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Results tagged with gt.geometric-topology
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user 1345
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).
5
votes
Accepted
Homologous curves and maps to surfaces
I suspect the answer to this question is no in general.
Suppose $Z$ has two components $Z_1 \cup Z_2$ with different multiplicities (as a formal linear combination of knots) and $Z'$ has one compone …
- 68.9k
7
votes
Dehn filling of hyperbolic 3-manifolds and Gromov volume
In Theorem 6.5.6 in Thurston's notes, he indicates that volume decreases strictly under hyperbolic Dehn filling, referring to Theorem 6.4 (volume rigidity, see the appendix to Dunfield's paper for mor …
3
votes
Accepted
Minimum dilatation pseudo-anosovs on non-orientable surfaces
One may construct upper bounds in the non-orientable case the same way as McMullen does in his paper (see p. 523 for a short description of his "renormalization" procedure, or Section 10).
- 68.9k
10
votes
Hakenness of Heegard splitting
Yes, there is a well-known condition of Casson & Gordon, that a Heegaard splitting which is weakly reducible, but not reducible, gives a manifold which is Haken. In terms of the curve complex, this st …
- 68.9k
11
votes
Properties of the n-dimensional Stereographic Projection
The circle and the north pole (or wherever the origin of the stereographic projection is) span a 3-dimensional subspace generically, such that the restriction to this subspace is the 2-dimensional ste …
- 68.9k
8
votes
Accepted
Do abelian spinorial prime three manifolds exist?
Yes, there exists such a manifold $M$. This follows if there exists aspherical $M$ with $Diff(M)\simeq 0$ (contractible) and $H^2(M;\mathbb{Z}/2\mathbb{Z})=0$. I claim there exists such manifolds. Let …
- 68.9k
15
votes
extension of surface homeomorphism
If you take a mapping class $\psi:\Sigma\to \Sigma$ whose characteristic polynomial of its action on homology $\psi_\ast: H_1(\Sigma;\mathbb{Z}) \to H_1(\Sigma;\mathbb{Z})$ is irreducible, then it can …
- 68.9k
4
votes
Accepted
regular tiling of a surface of genus 2 by heptagons
Addressing the second question positively, this is the Main Theorem of this paper:
Edmonds, Allan L.; Ewing, John H.; Kulkarni, Ravi S., Regular tessellations of surfaces and (p,q,2)-triangle groups, …
- 68.9k
6
votes
Accepted
A strong annulus theorem for 3-manifolds
I think this might follow from JSJ theory. Assume that $M$ is irreducible with incompressible boundary. Then any essential annulus is homotopic into an $I$-bundle region or a Seifert-fibered region of …
- 68.9k
5
votes
Translation distance in the curve complex
I don't know an algorithm, but here's a possible approach. As Richard and Lee have observed, one may assume that $\psi$ is pseudo-Anosov. In that case, the mapping torus $T_\psi$ is a hyperbolic 3-man …
- 68.9k
7
votes
Accepted
Constructing 4-manifolds with fundamental group with a given presentation.
Related to question (1), suppose you wanted to get a 4-manifold with boundary as a submanifold of $\mathbb{R}^4$ with fundamental group a finitely presented group, with presentation complex $K$. It i …
- 68.9k
7
votes
Local complexity of triangulations
I think that there's a fairly useless argument using Riemannian geometry for smoothable PL manifolds (so in dimensions at most $7$ this should work). Assume that $M$ is a PL manifold which admits a co …
- 68.9k
11
votes
the criteria for 3-dim manifolds diffeomorphic to $\mathbb{R}^3$
Husch and Price is the right reference, but I thought I could give a quick sketch of a proof.
Let $X^3$
be simply connected at infinity and contractible. For every compact
$C \subset X$, there exis …
- 68.9k
11
votes
4-dimensional h-cobordisms
Just a remark: it follows from geometrization that two closed 3-manifolds are simple-homotopy equivalent, then they are diffeomorphic. So for an $s$-cobordism, you know that at least the two ends are …
- 68.9k
4
votes
Is there a knotted torus in 4-sphere whose complement's fundamental group is infinite cyclic...
Now Kawauchi claims that knotted surfaces in $S^4$ whose complement has cyclic fundamental group are smoothly trivial (i.e., bound a handlebody).
See Corollary 1.3.
Note: This paper currently has a …