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Results tagged with gt.geometric-topology
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user 18263
Topology of cell complexes and manifolds, classification of manifolds (e.g. smoothing, surgery), low dimensional topology (e.g. knot theory, invariants of 4-manifolds), embedding theory, combinatorial and PL topology, geometric group theory, infinite dimensional topology (e.g. Hilbert cube manifolds, theory of retracts).
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
Accepted
Who first considered constructibility of simplicial complexes?
If you want the first use of the term "constructible" in this context, then your reference to Mel Hochster's work is right-on. But if you want the actual notion, then things get slightly hazy. I think …
- 15.5k
8
votes
Accepted
Is every triangulation of a Euclidean ball by convex tetrahedra shellable?
I haven't had time to check through the details of the construction, but the example B_3_9_18 found in the proof of Theorem 2 here by Frank Lutz appears to be embeddable in 3-space.
Frank specializes …
- 15.5k
4
votes
0
answers
219
views
Contractibility of regular CW sphere minus open star
Let $S$ be any regular CW decomposition of (a space homeomorphic to) the $n$-sphere, and consider a cell $\sigma$ of dimension $d \in \{0,\ldots,n\}$. Let $S'$ be the regular CW complex which remains …
- 15.5k
3
votes
Is barycentric subdivision of a collapsible, regular CW complex collapsible (non-evasive)?
The first question is answered (modulo some details) by Forman in
R Forman, Morse theory for cell complexes. Advances in Mathematics, 134 pp 90 - 145, (1998).
Theorem 12.1 shows that a discrete …
- 15.5k
2
votes
Using Discrete Morse Theory to represent hom classes
The answer to your question as stated is no.
What discrete Morse theory gives you, starting from a finite regular CW complex $X$ and a discrete Morse function $f:X \to \mathbb{R}$ (with discrete vec …
- 15.5k
14
votes
Good covers of manifolds
I am not aware of a general result regarding the existence of good covers (and would guess that the general answer is negative). However, if you are willing to make certain sacrifices in terms of addi …
- 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 …
3
votes
Accepted
Injective simplicial maps between Arc complexes
I'm not as familiar with this area as I should be, but it seems as though the desired result follows from the main theorem of the following paper:
Irmak, McCarthy. Injective simplicial maps of the ar …
- 15.5k
22
votes
Accepted
fixed point property for maps of compacts
Lovely question! Sadly, the answer is "no" in the sense that the fixed point property is not homotopy-invariant even in the category of finite polyhedra. In fact, it is also not invariant under the op …
- 15.5k
6
votes
Voronoi cells and the dual complexes in Riemannian manifolds
At least partial answers to your first two questions can be found in the brief article called Delaunay triangulations and Voronoi diagrams for Riemannian manifolds by Leibon and Letscher available her …
- 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 …
3
votes
0
answers
305
views
Are there CW structures on homotopy limits of CW maps?
Consider the diagram of finite CW complexes $X \stackrel{f}\leftarrow Y$ where $f$ is a cellular map and note that its homotopy colimit is precisely the mapping cylinder
$$C_H = \frac{X \sqcup (Y \tim …
- 15.5k
13
votes
3
answers
832
views
What fraction of n-point sets in the unit ball have diameter smaller than 1?
This question is inspired by a recent talk by Matt Kahle on random geometric complexes.
Some simple notation: let $\mathcal{B} \subset \mathbb{R}^d$ be the unit ball in $d$-dimensional Euclidean spa …
- 15.5k
28
votes
1
answer
1k
views
Is there a general theory of fiber theorems?
Here are three vague theorems rolled up in one.
Let $X$ and $Y$ be sufficiently nice topological spaces and $f:X \to Y$ a sufficiently nice surjection. If for each $y \in Y$, the fiber $f^{-1}(y) …
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