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Results tagged with differential-topology
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user 18263
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”.
12
votes
Can we define Whitney stratification algebraically?
There is a purely algebraic characterisation of Condition (B) due to Le and Teissier, see Proposition 1.3.8 of the paper
Lê Dũng Tráng; Teissier, Bernard, Limites d’espaces tangents en géométrie analy …
- 15.5k
7
votes
Good books on Geometric Theory of Dynamical Systems
Pick up (almost) anything by Ethan Akin. I particularly recommend "The General Topology of Dynamical Systems" available on Amazon. Although it is somewhat older than what you indicate you are looking …
4
votes
1
answer
358
views
Classifying smooth embeddings which yield Morse functions
Let $\mu:M \to \mathbb{R}$ be a fixed surjective smooth function on a smooth manifold $M$. Let $N$ be a smooth compact manifold that embeds smoothly into $M$ via $\iota:N \to M$.
What conditions o …
- 15.5k
3
votes
1
answer
320
views
Measuring almost-critical values of smooth functions.
Consider a compact sub-manifold $X \subset \mathbb{R}^n$ of Euclidean space and let $f:X \to \mathbb{R}$ be any smooth function. Recall that $x \in X$ is a critical point of $f$ if the gradient $\nabl …
- 15.5k
1
vote
Analogue of singularity theory in other categories
A brief account of PL Morse theory has been requested by Andras in the comments, so I am writing it down here. Note that this does not address the main question on PL singularity theory. Note also tha …
- 15.5k
1
vote
Lipschitz Approximation to a PW Smooth Map
Here is Ryan Budney's answer from the comments, I'm copying it here so that this question does not re-appear on the front page as unanswered.
Let $f:\mathbb{R}\to\mathbb{R}$ be the absolute value f …
1
vote
Accepted
$\varepsilon$-covering number after diffeomorphism
(Since the OP appears satisfied with my comment, I am reproducing it here as an answer.)
Fix $\epsilon > 0$. Since $\phi$ is $\mathscr{C}^1$, it must also be Lipschitz-continuous on compact subsets o …
- 15.5k