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Results tagged with differential-topology
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user 1345
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”.
3
votes
Accepted
a small questions about hopf theorem
I realize that this doesn't answer your question, but there is also an approach using the methods of homotopy theory and CW complexes. If $M$ is a closed smooth orientable $p$-manifold, then $M$ is ho …
- 68.9k
9
votes
On the generalized Gauss-Bonnet theorem
I'll make some general remarks. One may break down the generalized Gauss-Bonnet into two parts.
The first is to verify that it is a topological invariant. This means that
it should be independent of t …
- 68.9k
10
votes
Isotopy in 3-manifolds
If $\Sigma_1 \hookrightarrow M$ is an embedded $\pi_1$-injective surface, then any homotopic embedded surface will be isotopic to $\Sigma$. As Ryan and Allen point out, this is due to Waldhausen for i …
- 68.9k
43
votes
Is differential topology a dying field?
I don't think differential topology is a dying field.
I'll interpret this as the classification of smooth
manifolds and, more broadly, maps between them
(immersions, embeddings, diffeomorphism group …
- 68.9k
6
votes
Accepted
Irreducibility of 3-manifolds with (non)empty boundary
You're asking how reducibility/irreducibility behaves under drilling and filling. I think you've captured the essence of drilling: if a link is "sphere busting" in a reducible manifold (meets every es …
- 68.9k
5
votes
Accepted
what is the meaning of "inseparable" in this case
I think you mean "non-separating" rather than "inseparable" (or at least I'm not familiar with that terminology), which means as you say that when you cut along the submanifold, you get a connected sp …
- 68.9k
9
votes
Accepted
General position for map from surface to 3-manifold
In general, no you cannot. Consider one dimension down. Take two curves on a torus, intersecting transversely in a point. One of the curves cannot be homotoped to be disjoint from the pair of curves. …
- 68.9k
9
votes
Accepted
When can a surface in a 3-manifold be isotoped off a knot?
Let’s assume that the manifold $M$ is irreducible and orientable and the surface $S$ is orientable. This is to avoid 1-sided surfaces.
First let’s assume that the surface $S$ is fully compressible. T …
- 68.9k
4
votes
Accepted
Compact closed aspherical manifolds with vanishing second homology in all the covering spaces
I think that the answer to this question is unknown in general. If one had a closed aspherical manifold with this property, then it could not contain a Baumslag-Solitar subgroup since such a group has …
- 68.9k
5
votes
Classification of surface bundles over surfaces
For a), when $\chi(F)<0$ $F$-bundles over $B$ are classified by maps $\pi_1(B)\to Mod(F)$, where $Mod(F)$ is the mapping class group of $F$. This boils down to the fact that $Diff_0(F)$ is contractibl …
- 68.9k
7
votes
Embedded 2-tori in $S^1\times S^4$
I think this is true (homotopy implies isotopy). Consider a generic smooth torus embedding $f: T^2 \to S^1\times S^4$, then the projection to $S^1$ gives a circle-valued Morse function on $T^2$. Becau …
- 68.9k
8
votes
Accepted
Realizing Morse functions on $S^2$ as height functions
Any Morse function on $S^2$ may be realized by an embedding $S^2\hookrightarrow \mathbb{R}^3 \to \mathbb{R}$. For a Morse function $F:S^2\to \mathbb{R}$, take the equivalence relation with equivalence …
- 68.9k
7
votes
Accepted
Smooth map between oriented manifolds
This follows from a result of Hopf (see the exposition of Epstein). By this result, one may assume that there is a disk $D\subset N$ such that $f^{-1}(D)$ is $d$ disks mapping diffeomorphically to $D$ …
- 68.9k
5
votes
Homologically trivial submanifolds
This is true in codimension and dimension 1. In fact, in these dimensions, any
homology class is represented by an embedded manifold. In codimension 1, the
manifold $N$ will separate $M$ into (at lea …
- 68.9k
5
votes
Theorems similar to Tischler fibering theorem
On a $2k+1$-dimensional manifold, a 1-form $\alpha$ such
that $\alpha\wedge (d\alpha)^k \neq 0$ at each point gives
a contact structure. I believe that it is still open
which $2k+1$-manifolds for $k>1 …
- 68.9k