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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).
11
votes
extension of surface homeomorphism
You can find a proof of this in the paper
MR1885215 (2002m:57019)
Leininger, Christopher J.(1-TX); Reid, Alan W.(1-TX)
The co-rank conjecture for 3-manifold groups. (English summary)
Algebr. Geom. …
- 44.8k
26
votes
Thurston's "tinker toy" problem
It was published here:
M. Kapovich, J. Millson, Universality theorems for configuration spaces of planar linkages,
Topology 41 (2002), no. 6, 1051–1107.
- 44.8k
6
votes
Accepted
Homotopies with prescribed regular values
The answer is "yes" if $M_1$ is compact. Here's a sketch of a proof.
Consider any smooth homotopy $G:M_1 \times I \rightarrow M_2$ between $f_0$ and $f_1$. Since $M_1$ is compact, the set of regula …
- 44.8k
10
votes
Accepted
What does the matrix of a mapping class tell you about the 3-manifold?
In addition to the first homology group, it also determines the Seifert pairing on the torsion in the first homology group. What is more, in an appropriate sense this is all it determines. This is a …
- 44.8k
7
votes
Homology generated by lifts of simple curves
I just posted a paper with Justin Malestein entitled "Simple closed curves, finite covers of surfaces, and power subgroups of $\text{Out}(F_n)$" that gives a nearly complete answer to this question. …
- 44.8k
9
votes
Teichmuller theory and moduli of Riemann surfaces
I'll discuss things which are more applications of the mapping class group to moduli space rather than Teichmuller theory per se, but of course this is all tightly connected.
One of the big applicati …
- 44.8k
4
votes
Jordan Curve Homotopy
At least in dimension 2, results like this are true in great generality. A Jordan curve in the plane is simply an embedded curve. In other surfaces, we have that two simple closed curves $\gamma_1$ …
- 44.8k
5
votes
Accepted
Resource for Measured Foliations and Whitehead Equivalence
I don't know of a source other than FLP for understanding the Thurston compactification via measured foliations, but you might find it easier to understand this compactification using measured geodesi …
- 44.8k
10
votes
Maximal euler characteristic of surfaces bounding two fixed curves
For orientable surfaces of genus at least $2$, pretty sharp bounds are obtained in the unpublished PhD thesis of Ingrid Irmer, which is available here.
- 44.8k
4
votes
Surface automorphisms and conformal automorphisms
It was proven in
J. MacCarthy and A. Papadopoulos. Involutions in surface mapping class groups. Enseign. Math. 33 (1987), 275–290.
that the mapping class group is generated by the conjugates of a si …
- 44.8k
4
votes
Accepted
a CW-complex homotopic to a manifold
For this to be true, you need to assume that your $4$-manifold $M$ is not a compact nonorientable manifold. Otherwise, you would have $H_4(M;\mathbb{Q}) = 0$ but $H_4(M;\mathbb{Z}/2) \neq 0$, so ther …
- 44.8k
17
votes
Accepted
Homotopy type of the plane minus a sequence with no limit points
Yes. More generally, if $X$ is a proper closed subset of $\mathbb{R}^2$, then every path component $M$ of $\mathbb{R}^2 \setminus X$ is homotopy equivalent to a wedge of circles (observe that $X$ mig …
- 44.8k
5
votes
Accepted
Normal generation of Torelli
For the Torelli group of a surface, I gave a much easier and more geometric proof in my paper
MR2302503 (2008c:57049)
Putman, Andrew(1-CHI)
Cutting and pasting in the Torelli group. (English summary …
- 44.8k
7
votes
How many different knot types can have the same shadow (projection)
There exist projections with any number of crossings such that any choices of over/under crossings gives the unknot.
I am writing on my phone and cannot draw a picture, but here is a description of o …
- 44.8k
4
votes
Reference request: Reidemeister type moves for immersed curves on surfaces
This isn't true as stated. What is true is that if $\gamma$ and $\gamma'$ two homotopic curves that are in minimal position (i.e. each contains the minimal number of self-intersections in their homot …
- 44.8k