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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).
3
votes
Accepted
Mirzakhani's hyperbolic method generalized to moduli space of stable maps
Mirzakhani's computation of volumes of deformation spaces is very heavily based on the work of Greg McShane (McShane's identity). McShane's theory has been extended to other deformation spaces, see, f …
- 96.4k
4
votes
Lengths of closed geodesics on a flat vs hyperbolic punctured torus
Yes, this is true. It is shown (either in the paper you cite or the other McShane-Rivin paper) that the length of a simple closed geodesic is quasi-the-same as the combinatorial length ($m+n$) (this i …
- 96.4k
13
votes
SL(2, C)-representation of a knot
$(P)SL(2, \mathbb{C})$ is the isometry group of $\mathbb{H}^3,$ so $SL(2, \mathbb{C})$ representations are the natural generalization of hyperbolic structures on knot complements.There is a vast liter …
- 96.4k
0
votes
What is a geodesic in Outer space?
You should consult the oeuvre of Yael Algom-Kfir, in particular "Strongly Contracting Geodesics in Outer Space", whereupon enlightenment will ensue.
- 96.4k
5
votes
Are finite presentations of arithmetic groups computable?
You might want to study the work of Detinko, Flannery, and co-authors. For example:
Detinko, A.; Flannery, D. L.; Hulpke, A., Zariski density and computing in arithmetic groups, ZBL06825254.
Detink …
- 96.4k
2
votes
Find a surface or 3-manifold whose fundamental group is $(\mathbb{Z}/n\mathbb{Z}) \rtimes (\...
A connected sum of the appropriate lens space and $\mathbb{R}P^3$ will have fundamental group $\mathbb{Z}_m \ast \mathbb{Z}_2.$ Otherwise, the only abelian fundamental groups of $3$-manifolds are $\ma …
- 96.4k
3
votes
Any 3-manifold can be realized as the boundary of a 4-manifold
In this question it is shown how to see that two smooth manifolds are topologically cobordant if and only if they are smoothly cobordant, which answers some subset of the questions (the second manifol …
- 96.4k
11
votes
What is the weakest negative curvature condition ensuring a manifold is a $K(G,1)$?
For 1-3, yes, by the Cartan-Hadamard Theorem.5-... No. For example, every 3-manifold admits a metric of negative scalar curvature (I think this is actually true for any manifold, due to Lohkamp).
- 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
8
votes
What is the complexity of determining if a knot group is $\mathbb{Z}$?
The knot group is $\mathbb{Z}$ if and only if the knot is unknotted: see https://math.stackexchange.com/questions/1478171/knot-group-and-the-unknot
So, your first question is on the complexity of dec …
- 96.4k
7
votes
Accepted
How many simple closed geodesics in a given primitive homology class?
The thrice punctured sphere has no simple closed geodesics. The four-times punctured sphere has a unique simple geodesic in each homology class. In general, it is a result of I. Rivin that the number …
- 96.4k
3
votes
Is there a mathematical book on general relativity that uses exclusively a coordinate free l...
You might want to check out the classic paper by Tullio Regge: General Relativity without Coordinates (it is discussed in the Misner/Thorpe/Wheeler phonebook, but it is usually better to go to the sou …
- 96.4k
1
vote
Accepted
Is there a geometric interpretation of a Zariski dense surface subgroup?
I am interpreting the question as meaning that the OP asks for a geometric interpretation of a Zariski dense surface subgroup in $SL(n, \mathbb{Z})$ for $n=3, 4.$ The existence of such has been shown …
- 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
3
votes
Nielsen-Thurston decomposition from the product of Dehn twists
Not only is it possible, it is implemented by Mark Bell and Saul Schleimer as Twister.
- 96.4k