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Results tagged with gt.geometric-topology
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user 5010
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).
10
votes
Accepted
Wanted: differential coming from higher genus surface in Heegaard Floer homology
Yes, $S \to \Sigma$ can fail to be an immersion. The failure is a branch point of the map $S \to \Sigma$, since it is (close to) a holomorphic map. This is fairly common as soon as the multiplicity …
- 2,821
49
votes
Elegant proof that any closed, oriented 3-manifold is the boundary of some oriented 4-manifold?
I know of several different arguments. You can decide which one you think is most elegant...
Rohlin's argument, which is actually quite geometric. You start with an immersion of the 3-manifold in $ …
- 4,713
12
votes
Accepted
Maslov index and Heegard Floer homology
Robert Lipshitz has the nicest formula, described in Corollary 4.3 of this paper:
Robert Lipshitz, A cylindrical reformulation of Heegaard Floer homology, Geometry & Topology 10 (2006) 955–1096, DOI: …
- 35.5k
12
votes
1
answer
713
views
Injectivity of the Dehn-Nielsen-Baer map?
If $S$ is a closed hyperbolic surface, is there an easy proof of the injectivity of the Dehn-Nielsen-Baer map from $\mathrm{Mod}(S)$ to $\mathrm{Out}(\pi_1(S))$, taking an element of the mapping class …
CommunityBot
- 1
54
votes
5
answers
2k
views
Unusual symmetries of the Cayley-Menger determinant for the volume of tetrahedra
Suppose you have a tetrahedron $T$ in Euclidean space with edge lengths $\ell_{01}$, $\ell_{02}$, $\ell_{03}$, $\ell_{12}$, $\ell_{13}$, and $\ell_{23}$. Now consider the tetrahedron $T'$ with edge le …
- 12.9k
10
votes
2
answers
238
views
What are the possible linking matrices of a quasi-positive link?
I was surprised recently to come across a 3-component link where the linking number of two of the components was negative. For a while I thought I had made a mistake, then I thought a little more and …
- 890
8
votes
1
answer
392
views
To find a point in Teichmüller space or measured foliation, how many lengths of curves do yo...
To parametrize Teichmüller space, it suffices to measure the hyperbolic lengths of a finite number of curves. It is well-known that $9g-9$ curves suffice, by a standard pair-of-pants argument given in …
- 68.9k
6
votes
Tiling of genus 2 surface by 8 pentagons
Ian answered the second question as asked, but in case you meant to ask a different question: there is not always a symmetric tiling by regular polygons of the given type, even if those restrictions h …
- 10.1k
49
votes
2
answers
4k
views
Can knot diagrams be monotonically simplified using under moves?
It is well known that knot diagrams cannot be monotonically simplified using Reidemeister moves. For instance, the Goeritz unknot cannot be directly simplified. On the other hand, there is a stronger …
- 2,821
13
votes
2
answers
482
views
Geodesic current supported on a pencil?
Consider a geodesic current $\mu$ on a closed surface $\Sigma$, as defined by Bonahon ("The Geometry of Teichmüller space via geodesic currents"). These are $\pi_1(\Sigma)$-invariant measures on the s …
- 10.1k
6
votes
1
answer
236
views
Reference request: Can iterated torus links be mutated?
I believe that most iterated torus links cannot be changed non-trivially by a Conway mutation, as follows. If you look at the JSJ decomposition of the double-branched cover, then each satellite torus …
- 68.9k
10
votes
2
answers
579
views
Is the Lisca-Matic bound (aka slice-Bennequin bound) strictly stronger than the Bennequin bo...
The Bennequin bound [1] says that, for a transverse knot (or later link) $K$ in $S^3$,
$$\mathrm{sl}(K) \le - \chi(\Sigma)$$
for any Seifert surface $\Sigma$ for $K$, where $\mathrm{sl}$ is the self-l …
14
votes
1
answer
1k
views
Who proved that two homotopic embeddings of one surface in another are isotopic?
If $\Sigma_1$ and $\Sigma_2$ are two compact topological surfaces with boundary and $\phi, \psi : \Sigma_1 \hookrightarrow \Sigma_2$ are two orientation-preserving embeddings that are homotopic, then …
- 683
7
votes
1
answer
310
views
Can a surface group act on a finite-valence simplicial tree?
Question. Let $S$ be a closed surface of genus $> 1$. Can $\pi_1(S)$ act faithfully and minimally on a simplicial tree of finite valence? Here "minimal" means that there is no invariant sub-tree.
Thi …
- 11.6k
5
votes
1
answer
271
views
Is every triangulation of a Euclidean ball by convex tetrahedra shellable?
Suppose you are given a 3-ball $B$ in $\mathbb{R}^3$ that is bounded by a PL sphere, a triangulation $T$ of $B$ by Euclidean tetrahedra. Is that triangulation necessarily shellable?
I know that if $ …
- 15.5k