin morphisms in a cartesian square
Let $X, Y, Z$ be compact topological manifolds $f: Y \to X, g: Z \to X$ be embeddings of submanifolds meeting transversely and let $W = Y \times_X Z$:
$$
\begin{array}{ccc}
Y & \to^f & X \\...
- 4,070
6
votes
1
answer
359
views
Homotopy cosheaf?
Let $C$ be a site, $\mathbf{S}$ some ($\infty$-? homotopy?) category of spaces.
Question. What do you call a (covariant!) functor $F:C\to \mathbf{S}$ enjoying the following property: for every ...
- 16.1k
6
votes
1
answer
328
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
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