2
votes
0answers
129 views

Construction of generalized Eilenberg-MacLane spaces

The Eilenberg-MacLane spaces $K(G,q)$ are readily generalized to study cohomology with local coefficients.The generalized Eilenberg-MacLane space $K_{\pi}(G,q)$ are spaces with only two nnvanishing ...
4
votes
2answers
559 views

Why Cech cohomology does not compute sheaf cohomology on an open annulus

Let $A=\{z\in\mathbf{C}:1/2<|z|<1\}$ be an open annulus. Let us cover $A$ by 3 open sets: $U_0,U_1$ and $U_2$ which we assume to be all homeomorphic to a 2 dimensional open disc. Moreover, we ...
3
votes
1answer
246 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 ...
3
votes
1answer
176 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 ...
8
votes
1answer
518 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 ...
6
votes
0answers
410 views

Eilenberg-Steenrod axioms of sheaf cohomology

Cohomology of a space is often defined axiomatically: a cohomology theory is a functor from pairs of spaces to abelian groups satisfying the Eilenberg-Steenrod axioms. Is there a similar ...
1
vote
0answers
66 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 ...
2
votes
2answers
504 views

Top cohomology detecting compactness

I am looking for a reference for the fact that the top cohomology $H^n(X;A)$ of an $n$-dimensional manifold $X$ is non-trivial precisely when $X$ is compact. I tried to ask this question on ...
2
votes
0answers
296 views

Weil Kostant Integrality Result as Stated by Brylisnki

I'm reading through Brylinski's "Loop Spaces, Characteristic Classes and Geometric Quantization" and I am stuck on a piece of Theorem 2.2.15, which asserts that If $K$ is a closed complex-valued ...
4
votes
1answer
322 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 ...
7
votes
0answers
274 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 ...