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Results tagged with differential-topology
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user 482183
The study of differentiable manifolds and differentiable maps. One fundamental problem is that of classifying manifolds up to diffeomorphism. Differential topology is what Poincaré understood as topology or “analysis situs”.
25
votes
1
answer
542
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Does every oriented $3$-dimensional submanifold of $\mathbb{R}^6$ bound an oriented $4$-dime...
In my recent research, I encountered the following problem about embeddings. Let $M^3$ be a closed compact oriented smooth $3$-dimensional submanifold of $\mathbb{R}^6$. Does there exist a compact ori …
- 587
5
votes
0
answers
150
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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 …
- 587
13
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1
answer
503
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Impossibility of realizing codimension 1 homology classes by embedded non-orientable hypersu...
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 $2$ …
- 587
16
votes
0
answers
408
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Is the oriented bordism ring generated by homogeneous spaces?
I am trying to find a Riemannian geometrically well-understood set of generators of the oriented bordism ring, including the torsion parts. By a set of generators, I mean that the set generates the ri …
- 587
13
votes
1
answer
368
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Realizing integral homology classes on non-orientable manifolds by embedded orientable subma...
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 …
- 587
20
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2
answers
869
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Integral homology classes that can be represented by immersed submanifolds but not embedded ...
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 $z$ can …
- 587
8
votes
1
answer
212
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Integral homology classes of which no multiples admit embedded representatives with trivial ...
Let $M$ be a closed smooth manifold of dimension $n$ and $z\in H_l(M,\mathbb{Z})$ a $k$-dimensional integral homology class. Theorem II.4 of Thom's classical 1954 paper states that for $l< n/2$ or $n- …
- 587
13
votes
1
answer
447
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Compact closed aspherical manifolds with vanishing second homology in all the covering spaces
I wonder if there exists a compact closed smooth aspherical manifold $M$ of dimension at least $4$, so that for any covering space $\tilde{M}$ over $M,$ we always have $H_2(\tilde{M},\mathbb{Z})=0$ an …
- 587
12
votes
1
answer
309
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Are the symmetric spaces $\operatorname{SU}(n)/{\operatorname{SO}(n)}$ always nontrivial in ...
In my recent research, I need to know if the symmetric spaces $\operatorname{SU}(n)/{\operatorname{SO}(n)}$ are always nontrivial in the unoriented and oriented bordism rings for $n>2$. (For the motiv …
- 587