All Questions
Tagged with cohomology homology
69
questions
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
4
votes
1
answer
440
views
Cohomology of finite symmetric products of manifolds
Let $M$ be a closed, orientable manifold of dimension $k$. I am looking for results to determine explicitly the (co)homology groups and/or cohomology ring structure (in integer or rational ...
- 506
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
20
votes
2
answers
802
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
2
votes
0
answers
116
views
Triple insersection number of a surface in three-manifolds
I heard something about the triple intersection number $\text{mod}(2)$ (but maybe also $\text{mod}(n)$) of a surface in an orientable three-manifold but I couldn't find a precise definition. My guess ...
- 761
3
votes
3
answers
383
views
Pairing between cohomology and the image of the Hurewicz homomorphism
Let $X$ be a compact manifold of dimension $\geq k$. Denote by
\begin{equation}
h: \pi _k(X) \rightarrow H_k(X,\mathbb{Z})
\end{equation}
be Hurewicz homomorphism and by $\Gamma _k(X)\subset H_k(X,\...
- 761
3
votes
2
answers
640
views
Does the cohomology Bockstein homomorphism map to the homology Bockstein homomorphism under Poincarè duality?
Given a manifold $X$ and short exact sequence of abelian groups
$$
1\rightarrow A_1\overset{\iota}{\rightarrow} A_2\overset{\pi}{\rightarrow} A_3\rightarrow 1
$$
we get the Bockstein map in cohomology ...
- 761
5
votes
1
answer
372
views
Computation of the linking invariant on Lens spaces
Let $L_n(p)$ be the $2n+1$ dimensional Lens space
$$
S^{2n+1}/\mathbb{Z}_p
$$
where the action is given as $z_i\rightarrow e^{\frac{2\pi}{p}}z_i$, $i=1,...,n+1$, with $z_i$ the coordinates of $\mathbb{...
- 761
5
votes
1
answer
270
views
Reference for Künneth Theorem in (co)homology with local coefficients
Is there a discussion in the literature of Künneth-type theorems for (co)homology with local coefficients? The sources I know of that discuss local coefficients (Whitehead's Elements of Homotopy ...
- 8,568
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
1
vote
0
answers
77
views
Optimality condition of the harmonic form representatives of a homology class
In "Hodge theory on metric spaces, Smale et al." the $d$-th harmonic forms of the Hodge Laplacian $\Delta_d=\delta^* \delta+\delta \delta^*$ satisfying $\Delta_d(f)=0$ are claimed to be ...
- 11
4
votes
0
answers
162
views
Have mod $p^k$ Dyer Lashof operations been studied?
Here is one of the motivations for my question, when $p=2$. The homology of the spectrum $H\mathbb F_2$ as an algebra is generated by the Dyer Lashof operations on the single generator $\xi_1$ (and it ...
- 283
2
votes
0
answers
75
views
Analog of Cartan model for equivariant homology
Let $X$ be a manifold, acted on by a Lie group $G$.
(For example $X$ real-even-dimensional acted on by $G=U(1)$ with only finitely many isolated fixed points.) The Cartan model for $G$-equivariant ...
- 533
2
votes
1
answer
360
views
Alexander duality for Homology sphere which is the Geometric realization of a finite simplicial complex
The Alexander duality Theorem is usually stated for a triangulable pair $(\mathbb S^n, Y)$ where $Y$ is a subset of the standard sphere $\mathbb S^n$. My question is: Does the duality also hold if we ...
- 280