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Results tagged with gt.geometric-topology
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user 1650
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).
29
votes
Drawing of the eight Thurston geometries?
Here is a nice cyclic ordering of the eight geometries:
$$\Bbb H^2\times \Bbb R,\quad \Bbb S^2\times \Bbb R,\quad \Bbb E^3,\quad \mathsf{Sol},\quad \mathsf{Nil},\quad \Bbb S^3,\quad \mathsf{PSL},\quad …
- 28.2k
21
votes
Accepted
Obtain any 3-manifold from repeating surgeries on knots in $S^3$
This is a result of Lickorish, in his paper "A representation of orientable combinatorial 3-manifolds". The paper is only eleven pages, and is very readable. In his proof, Lickorish rediscovers some …
- 28.2k
20
votes
Accepted
Teichmuller Theory introduction
The primer on mapping class groups, by Farb and Margalit.
19
votes
3-manifold with fundamental group $\mathbb Z$
No. For example, take a copy of $S^1 \times S^2$ and remove the interior of a closed, nicely embedded, three-ball.
You will need to add the hypothesis of irreducibility (to rule out "punctures" as in …
- 28.2k
19
votes
Is there a continuous partition of space into circles?
As a partial answer, there is no smooth partition of $\mathbb{R}^3$ into smooth loops. For, consider the quotient space; it is necessarily a two-orbifold (locally a surface, possibly having isolated …
- 28.2k
17
votes
Self-covering spaces
Here are some pretty examples of self covering manifolds: Suppose that $F$ is a manifold and $f \colon F \to F$ is a periodic homeomorphism of period $k$. We define $M_f = F \times I \,/\, f$ to be t …
- 28.2k
17
votes
Classification of homology 3-spheres?
A nice historical note - Dehn observed that if $M$ and $N$ are knot complements and if you glue $M$ to $N$ switching meridian and longitude then the result is a homology sphere. Of course this is a s …
- 28.2k
16
votes
How would a topologist explain "every Riemann surface of genus $g$ is hyperelliptic if and o...
Among closed, oriented, connected surfaces $S_g$ of genus two or higher, the genus two surface $S_2$ is the only one whose mapping class group has non-trivial centre. This centre is a copy of $\mathb …
- 28.2k
16
votes
Topology of the space of embedded genus $g$ surfaces in $S^3$
The first observation is that $\mathcal{E}_g$ is not connected when $g > 0$. This is due to the existence of "knotting". For example, in genus one, let $K$ and $K'$ be smooth knots. Let $T$ and $T' …
- 28.2k
16
votes
Accepted
Fixed-point free diffeomorphisms of surfaces fixing no homology classes
Goodwillie's construction (in genus two) generalises to all higher genus as follows.
Let $P_n$ be the regular $n$-gon in the plane with vertices at roots of unity. When $n$ is even, we can glue opposi …
- 28.2k
16
votes
In knot theory: Benefits of working in $S^3$ instead of $\mathbb{R}^3$?
Here is a way to say it directly in the language of three-manifolds. By Alexander's theorem, both $R^3$ and $S^3$ are irreducible. However, as Mark indicates, a knot complement in $R^3$ is reducible. …
- 28.2k
15
votes
Accepted
Is there a contractible hyperbolic 3-orbifold of finite volume?
Yes. For example, let $M$ be the figure-eight knot complement. So $M$ is a hyperbolic manifold with volume a bit more than 2. The manifold $M$ has a two-fold symmetry $\tau$ that fixes, pointwise, a …
- 28.2k
14
votes
Accepted
Definition of Thurston's skinning map
Let’s simplify to the case where $M$ has exactly one boundary component, say $\partial M = S$. So the hyperbolic structures on $M$ are parametrised by the conformal structures on $S$. Fix one such co …
- 28.2k
13
votes
Accepted
How many knots are there with hyperbolic volume less than a given constant
Here is an expansion of Ian's answer.
Dehn surgery on a hyperbolic knot or link generically gives another hyperbolic manifold. This follows from the Dehn surgery theorem; see Theorem 5.8.2 in chapt …
- 28.2k
13
votes
Examples of the Thurston geometries with transitive Lie group action
Here is a cool fact about $\mathrm{SL}(2, \mathbb{R}) / \mathrm{SL}(2, \mathbb{Z})$; it is homeomorphic to the complement of the trefoil knot in the three-sphere. Apparently this was first proved by …
- 28.2k