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Results tagged with gt.geometric-topology
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user 943
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).
4
votes
Accepted
Can you explain a step in a proof about 2-sided surfaces in 3-manifolds?
We want to prove that in the case F is not one-sided, we may replace J by a curve J' that is contained in a small neighborhood of F and interesects F in the same way as J. By assumtion F is one sided. …
- 28.9k
4
votes
Necessary and sufficient criteria for a surface to cover a surface
I don't know the reference for this question, but I am pretty sure that it should follow from some known statement. Anyway let me give the answer in the case when S has negative Euler characteristic, …
- 28.9k
3
votes
limiting behaviour of converging loops on a torus
EDITED. The following example should not be conisdered as a contre-example,
it is just a well known example of a pathology that can happen.
EXAMPLE. $L$ is composed of a union of the vertical circl …
- 28.9k
7
votes
Accepted
If a polyhedron is homeomorphic to a simplex, is it piecewise-linear homeomorphic?
If you assume that your polyhedron has only finite number of faces, I think the answer to your question is unknown. Moreover any answer to such question would give a solution to Smooth Poincare conjec …
- 28.9k
2
votes
Hamiltonian actions and contractible loops
There are counter-examples, hope they answer your question completely, just take
any non-simply connected $G$ and consider its action on $T^*G$. The simplest case is:
Let $M$ be the cylinder $S^1\tim …
- 28.9k
9
votes
Classification of symplectic surfaces
This is not a complete answer to the question, and I don't know if a complete answer is written down anywhere in the literature. In the first revison of the answer I tried to adress all the 10 commen …
- 28.9k
6
votes
Accepted
A question about regularity of foliations
As far as I understand, the answer to the question is no, you can check this in
Handbook of Dynamical Systems, Volume 1, Part 1 By Boris Hasselblatt, Anatole Katok, page 173. This question is identica …
- 28.9k
5
votes
Accepted
Embeddings of magnetic cotangent bundles over surfaces into closed symplectic 4-manifolds
This can always be done.
Let's first treat the case when $\Sigma$ is not a torus. Then take any symplectic $4$-manifold $(M,\omega)$ where $\Sigma$ can be embedded as a Lagrangian surface. Now, take a …
- 28.9k
5
votes
Accepted
Negative intersection of symplectic submanifolds
The answer to this question is YES. I assume you want $A$ and $B$ to be connected.
Already in the case of four manifolds two symplectic surfaces can have negative intersection.
To construct an examp …
- 28.9k
6
votes
Topology of plane curve complements after blow-ups
Corrected according to the comment of Scott.
Let me show that in the case you blow up $\mathbb P^2$ in one point an throw away preimages of three lines trough the point you get something with non-abe …
- 28.9k
16
votes
Closed 3-manifolds with free abelian fundamental groups
Only $\mathbb Z$ and $\mathbb Z^3$ (for $T^3$) are free abelian groups that appear as fundamental groups of $3$-manifolds. Hopefully the following is an approximative proof.
The manifold must be prim …
- 28.9k
11
votes
Accepted
Osculating conics and cubics and beyond
These highly osculating curves were studied, in particular by V.I. Arnol'd. One of the important references will be:
Topological invariants of plane curves and caustics.
Dean Jacqueline B. Lewis Me …
- 28.9k
13
votes
Classification problem for non-compact manifolds
This answer complement the answers of Henry and Algori. I think, it is worth to strees, that a classification of open manifolds does not follow from a classification of compact manifolds. Open surface …
- 28.9k
23
votes
Gromov's list of 7 constructions in differential topology
Unfortunately I missed the talk, but on the other hand Gromov have just produced a new paper called
Manifolds : Where do we come from ? What are we ? Where are we going ?
It can be found on his we …
- 28.9k
4
votes
Mapping Class Groups of Punctured Surfaces (and maybe Billiards)
Here is an arcticle of Luo,
A Presentation of the Mapping Class Groups
http://arxiv.org/PS_cache/math/pdf/9801/9801025v1.pdf
I remember seen somewhere a neat presentation of the hyperelliptic mapp …
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