All Questions
Tagged with cohomology gt.geometric-topology
50
questions
6
votes
1
answer
355
views
Fivebrane bordism $\Omega_d^{\mathrm{Fivebrane}}$
$\newcommand{\Fr}{\mathrm{Fr}}\newcommand{\Fivebrane}{\mathrm{Fivebrane}}\newcommand{\String}{\mathrm{String}}\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\SO{SO}\DeclareMathOperator\GL{GL}$What ...
- 10.4k
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
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
6
votes
2
answers
382
views
Dual surfaces of a first cohomology class of a 3-manifold
Let $M$ be closed 3-manifold and $\alpha\in H^1(M;\mathbb Z_2)$ an arbitrary element. (In my case we know that $M$ is non-orientable and $\alpha^3=0$.) It is well known that there is a closed 2-...
- 1,085
3
votes
1
answer
121
views
Cohomology of the coned off space
Let $X$ be a compact manifold with boundary $\partial X$ with $
\dim X\setminus \partial X=n$. Moreover, $X$ and $\partial X$ are both aspherical. Then what's the $H^n(X\cup_{\Sigma\subset \partial X} ...
- 199
1
vote
0
answers
284
views
Boundary map in Mayer-Vietoris sequence of cohomology
Suppose $M$ is a 3-manifold with connected boundary. Let $T$ be a tangle in $M$, i.e., $T$ is a embedded connected 1-submanifold whose boundary is on $\partial M$ ($T$ is not closed). Moreover, ...
- 653
5
votes
2
answers
740
views
Complement to a union of spheres in a sphere
Take $S^n$ and consider the union $Z$ of $k_1$ circles, $k_2$ 2-dimensional spheres, ..., $k_{n-2}$ $(n−2)$-dimensional spheres, embedded in $S^n$ in an unknotted way, with no mutual intersection and ...
- 243
3
votes
0
answers
276
views
Evaluating the Euler class of a circle bundle on fibers
I am trying to understand what kind on information the Euler class provides about certain submanifolds of a given circle bundle.
This might be completely obvious, but I don't see how to answer the ...
- 1,227
18
votes
1
answer
857
views
Oriented cobordism classes represented by rational homology spheres
Any homology sphere is stably parallelizable, hence nullcobordant. However, rational homology spheres need not be nullcobordant, as the example of the Wu manifold shows, which generates $\text{torsion}...
- 11.9k
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
3
votes
2
answers
391
views
Alexander duality and homology equivalence
While reading the paper of Kauffman and Taylor "Signature of links" I found the following situation.
In the proof of Theorem 2.6 they suppose that two links $L_1, L_2\in \mathbb{S}^3$ are ...
- 511
9
votes
0
answers
261
views
Cohomology of $\operatorname{SO}(p,q;\mathbb{Z})$ with $p=3,q=19$
I would like to understand the topology of the moduli space of Einstein orbifold metrics on the $K3$-surface. It is known that this space is given by the bi-quotient $SO(3,19;\mathbb{Z})\setminus SO(3,...
- 423
18
votes
1
answer
985
views
Wu formula for manifolds with boundary
The classical Wu formula claims that if $M$ is a smooth closed $n$-manifold with fundamental class $z\in H_n(M;\mathbb{Z}_2)$, then the total Stiefel-Whitney class $w(M)$ is equal to $Sq(v)$, where $v=...
- 1,329