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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).
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 …
- 10.1k
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 $ …
- 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 …
- 10.1k
27
votes
Accepted
Usefulness of using TQFTs
All the answers so far have focused on 3 dimensions, but the answer is much more striking in 4 dimensions. Freedman's theorem tells you that classical homology invariants give you complete informatio …
- 10.1k
18
votes
3
answers
630
views
Classification of tangles?
Has anybody done any work on making a classification of low-complexity tangles, analogous to the work for knots and links? I expect most of the small ones to be rational, and those that aren't rationa …
- 10.1k
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 …
- 10.1k
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
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: …
- 10.1k
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 …
- 10.1k
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 …
- 10.1k
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 …
- 10.1k
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 …
- 10.1k
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 …
- 10.1k
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 …
- 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 …
- 10.1k