All Questions
Tagged with homology dg.differential-geometry
10 questions
5
votes
1
answer
507
views
Invariance of morse homology, doubt in proof in book "Morse Theory and Floer homology"
I am reading the book "Morse theory and Floer Homology" by Michele Audin and Mihai Damian. Now I am reading the proof of the following theorem.
Link to the statement of the theorem
...
- 53
7
votes
2
answers
2k
views
Is there a theorem showing that de Rham homology is isomorphic to singular homology?
The only exposition of de Rham homology I've found is an appendix to Uranga and Ibanezs book on String Phenomenology. It was brief and gave only basic outline of how to construct this homology.
Now de ...
- 2,369
11
votes
1
answer
580
views
Smallest volume representatives of homology
Given a Riemannian manifold, I have a notion of volume for each of my chains, so it makes sense to ask for a representative of a homology class with the smallest volume. Are there conditions for when ...
- 353
4
votes
0
answers
69
views
Obstructions to symplectically embedding compact manifolds of dimension $4$ or higher
It is known in Li's paper (http://arxiv.org/pdf/0812.4929v1.pdf) that in compact symplectic manifolds $(X^{2n},\omega)$ of dimension at least $2n\geq 4$, an immersed symplectic surface represents a $2$...
- 191
6
votes
1
answer
291
views
Strange problem about triplets of differential forms
Suppose we have the following map:
$$(\Omega^1(\mathbb{R}^n))^3\longrightarrow(\Omega^2(\mathbb{R}^n))^3$$
$$(\alpha,\beta,\gamma)\longmapsto(\mathrm{d}\alpha+\beta\wedge\gamma,\mathrm{d}\beta+\...
- 2,091
16
votes
5
answers
1k
views
Take contraction wrt a vector field twice and define kernel mod image. Does that give anything interesting?
First, we make the following observation: let $X: M \rightarrow TM $ be a vector
field on a smooth manifold. Taking the contraction with respect to $X$ twice gives zero, i.e.
$$ i_X \circ i_{X} =0.$$...
- 3,245
1
vote
1
answer
414
views
When do submanifolds lie in the same homology class? [closed]
Hello,
this may be a trivial question, but I am not very familiar with the topic.
Let (M,g) be a Riemannian Manifold. (In fact, we don't need the metric here.)
What exactly does it take for two k-...
- 329
1
vote
1
answer
388
views
When is there a deRham duality relation between the fundamental class and a top form.?
Hi, everyone:
I am reading a small expository paper on properties of CP2,
in which the intersection form is defined as an integral of
the wedge of two forms $w_1$, $w_2$, and these forms $...
- 361
9
votes
1
answer
3k
views
De Rham homology
Suppose M is an arbitrary smooth manifold and D is its bundle of 1-densities.
On the category of finite-dimensional vector bundles over M and linear differential operators between them
there is a ...
- 37.8k
19
votes
7
answers
6k
views
CW-structures and Morse functions: a reference request
The following is probably well known, but I wasn't able to locate a reference in the literature.
Let $f$ be a Morse function on a smooth compact manifold $M$ without boundary and let $\rho$ be a ...
- 23.5k