All Questions
8 questions
3
votes
0
answers
84
views
de Rham cohomology relative to a closed subset
I am interested whether there exists a versions of de Rham relative cohomology $H^\bullet(M, N)$ in which $N$ does not need to be a manifold. I know there are two main definitions in literature as ...
0
votes
0
answers
85
views
Existence of covering space with trivial pullback map on $H^1$
I have seen somewhere the following claim (but can't remember where): let $M$ be a connected orientable closed smooth manifold with $b_1(M)=1$, then there exists a connected covering space $p:\tilde{M}...
5
votes
1
answer
246
views
Gysin isomorphism in de Rham cohomology using currents
I'd like to find a reference for the following fact.
First, some background: we can define de Rham cohomology of a smooth manifold $X$ of dimension $d$ using the de Rham complex
$$
\Omega^0_X\to \...
- 2,044
6
votes
1
answer
859
views
De Rham's theorem for top-forms in manifolds with boundary
In page 79 of Bott-Tu, "Differential Forms in Algebraic Topology", they define the relative de Rham theory as follows:
Let $f:S\to M$ be a smooth map. Define the complex $\Omega^*(f)$ by
$$\...
- 101
17
votes
1
answer
1k
views
Direct proof that Chern-Weil theory yields integral classes
Suppose $E$ is a complex vector bundle of rank $n$ on a compact oriented manifold (both assumed smooth). Let $h$ be a Hermitian metric on $E$, and let $A$ be a Hermitian connection on $E$ and $F_A$ ...
- 1,761
2
votes
0
answers
152
views
When are automorphisms of the cohomology ring realized by isometries?
Let $(M,g)$ be a closed smooth Riemannian manifold, and denote by $G$ a closed subgroup of its isometry group. By considering the maps $g^*$ induced by elements $g\in G$ in the (de Rham) cohomology $H^...
- 5,891
7
votes
1
answer
2k
views
What is the scope of validity of Kunneth formula for de Rham?
In books like Bott-Tu or all pdf texts I have found on internet, the Kunneth formula for manifolds $M$ and $N$ and their de Rham cohomology
$$ H^{\bullet}_{dR}(M \times N) \simeq H^{\bullet}_{dR}(M) \...
- 1,346
3
votes
1
answer
577
views
How does one introduce characteristic classes [closed]
How does one introduce, or how were you introduced to characteristic classes?
You can assume that the student is comfortable with principal bundles and connections on principal bundles.
I am not ...