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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).
8
votes
Accepted
Preserving non-conjugacy of loxodromic isometries in a Dehn filling
It follows from the comment of Moisha Kohan above.
Another way to see it: take the closed oriented geodesics $\gamma, \eta$ in $M$ realizing the monodromy of $g$ and $h$ respectively. Since $g$ and $h …
- 68.9k
7
votes
Does every mapping class group embed into some $\mathrm{Out}(F_n)$?
$\DeclareMathOperator\Mod{Mod}\DeclareMathOperator\GL{GL}\DeclareMathOperator\Out{Out}$It is true for $g=1,2$. $\Mod(S_1) \cong \GL_2(\mathbb{Z})\cong \Mod(S_{1,1})\cong \Out(F_2)$.
In $\Mod(S_2)$, th …
- 68.9k
1
vote
Necessary condition for invertible knot concordance from both ends
Let $C$ denote a concordance from $K_1$ to $K_2$ in $S^3\times [0,1]$, and let $C_1, C_2$ be concordances from $K_2$ to $K_1$.
Let $C_1 \cdot C \sim K_2 \times [0,1]$, $C\cdot C_2\sim K_1\times [0,1]$ …
- 68.9k
4
votes
Accepted
Slice-ribbon conjecture in other 3-manifolds
One may still ask for the disk to be “ribbon” in $M\times [0,1]$ in the sense that the projection of $M×[0,1]$ to $[0,1]$ is a Morse function when restricted to the disk with only index 1 and 2 critic …
- 68.9k
1
vote
Classifying nested 3-manifolds with fundamental group property
The only sort of examples that I can imagine are when there is a handlebody $H$ such that $M_1\subset H \subset M_2$. It holds more generally when $\pi_1(M_2-M_1)\to \pi_1(M_2)$ is onto. This might al …
- 68.9k
4
votes
Is there a way to classify incompressible surfaces in $\Sigma \times [0,1]$ ?
Since you are asking for incompressible but not $\partial$-incompressible, the classification is more complicated. As pointed out in Sam Nead's answer, the classification of incompressible and boundar …
- 68.9k
4
votes
Accepted
The diameter of the projection of a convex core
Take a Schottky group $\Gamma$ with convex core $N\subset \mathbb{H}^3/\Gamma$ with $diam(N)$ large, but $diam(\partial N)$ bounded. Then by a theorem of Robert Brooks, there is a small deformation $\ …
- 68.9k
23
votes
$3$-manifold that is a surgery on a knot
This is an extensively studied question and is far from being understood in general. Here are some other conditions beyond the fact that $H_1(M)$ is cyclic.
the fundamental group should have weight 1 …
- 68.9k
5
votes
Existence of a surface group ensures the existence of a $\pi_1$-injective immersed surface
Given a $CW$-complex $X$ and a closed surface group $\pi_1(\Sigma,v) < \pi_1(X,x)$, there exists a map $\phi: (\Sigma,v) \to (X,x)$ such that the image of the fundamental group is this subgroup. Take …
- 68.9k
3
votes
Residual finiteness of hyperbolic 3-manifold groups
The answer to Q1 is negative in general (allowing infinitely generated fundamental group). See Example 2 which is a discrete torsion-free subgroup $G< PSL_2(\mathbb{C})$, hence $\mathbb{H}^3/G$ is a h …
- 68.9k
5
votes
Accepted
Knotted concordances of slice links
I think this is likely an unknown question. Namely, the negation of 3) would follow from 1) and 2) if
strongly slice links are strongly ribbon (which seems to be open)
ribbon disks bounding the unkno …
- 68.9k
2
votes
Accepted
Reference request and prerequisites for understanding the Sphere Theorem and the Loop Theore...
On Marc Lackenby’s webpage you can find notes on 3-manifold topology (Michaelmas 1999). The proof of the loop theorem in Chapter 9 uses special hierarchies (instead of Papakyriakopoulos’ towers) follo …
- 68.9k
3
votes
Accepted
Does the inner automorphism group of the fundamental group of a closed aspherical manifold a...
If $G=Inn(\pi_1)$ is a 2-group, then it must be finite (and then you are done by Schur’s argument above).
As you describe, $\pi_1$ is a central extension of $G$ by $\mathbb{Z}^k$ for some $k\leq n$, s …
- 68.9k
7
votes
Embedded 2-tori in $S^1\times S^4$
I think this is true (homotopy implies isotopy). Consider a generic smooth torus embedding $f: T^2 \to S^1\times S^4$, then the projection to $S^1$ gives a circle-valued Morse function on $T^2$. Becau …
- 68.9k
6
votes
Do we get a instanton $S^{3}$ if we do $1/n$ surgery on a knot in $S^{3}$?
This follows now from Theorem 1.15 of this paper for $n>1$. To clarify, Theorem 1.15 is stated for framed instanton homology $I^\#(S^3_{1/n}(K))$. If instanton Floer homology $I(S^3_{1/n}(K))=0$, then …
- 68.9k