All Questions
Tagged with at.algebraic-topology sheaf-theory
111 questions
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
3
votes
0
answers
341
views
Descent of singular cohomology
When proving that singular cohomology of an appropriate space $X$ equals sheaf cohomology of $X$ with "values" (does one say that?) in the sheaf $\mathbb{Z}_X$ of locally constant functions, the ...
- 319
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
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
5
votes
2
answers
987
views
The equivalence of category of equivariant sheaves on principal bundle and category of sheaves on base space
Let $\pi:P\to B$ is a $G$-principal bundle, which means $G$ acts on $P$ freely and $\pi$ is a locally trivial fibration. Here is a well-known theorem:
THeorem: The inverse image functor $\pi^{*}$ ...
- 1,457
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
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
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
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
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
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