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Results tagged with differential-topology
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user 1465
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”.
89
votes
Accepted
Can every manifold be given an analytic structure?
(similar to Mariano's post)
Q1: no. There are topological manifolds that don't admit triangulations, let alone smooth structures. All smooth manifolds admit triangulations, this is a theorem of White …
- 44.4k
53
votes
Theoretical physics: Why not just $\mathbb{R}^4$?
To play devil's advocate, you could easily turn around this line of thought. Since we live in a 3+1-dimensional space-time, we develop concepts that are sensitive to our experiences. So calculus and …
- 44.4k
29
votes
Accepted
Smooth homotopy theory
Yes, the map you mention is an isomorphism. I think the main reason people rarely address your specific question in literature is that the technique of the proof is more important than the theorem. …
- 44.4k
28
votes
Accepted
Can we decompose Diff(MxN)?
The homotopy-type of the group of diffeomorphisms of a manifold are fairly well understood in dimensions $1$, $2$ and $3$. For a sketch of what's known see Hatcher's "Linearization in three-dimension …
- 44.4k
26
votes
Accepted
Strong Whitney embedding theorem for non-compact manifolds
Regarding question 1, yes you can always ensure the image is closed. You prove the strong Whitney by perturbing a generic map $M \to \mathbb R^{2m}$ to an immersion, and then doing a local double-poi …
- 44.4k
20
votes
Two kinds of orientability/orientation for a differentiable manifold
Your main question was answered by Emerton. Regarding other notions of orientability, there's many. A popular one is the obstruction-theoretic approach:
1) A manifold $M$ is orientable if the tange …
- 44.4k
19
votes
explicit diffeomorphim between open simplex and open ball
If the compact simplex is
$$\Delta_n = \{ (x_0,\cdots,x_n) : x_i \geq 0, x_0+x_1+\cdots+x_n=1\} \subset \mathbb R^{n+1}$$
then consider this function $f : \Delta \to \mathbb R \cup \{\infty\}$ def …
- 44.4k
16
votes
Accepted
A Compact Manifold with odd Euler characteristic whose tangent bundle admits a field of lines
I believe there is no example satisfying all your constraints. If I recall (my memory is a little foggy on this) the result likely goes back to Hopf, and one of his variations on the Poincare-Hopf in …
- 44.4k
15
votes
Topology of function spaces?
A standard reference for this is Hirsch's Differential Topology textbook. If $X$ is compact near all the topologies you'd like to consider are essentially the same. Sometimes they're called the Whit …
- 44.4k
15
votes
Accepted
Diffeomorphism group of the projective plane
Two different answers using almost identical techniques! Allen's response got me to think through my response more carefully. Let me edit in a comment to point out my sloppiness, as it points out a …
- 44.4k
14
votes
Accepted
Isotopic diffeomorphisms of the sphere
This is Cerf's pseudoisotopy-implies-isotopy theorem.
Cerf's result is true in high dimensions, while it's independently known in a low-dimensional range. In dimension $2$ it goes back to the Earle–E …
- 44.4k
13
votes
Accepted
Reference for a fact (?) on homeomorphic knot complements
The result they use is Moise's theorem:
Moise, Edwin E. (1952), Affine structures in 3-manifolds. V. The triangulation theorem and Hauptvermutung, Annals of Mathematics. Second Series 56: 96–114,
Th …
- 44.4k
13
votes
Accepted
First cohomology of the space of long knots in ℝ⁴
Long knots in $\mathbb R^4$ form a simply-connected space. I pointed this out in my survey paper A Family of Embedding Spaces. The primary tool used to prove it is what's called the embedding calcu …
- 44.4k
12
votes
Is the space of immersions of $S^n$ into $\mathbb R^{n+1}$ simply connected?
Let me just fill-in the gap in j.c.'s exposition. Smale-Hirsch states that the derivative from the space of immersions $Imm(S^n, \mathbb R^{n+1})$ to the space of bundle monomorphisms $Mono(TS^n, T\ma …
- 44.4k
12
votes
Accepted
Unknotted $S^{n-2}$ in $S^n$
My understanding is this remains an open problem in the smooth category.
I believe there have been a few claims of proofs of this statement in the literature over the years, but as far as I know none …
- 44.4k