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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”.

4 votes

Fundamental group of the complement of a codimension two submanifold

To your first question, the answer is yes. Take a $k$-component trivial link in $S^n$, i.e. the boring, linear embedding $$\sqcup_k S^{n-2} \to S^n$$ that is the boundary of a linear embedding $$\sqcu …
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4 votes

Lipschitz bounds and homotopy groups of diffeomorphism groups

I believe the answer is no, at least for the manifold $M = S^1 \times D^{m-1}$, when $m \geq 4$. At present I know it to fail for $k=0$ and $k=m-4$, but it likely fails for a broad range of values o …
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7 votes
Accepted

Can a nontrivial $n$-sphere bundle over $M$ embed in $M\times \mathbb{R}^{n+1}$?

A variation of your question has a positive answer. If you take any compact manifold that is a smooth bundle over another compact manifold $\pi : M \to N$, there is a smooth embedding $$f : M \to N \t …
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11 votes
Accepted

Homotopy groups of the space of diffeomorphisms

There is no stability of the sort you are looking for. The reason is fairly simple-minded. For example, the orthogonal groups do not have the homotopic stability you are looking for, and diffeomorph …
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8 votes

Isotopies of codimension-1 disks relative to boundary

This is a little different than the Schoenflies problem. You can rephrase your question to be about the space of embeddings $$D^{n-1} \to S^1 \times D^{n-1}$$ that agree with the standard embedding $\ …
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3 votes

How to chart tubes around manifolds with boundary/corners?

From the comments I think the theorem you are looking for is this. I'll be a little fast and loose just to make it easier to state. Let $M$ be a manifold with corners and $N$ a submanifold, potential …
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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 …
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4 votes
Accepted

Does every simply connected, orientable, non-compact, 3-manifold embed in $\mathbb{R}^3$?

When the manifold is the universal cover of a compact $3$-manifold $M$ (to begin with, lets say without boundary) then you construct the embedding by hands, using geometrization. In your question let …
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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 …
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1 vote

applications of Sard's to differential topology

It's been a long time, but isn't your suggestion roughly Whitney's original approach to this problem? I don't have Whitney's papers in front of me but this is roughly how I think his arguments went. …
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7 votes

The purpose of connections in differential geometry

On your more general question about differential geometry, i.e. why do people study it? There are many answers, some having little to do with each other. In my opinion differential geometry is perha …
4 votes

Analog of Cerf theory in PL

PL manifolds are triangulable. The theorem you want is Pachner's theorem, that any two triangulations are related by stellar moves. You can turn triangulations into special PL handle decompositions, …
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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 …
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2 votes

About the homotopy type of diffeomorphism groups

$\newcommand{\Diff}{\mathrm{Diff}} \newcommand{\Emb}{\mathrm{Emb}} \newcommand{\Homeo}{\mathrm{Homeo}} \newcommand{\HomEq}{\mathrm{HomEq}}$This is an ancient question, but I suppose it has many answer …
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10 votes

Rational homotopy groups of $S^2\vee S^2$

Here is an example of how to detect elements of $\pi_5 (S^2 \vee S^2)$. Let $\omega_i : S^2 \to S^2 \vee S^2$ be the inclusion of the $i$-th wedge summand. Then how do we detect $[[[\omega_1,\omega_2 …
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