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Results tagged with gt.geometric-topology
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user 1335
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).
36
votes
Compactification theorem for differentiable manifolds ?
No. A surface of infinite genus is not a submanifold of a compact surface.
- 10.6k
29
votes
Fundamental group of 3-manifold with boundary
No. The Baumslag--Solitar groups $\langle a, b | ab^m a^{-1} = b^n \rangle$ are not three-manifold groups when $m \neq n$.
See:
Heil, Wolfgang H. Some finitely presented non-$3$-manifold groups. Proc …
- 10.6k
26
votes
Accepted
Diffeomorphisms vs homeomorphisms of 3-manifolds
$\mathsf{Diff}(S^3)$ is homotopy equivalent to $O(4)$, by Hatcher's solution to the Smale Conjecture: Hatcher, Allen E. A proof of the Smale conjecture, Diff(S3)≃O(4). Ann. of Math. (2) 117 (1983), no …
- 10.6k
23
votes
Elegant proof that any closed, oriented 3-manifold is the boundary of some oriented 4-manifold?
Maybe overkill, but elegant:
By a theorem of Hirsch, an oriented $3$-manifold $M$ embeds in the $5$-sphere (nonorientable case: Rohlin and Wall, independently). By Alexander duality, $M$ bounds a "S …
- 10.6k
21
votes
Accepted
For which surfaces is Penner's conjecture known to be true?
Shin and Strenner have shown that the conjecture is false when 3g + n > 4.
See http://arxiv.org/abs/1410.6974
19
votes
Fibers of fibrations of a 3-manifold over $S^1$
As you suspect, the answer is no. There are $3$-manifolds that fiber over the circle in infinitely many ways, with fibers of unbounded genera.
Thurston constructed a seminorm on $H_2(M,\partial M; \ …
- 10.6k
17
votes
Accepted
Homology generated by lifts of simple curves
As far as I know, this is open.
In fact, I think the following weaker question is open.
Let $\Theta$ be the set of loops $\gamma$ in $\widetilde \Sigma$ such that the image of $\gamma$ in $\Sigma$ …
- 10.6k
15
votes
Accepted
Word problem for fundamental group of submanifolds of the 4-sphere
Update:
My memory was quite blurry about this when I originally answered.
See Gonzáles-Acuña, Gordon, Simon, ``Unsolvable problems about higher-dimensional knots and related groups,'' L’Enseigneme …
- 10.6k
15
votes
Accepted
Is the spectrum of closed geodesics in a closed hyperbolic 3-manifold asymptotically homogen...
I think this should just follow from the exponential mixing of the geodesic flow (due to Pollicott).
Exponential mixing says that there is a constant $q$ such that if you have two smooth functions $f …
- 10.6k
14
votes
When is a extension of $\mathbb{Z}$ by a free group a CAT(0) group?
You'll find some examples of CAT(0) free-by-cyclics in
Samuelson, "On CAT(0) structures for free-by-cyclic groups"
and
Barnard and Brady, "Distorsion of surface groups in CAT(0) free-by-cyclic grou …
- 10.6k
14
votes
Accepted
Do different Dehn fillings produce homeomorphic 3-manifolds ?
This phenomenon is called "cosmetic surgery."
If $K$ is an amphichiral knot in the $3$--sphere with exterior $M_K$, then $M_K(p/q) \cong - M_K(-p/q)$. So if $p/q$ is a hyperbolic filling slope, the …
- 10.6k
14
votes
Torsion in homology or fundamental group of subsets of Euclidean 3-space
I'll assume that the subset is compact.
Then, if you use Cech cohomology, Alexander duality turns this into a question about the complement, which is a 3-manifold.
So, I answer with another question: …
- 10.6k
13
votes
Accepted
Hyperbolic structures on $S\times\mathbb{R}$
It follows from Thurston's Covering Theorem that there are no such examples.
The covering theorem says that if a degenerate end is infinite-to-one under a covering map, then you are (virtually) in th …
- 10.6k
12
votes
Accepted
Topological rigidity of compact manifolds in dimension three
Yes.
When the manifolds are Haken this is a theorem of Waldhausen. See Ian Agol's answer here.
Since your manifolds are aspherical, they are irreducible by the Poincaré conjecture. Since they have …
- 10.6k
12
votes
Accepted
Examples of acylindrical 3-manifolds
(source)
The exterior of Suzuki's Brunnian graph on $n$-edges, here pictured with $n=7$, is irreducible, atoroidal, boundary incompressible, and acylindrical. See
Luisa Paoluzzi and Bruno Zi …
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