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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).
28
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1
answer
1k
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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) …
- 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
15
votes
3
answers
2k
views
Combinatorial analogues of curvature
There appear to be many "combinatorial" definitions of curvature as applied to finite simplicial (or regular CW) complexes. For instance, we have the ideas of Cheeger, Muller and Schrader, of Forman a …
- 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
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
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
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
8
votes
Is there a discrete Cerf theory?
I realize that I am several months late to the Cerf theory party, but this paper of Chari and Joswig might be of interest to the original poster and certainly deserves a mention in the context of this …
- 15.5k
8
votes
Homological computations
What is the fundamental domain of the action? In case you can easily create a cubical or simplicial decomposition of this region, there is tons of software out there to help you with the grunt-work.
…
- 15.5k
8
votes
3
answers
481
views
Does the metric space of compact metric spaces satisfy the binary intersection property?
A metric space $Y$ has the binary intersection property provided that whenever a collection of closed balls in $Y$ intersects pairwise, then there is a common intersection point.
Does the metric s …
- 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
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 …
7
votes
1
answer
817
views
Which Abelian Group sequences arise as the Homology of Embedded CW Complexes?
Background
Let $\mathcal{A} = \lbrace A_0, \ldots, A_M \rbrace$ be an arbitrary sequence of finitely generated Abelian groups. It is well-known that a finite CW complex $X_\mathcal{A}$ may be construc …
- 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
4
votes
Accepted
Discrete Morse function from smooth one
This is a rapidly developing area, and there are many short-cuts if all you want to do is compute the homology of sub-level sets of $f$. To answer your main question, as Liviu has already mentioned: t …
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