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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”.
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
2
votes
Accepted
Reference request and prerequisites for understanding the Sphere Theorem and the Loop Theore...
On Marc Lackenby’s webpage you can find notes on 3-manifold topology (Michaelmas 1999). The proof of the loop theorem in Chapter 9 uses special hierarchies (instead of Papakyriakopoulos’ towers) follo …
- 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
7
votes
Accepted
Does every triangulable manifold have a vertex-transitive triangulation?
There exists many closed connected hyperbolic 3-manifolds $M$ with trivial symmetry group, and hence trivial mapping class group. $M$ cannot be homeomorphic to a simplicial complex $\tau$ which admits …
- 68.9k
6
votes
Doubles of 2-handlebodies
A hyperbolic 4-manifold has zero signature and hence is null-cobordant. However there exist hyperbolic 4-manifolds with trivial isometry group, and hence which cannot be a double of a 2-handlebody (su …
- 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
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
4
votes
Under what conditions can an orientable Riemannian 3-manifold be defined implicitly?
This is too long for a comment.
If you want $0$ to be a regular value of $f$, then this should be possible iff there is a smooth isometric embedding to $R^n$ with trivial normal bundle (for any dimens …
- 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
18
votes
Accepted
Very particular kind of 4-manifolds. Classification
There are plenty of such manifolds, but as Danny indicates in his answer, there is not a known classification.
Take any acyclic group $G$ with a finite aspherical 2-complex $C$ with $\pi_1(C)=G$. Then …
- 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
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
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
47
votes
Parallelizability of the Milnor's exotic spheres in dimension 7
The tangent bundle to a smooth structure on $S^7$ is classified by a map $S^7 \to G_7(R^{\infty})$. By the exact sequence for a fibration for the fiber bundle $O(7)\to V_7(R^\infty)\to G_7(R^\infty)$, …
- 68.9k
4
votes
Classification of oriented vector bundles of rank 5 over closed oriented 5-manifolds
I don't know how to answer this question completely, but I'll make some observations.
The oriented vector bundles over $M$ are classified by homotopy classes of maps $f:M\to \tilde{G}_5$, the orient …
- 68.9k