Tagged Questions
7
votes
1answer
275 views
How calculate the De Rham cohomology group of $3$-torus: $T^3$?
How do I calculate the De Rham cohomology group of the $3$-torus $T^3$? Here $T^3=S^1 \times S^1 \times S^1 $.
Using the Mayer-Vietoris sequence, I can show that $\dim H_3(T^3)=\dim H_0(T^3)=1$. But ...
4
votes
1answer
117 views
Highest DeRahm Cohomology
Let $X$ be a $C^\infty$ manifold, compact oriented and connected of dimension $n$. How do you prove that the integration map $$\int_X: \omega \mapsto \int_X \omega $$ from $H^n_{DR}(X)$ to ...
7
votes
1answer
177 views
Confusion on Cech cohomology
From Harvard math qualification exam, 1990.
Let $X$ be a smooth manifold with an open cover $N<\infty$ sets $\{B_{n}\}^{N}_{1}$ which are contractible. Assume that $$\pi_{0}(B_{n}\cap B_{m})\le ...
2
votes
0answers
275 views
Homology and cohomology: why does Poincaré duality fail for domains with boundary?
Poincaré duality says that for a compact, orientable manifold without boundary the $k$th and $(n-k)$th homology groups are isomorphic.
For domains with boundary, it's easy to construct examples where ...
4
votes
2answers
539 views
Cohomology of complex projective plane
How can I compute Cohomology of complex projective plane $CP^2$?
Any magic like the one here?
13
votes
1answer
755 views
Cohomology of projective plane
How I can compute cohomology de Rham of the projective plane $P^{2}(\mathbb{R})$ using Mayer vietoris or any other methods?
7
votes
2answers
239 views
Manifold with 3 nondegenerate critical points
Suppose $M$ is a n-dimensional (compact) manifold and $f$ is a differentiable function with exactly three (non-degenerate) critical points. Then one can show, using Morse theory, that $M$ is ...
