All Questions
Tagged with cohomology differential-topology
34
questions
12
votes
0
answers
288
views
Is there a differential form which corresponds to an eigenvalue of the homomorphism in cohomology?
Let $M$ be a closed manifold and $f:M\to M$ be a diffeomorphism. Suppose the homomorphism $f^*:H^k(M;\mathbb R)\to H^k(M;\mathbb R)$ has an eigenvalue $\lambda\in\mathbb{R}$. Note that $\lambda$ is ...
- 1,085
5
votes
0
answers
147
views
Representing some odd multiples of integral homology classes by embedded submanifolds
Consider an $m$-dimensional compact closed orientable smooth manifold $M$ and an $n$-dimensional integral homology class $[\Sigma]$ on $M$, with $1 \le n \le m-1$. Then does there exist an odd integer ...
- 467
13
votes
1
answer
490
views
Impossibility of realizing codimension 1 homology classes by embedded non-orientable hypersurfaces
Suppose we have an $n+1$-dimensional compact closed oriented manifold $M$ and an $n$-dimensional integral homology class $[\Sigma]\in H_n(M,\mathbb{Z})$ on $M.$ Then is it true that $[\Sigma]$ mod $2$ ...
- 467
13
votes
1
answer
353
views
Realizing integral homology classes on non-orientable manifolds by embedded orientable submanifolds
Let $M^m$ denote a compact, non-orientable smooth manifold and $\nu$ an integral homology class of dimension $n$. I am interested in understanding the representability of $\nu$ by embedded, orientable ...
- 467
2
votes
0
answers
83
views
Turning cocycles in cobordism into an inclusion or a fibering
By the classical Pontryagin-Thom construction, we know that the cobordism group $\Omega^n_U(X)$ is represented by cocyles
$$ M\hookrightarrow X\times \mathbb{R}^{2k}\rightarrow X,$$
where $M$ is a ...
- 171
20
votes
2
answers
803
views
Integral homology classes that can be represented by immersed submanifolds but not embedded submanifolds
Let $M$ be an $m$-dimensional compact closed smooth manifold and $z\in H_n(M,\mathbb{Z})$ an $n$-dimensional integral homology class, with $m>n.$ Does there exist a pair of $M$ and $z$ so that $z$ ...
- 467
5
votes
1
answer
1k
views
Basic question on the de Rham theorem
There is a modern nice proof of the de Rham theorem based on sheaf theory.
The de Rham theorem says that for a smooth manifold $M$ there is a canonical isomorphism
$$H^i_{dR}(M,\mathbb{R})\simeq H^i_{...
- 21.3k
6
votes
0
answers
271
views
Regarding homology of fiber bundle
Let $f: X\to Y$ be a smooth map between smooth manifolds, both connected. Let $Y=\cup_{i=1}^k Y_i$ be a finite union of disjoint locally closed submanifold $Y_i$ such that $f^{-1}(Y_i)\to Y_i$ is ...
- 585
1
vote
0
answers
105
views
Modular cycles?
It is well known that cocycles (differential forms) and cycles share many properties through duality (e.g., de Rham). I've been reading about modular forms recently and I came with a very naive ...
- 211
3
votes
1
answer
249
views
Relation between cohomological dimensions of manifolds
$\DeclareMathOperator\Ch{Ch}$Let $M$ be a connected manifold of finite type. We denote $\Ch_{\mathbb{Q}}(M),$ $\Ch_{\mathbb{Z}}(M)$ and $\Ch_{\mathbb{\pm}\mathbb{Z}}(M)$ by cohomological dimensions of ...
- 173
23
votes
3
answers
2k
views
Next steps for a Morse theory enthusiast?
I don't know if this question is really appropriate for MO, but here goes: I quite like Morse theory and would like to know what further directions I can go in, but as a complete non-expert, I'm ...
3
votes
0
answers
106
views
Reference request: $K$-theoretic wrong-way map for a boundary inclusion
Let $W$ be a compact manifold with boundary. Let $i:\partial W\hookrightarrow W$ be the natural inclusion. We have a long exact sequence in complex $K$-theory:
$$\ldots\to K^*(\partial W)\xrightarrow{...
- 1,901
4
votes
1
answer
221
views
A differential form whose support is in a tubular neighborhood of $T^k\times \{0\}^{n-k}\subset T^n$
Let $\alpha$ be a differential form on the torus $T^n$ whose support $\mathrm{supp}(\alpha)$ is contained in a small neighborhood of the subtorus $T^k\equiv T^k\times \{0\}^{n-k}$.
Question:
Suppose $...
- 2,739
4
votes
0
answers
294
views
Holomorphic covers pulling back the volume form to any integer multiple
Let $M$ be a closed connected complex manifold with $\mathrm{dim}\:M=n$. Can there exist holomorphic covering maps $\phi_k:M\to M$ for all integers $k\geq 1$ such that $\phi_k^*:H^n(M, \mathbb{Z})\to ...
user145520
5
votes
0
answers
175
views
Clarify formula for Steifel-Whitney (Poincaré dual) homology classes in a barycentric subdivision?
Let $X$ be a triangulated manifold of dimension $n$. Let $[W_{n-p}] \in H_{n-p}(X,\mathbb{Z}_2)$, be the homology class that's Poincaré dual to the $p$-th Stiefel-Whitney class $[w_p] \in H^p(X,\...
- 535