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Results tagged with differential-topology
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user 23571
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”.
41
votes
Parallelizability of the Milnor's exotic spheres in dimension 7
Here's another way to answer the original question. There is a theorem of Bredon and Kosinski (Annals, 1966) which says that if a manifold $M^n$ is stably parallelizable, then either $M^n$ is parallel …
- 20k
24
votes
Reference for a fact (?) on homeomorphic knot complements
This is a question that I remember worrying about when I first started learning about knot theory. Older books have a tendency to skim over this point rather lightly, perhaps because the resolution of …
- 20k
19
votes
Closed manifold with non-vanishing homotopy groups and vanishing homology groups
As suggested by Lennart Meier, the connected sum $M=P\#P$ of two copies of the Poincaré homology sphere gives an example. The homotopy groups $\pi_n(M)$ are nonzero for all $n>1$ because the universa …
- 20k
14
votes
Accepted
Homotopically trivial vs isotopically trivial diffeomorphisms
The quotient group $Diff_1(M)/Diff_0(M)$ is a discrete group since $ Diff_0(M)$ is a path component of $Diff(M)$, hence also a connected component since $Diff(M)$ is locally path-cconnected, and $Diff …
- 20k
12
votes
Accepted
Does there exist a Haken manifold where all its incompressible surfaces are non-separating?
There exist closed orientable hyperbolic 3-manifolds that are surface bundles such that the fiber is the only incompressible surface in the manifold (up to isotopy). Such manifolds can be obtained by …
- 20k
11
votes
Diffeomorphism group of the projective plane
It is a theorem of A. Gramain from 1973 (Annales Sci. E.N.S.) that the diffeomorphism group of the projective plane has the homotopy type of $SO(3)$, the subgroup of isometries of the standard constan …
- 20k
10
votes
Accepted
Generalized Schoenflies - formalizing step in proof?
For an interval $[a,b]\subset{\mathbb R}$ in which the height function $f:S\to {\mathbb R}$ has no critical values one obtains a product structure on $f^{-1}([a,b])$ by following flow lines of the gra …
- 20k
9
votes
Non-zero homotopy/homology in diffeomorphism groups
If ${\rm Diff}(M)$ is contractible then the question of course has a negative answer. Examples where this happens are known in dimension three but not in higher dimensions. For $M$ a closed hyperboli …
- 20k