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Results tagged with differential-topology
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user 317
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”.
5
votes
Accepted
Extending diffeomorphisms between surfaces
The way you phrased this makes it sound harder than it is. Your two surfaces are diffeomorphic, so we can identify them both with a single surface $M$. Do this in a way that reflects the identificat …
- 44.8k
8
votes
Accepted
Linking number and intersection number
$\DeclareMathOperator\tX{\widetilde{X}}\DeclareMathOperator\tB{\widetilde{B}}\DeclareMathOperator\tD{\widetilde{D}}\DeclareMathOperator\Z{\mathbb{Z}}$
In fact, $B$ must intersect $D$ at least $|\text{ …
- 44.8k
3
votes
Classification of surface bundles over surfaces
About 6 years ago there was an Oberwolfach meeting on surface bundles, and most of the talks were recorded and can be seen here. If I remember correctly, Benson Farb’s overview talk was particularly …
- 44.8k
21
votes
Accepted
Manifolds with two coordinate charts
I'll only discuss the first question (EDIT: Actually, I address the second question at the end). As Agol pointed out in the comments, for $n \geq 5$ this is an easy consequence of Newman's 1966 proof …
- 4,195
12
votes
Conceptual proof of classification of surfaces?
The proof of Zeeman described in this note is by a substantial margin the easiest and most conceptual proof I know. To simplify the exposition I restrict to orientable surfaces in the note, but it is …
- 44.8k
17
votes
Accepted
Foliation of $\mathbb R^n$ by connected compact manifolds
There does not, even if you don’t require the fiber and base to be manifolds (or even connected, just that $F$ is not a single point). See
Borel, Armand; Serre, Jean-Pierre,
Impossibilité de fibrer …
- 44.8k
11
votes
Accepted
Do an unlinked trefoil and figure-eight cobound an annulus in $B^4$?
The relationship you're asking for is called concordance. Determining if knots are concordant is quite difficult: there are many concordance invariants, but no kind of global picture of what it means …
- 44.8k
13
votes
Fundamental groups of compact Kähler manifolds
It is still open whether or not all Kahler groups occur as the fundamental groups of smooth complex projective varieties. However, there has been some interesting work on which groups can occur as th …
- 44.8k
16
votes
Accepted
Who proved that two homotopic embeddings of one surface in another are isotopic?
First, for simple closed curves, this was known long before Freedman-Hass-Scott. For closed surfaces, it was first proved by Baer in
Baer, R., Kurventypen auf Flächen. J. reine angew. Math., 156 (19 …
- 683
12
votes
Is $\mathrm{Diff}_0(S_g)$ torsion-free?
This theorem was proved by Hurwitz in the 19th century, who in fact showed the stronger theorem (also mentioned in Danny Ruberman's answer) that any finite-order diffeomorphism of a surface of genus …
- 44.8k
19
votes
Exotic $C^k$ manifolds
There are no such exotic manifolds. Whitney proved in
Whitney, Hassler Differentiable manifolds. Ann. of Math. (2) 37 (1936), no. 3, 645–680.
that every $C^1$-differential structure can be uniquely …
- 44.8k
19
votes
Accepted
When a homeomorphism is a diffeomorphism w.r.t to a suitable smooth structure?
Let me first answer your last question in the negative: there exist homeomorphisms $f:M \rightarrow M$ of smoothable manifolds $M$ such that neither $f$ nor $f^{-1}$ are smooth with respect to any smo …
- 44.8k
17
votes
Accepted
Existence of sections of the evaluation map for the diffeomorphism group
If a section $\sigma : M \rightarrow \text{Diff}_{+}(M)$ to $\text{Diff}_{+}(M) \rightarrow M$ exists, then $M$ must be parallelizable (i.e. the tangent bundle of $M$ must be trivial). Indeed, if $\v …
- 44.8k
15
votes
Extending a diffeomorphism of the sphere $S^2$ to the ball $D^3$
Smale's proof is actually pretty simple and geometric -- I highly recommend reading his paper.
There's also a beautiful short proof of Smale's theorem using the measurable Riemann mapping theorem in
…
- 44.8k
10
votes
Can cotangent bundles see exotic smooth structures?
The answer to your second question is yes (in some cases). See Abouzaid's paper "Framed bordism and Lagrangian embeddings of exotic spheres", available here.
- 44.8k