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Results tagged with gt.geometric-topology
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user 11142
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).
43
votes
Accepted
To which extent can one recover a manifold from its group of homeomorphisms
Answer is: Yes, one can recover $M$ if it is a compact manifold. See J. V. Whittaker: On Isomorphic groups and homeomorphic spaces, Annals of Math 1963.
EDIT Actually, one knows a lot more, see, for …
- 96.4k
39
votes
Accepted
Unusual symmetries of the Cayley-Menger determinant for the volume of tetrahedra
These are the so-called Regge symmetries, described by T. Regge in a 1970-ish paper. For a bit on it, with references, see the paper
Philip P. Boalch, MR 2342290 Regge and Okamoto symmetries, Comm. M …
- 96.4k
26
votes
Why should I care about the Jones polynomial?
As a historical note (others may have had a different perspective - I was a graduate student when the Jones polynomial made its appearance), when it came out there was some mild excitement because the …
- 96.4k
25
votes
Thurston's "tinker toy" problem
The result comes by way of Nash's theorem which states that every smooth manifold is a component of a real algebraic variety.
Nash, John, Real algebraic manifolds, Ann. Math. (2) 56, 405-421 (1952). …
- 96.4k
22
votes
Accepted
Which closed orientable $4$-dimensional manifolds cannot be embedded in $6$-space?
This is true if and only if $X^4$ is spin and its signature vanishes. This is on p. 345 in Gompf/Stipsicz (4-manifolds and Kirby calculus), who cite Ruberman: Imbedding four-manifold and slicing links …
- 96.4k
22
votes
Accepted
A manifold is a homotopy type and _what_ extra structure?
You are talking about the (much studied) Poincare duality spaces. For a survey, see the very nice one by John Klein: (seems to be unpublished, but dates to April 2010).
- 96.4k
19
votes
Accepted
Random links and $3$-manifolds
There is no universally accepted model of random three-manifolds (or random knots/links) for that matter, however, hyperbolicity is pervasive in all known models. The most popular (but not really sati …
- 96.4k
18
votes
18
votes
Accepted
Is there a combinatorial analogue of Ricci flow?
Actually, there are a number of references by Ben Chow, Feng Luo, and D. Glickenstein on this subject, mostly in two dimensions. Glickenstein's work (Glickenstein was a student of Ben Chow's) is more …
- 96.4k
17
votes
Accepted
certain trigonometric homeomorphisms
The magic words are $\tan(\theta/2).$ That substitution reduces your question to asking which rational functions $\mathbb{R} \rightarrow \mathbb{R}$ are homeomorphisms. Those are precisely the functio …
- 96.4k
15
votes
Can an oriented closed $n(\geq 2)$-dimensional manifold be smoothly embedded in $\mathbb{R}^...
Every closed orientable (smooth) $n$-manifold (smoothly) embeds in $\mathbb{R}^{2n-1}.$ This is true for $n> 4$ by Haefliger-Hirsch, $n=3$ by C.T.C. Wall (Wall's paper has the reference to Haefliger-H …
- 96.4k
15
votes
Accepted
Are there congruence subgroups other than $\operatorname{SL}_2(\mathbb Z)$ with exactly 1 cusp?
Yes, there are plenty of others with the best known being the one corresponding to the hexagonal torus. For a list of low-genus congruence subgroups see here. (Sebastian Pauli at UNC Greensboro).
- 96.4k
14
votes
When are (finite) simplicial complexes (smooth) manifolds?
A simplicial complex is a manifold if the links of all vertices are simplicial spheres. Recognizing the $n$-sphere is easy for $n=1, 2$, tractable for $n=3$ (the Rubinstein-Thompson algorithm and refi …
- 96.4k
14
votes
If $X$ and $Y$ are homotopy equivalent, then are $X \times \mathbb{R}^{\infty}$ and $Y \time...
I believe the answer is YES, where reasonable = finite simplicial complex, based on the work of Chapman on Hilbert Cube manifolds. Namely, I believe that it is a 1973 definition of Chapman that a Hil …
- 96.4k
13
votes
Accepted
Is $\partial X$ a sphere for $X$ a complete CAT$(0)$ space?
For a piecewise Euclidean or piecewise hyperbolic metric on a PL manifold, the answer is YES. This is proved (p348 in published version) by M. Davis and T. Januszkiewicz in
M. Davis, T.Januszkiewicz, …
- 96.4k