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Results tagged with gt.geometric-topology
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user 317
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).
9
votes
$\mathrm{GL}_n(\mathbb{Z}_2)=\mathrm{Out}(F_n)/\langle\langle \epsilon_1,\dots,\epsilon_n\ra...
$\DeclareMathOperator{Out}{Out} \DeclareMathOperator{Aut}{Aut} \DeclareMathOperator{IA}{IA}$
The key thing that is missing is that the Torelli subgroup of $\Out(F_n)$ is contained in the normal closur …
- 44.8k
5
votes
Accepted
Extending diffeomorphisms between surfaces
The way you phrased this makes it sound harder than it is. Your two surfaces are diffeomorphic, so we can identify them both with a single surface $M$. Do this in a way that reflects the identificat …
- 44.8k
3
votes
Finite normal subgroup of mapping class group
When the genus of the surface is at least 3, Lanier-Margalit proved something much stronger than the fact that there are no finite normal subgroups: aside from the hyperelliptic involution, the normal …
- 44.8k
8
votes
Accepted
Linking number and intersection number
$\DeclareMathOperator\tX{\widetilde{X}}\DeclareMathOperator\tB{\widetilde{B}}\DeclareMathOperator\tD{\widetilde{D}}\DeclareMathOperator\Z{\mathbb{Z}}$
In fact, $B$ must intersect $D$ at least $|\text{ …
- 44.8k
12
votes
Accepted
If the universal cover has three boundary components, does it have infinitely many?
Let $M$ be a compact connected 3-manifold with nonempty boundary such that $\pi_1(M)$ is infinite and the universal cover $\tilde{M}$ has finitely many boundary components. I will prove that $\tilde{ …
- 44.8k
8
votes
Proof of Giroux's correspondence
I haven't read it carefully, but the new paper here by Licata-Vértesi appears to be (finally) a complete proof, albeit one that is different from the original.
EDIT: As the comments pointed out, the …
- 44.8k
7
votes
Accepted
Dualizing module for $\operatorname{Aut}(F_n)$
In our paper here, Himes, Miller, Nariman, and myself prove that Hatcher-Vogtmann’s question has a negative answer, at least for $n=5$. It also probably has a negative answer for larger $n$, but our i …
- 44.8k
3
votes
Fibration of hyperbolic 3-manifold
This isn't true. Here's one easy construction. For some $g \geq 2$, let $\Sigma_g$ be a closed oriented genus $g$ surface and let $f\colon \Sigma_g \rightarrow \Sigma_g$ be a homeomorphism represent …
- 44.8k
8
votes
Accepted
Generate $\mathrm{Mod}(S_g)$ by two Dehn twists
In
Humphries, Stephen P.
Generators for the mapping class group. Topology of low-dimensional manifolds (Proc. Second Sussex Conf., Chelwood Gate, 1977), pp. 44–47,
Lecture Notes in Math., 722, Springe …
- 44.8k
8
votes
How to prove that Lie group framing on S^1 represents the Hopf map in framed cobordism
I wrote out careful proofs of all of this and more in my note "Homotopy groups of spheres and low-dimensional topology", available here.
- 44.8k
3
votes
Accepted
Do taut foliations leafwise branch covering S^2 yield foliations by circles?
To see what is going on, I think it is helpful to carefully work out an explicit example. In Section 2.1 of Calegari's paper, he explains how to deal with 3-manifolds that fiber over the circle. Her …
- 44.8k
4
votes
Accepted
Two details from Stallings's proof of the sphere theorem
What you need for your second question is that $H^1_c(\tilde{M})$ is a free abelian group. As you noted, this is isomorphic to $H_2(\tilde{M})$. Now, $\tilde{M}$ is a connected non-compact 3-manifol …
- 44.8k
5
votes
Is there a simple formula to compute the Casson invariant of an homology $3$-sphere from its...
I don’t know a nice recipe from a Heegaard diagram, but every Heegaard splitting of a homology 3-sphere is equivalent to one whose gluing map lies in the Torelli subgroup of the mapping class group (i …
- 44.8k
7
votes
Index of the mapping class group $\Gamma_{g,n}$ inside $\text{Out}(\Pi_{g,n})$
It only has finite index in very low-complexity degenerate cases.
Here's a proof that it always has infinite index for $\Sigma_{g,1}$ with $g \geq 2$. This proof generalizes in an obvious way to deal …
- 44.8k
8
votes
Generators for commutator subgroup of surface group
Before I answer this old question, a confession: the user "Linda" is actually me! I asked this basically to make sure that something I proved was not already known. This is such a classical subject …
- 44.8k