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Results tagged with gt.geometric-topology
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user 1463
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
What are some of the big open problems in 3-manifold theory?
ADDED (29 May, 2013)
As has been pointed out in the comments, there has been great progress since this answer was first written, and the conjectures below have now been proved, thanks to ground-breaki …
39
votes
finite generated group realized as fundamental group of manifolds
Theorem. Every finitely presentable group is the fundamental group of a closed 4-manifold.
Sketch proof. Let $\langle a_1,\ldots,a_m\mid r_1,\ldots, r_n\rangle$ be a presentation. By van Kampen, th …
- 25.2k
25
votes
Accepted
Fundamental group of 3-manifold with boundary
A couple of extra points.
Any compact 3-manifold with boundary $M$ can be doubled to give a closed 3-manifold $D$. As $M$ is a retract of $D$, it follows that $\pi_1(M)$ injects into $\pi_1(D)$. The …
- 25.2k
21
votes
When is a extension of $\mathbb{Z}$ by a free group a CAT(0) group?
An example of a free-by-cyclic group that is not CAT(0) was given by Gersten. It is constructed from the automorphism of $F_3\cong\langle a,b,c\rangle$ that sends
$a\mapsto a,~b\mapsto ba,~ c\mapsto …
- 25.2k
19
votes
Acyclic group and finite CW-complex
I presume by "acyclic" you are referring to homology with $\mathbb{Z}$ coefficients. There are many such examples.
For instance, you can take two elements $u,v$ in the free group $F_2$ of rank 2 tha …
- 25.2k
19
votes
Accepted
What are the implications of the simple loop conjecture?
I would motivate the simple loop conjecture as follows. (I'm fairly idiosyncratic about this; I fear I'm going to turn off many 3-manifold topologists.)
As well as understanding spaces, we want to u …
- 25.2k
17
votes
When is a finitely generated group finitely presented?
An often-used method is to compute $H_2$. If the group is finitely presentable then $H_2$ is of finite rank with any coefficients.
For instance, you can use this technique to show that if $q:F\to\ma …
16
votes
Accepted
Conjugacy classes of the mapping class group
An exponential-time solution to the conjugacy problem in the mapping class group was given by Jing Tao, in:
Tao, Jing(1-OK)
Linearly bounded conjugator property for mapping class groups. (English sum …
- 25.2k
15
votes
Accepted
Judging whether a finitely presented group is a 3-manifold group?
Apologies for the shameless self-promotion, but as you ask for necessary conditions, you seem to want a list of theorems of the form 'If G is a 3-manifold group then G has property P'.
Aschenbrenner, …
14
votes
Accepted
Does any surface of constant curvature admit a cocompact group action?
As Uri Bader says in the comments, covering-space theory implies that this happens if and only if $\pi_1S$ is a normal subgroup of $\pi_1\Sigma$, where $\Sigma$ is some compact surface.
The cases of …
- 25.2k
14
votes
Accepted
Topology of boundaries of hyperbolic groups
There are plenty of other possibilities. Here are a few examples:
The boundary of the fundamental group of an acylindrical hyperbolic 3-manifold with totally geodesic boundary is homeomorphic to a S …
- 25.2k
13
votes
Accepted
Walls of CAT(0) cube complex sufficiently far apart implies intersection of stabilizers finite
Here's a proof.
Lemma: Suppose $G$ is a (word-)hyperbolic group acting properly discontinously, cocompactly and faithfully on a CAT(0) space $X$. Then there is a uniform bound $R_0$ on the width …
- 25.2k
13
votes
The work of Thurston
This isn't a direct answer to the actual question, but in your first sentence I think you're alluding to Thurston's article On proof and progress in mathematics. In section 6, entitled "Some persona …
- 25.2k
13
votes
Applications of infinite graph theory
Bass--Serre theory translates the algebraic notion of a `splitting' of a group $G$ into an action of $G$ on a (usually infinite) tree. See Serre's classic Arbres, Amalgames, $SL_2$.
- 25.2k
13
votes
Accepted
Does every retraction of free groups arise from projection to a subset of a freely generatin...
No. This is explicitly stated in the paragraph above Theorem 1 of:
Turner, Edward C, Test words for automorphisms of free groups.
Bull. London Math. Soc. 28 (1996), no. 3, 255--263.
The author refer …
- 25.2k