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Results tagged with gt.geometric-topology
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user 2233
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).
20
votes
2
answers
2k
views
Simple curves on non-orientable surfaces.
Given an element in the (first) homology group of a surface, I would like to know if it can be represented as a simple closed curve. For orientable surfaces, this is well-known, but I wasn't able to …
- 32.1k
19
votes
3
answers
995
views
Drawing planar graphs with integer edge lengths
It is well known that every planar graph has an embedding such that every edge is drawn as a straight line segment (Fáry's Theorem). Kemnitz and Harborth made the following stronger conjecture
Conje …
- 32.1k
18
votes
2
answers
4k
views
Turning pants inside-out (or backwards) while tied together
An entertaining topological party trick that I have seen performed is to turn your pants inside-out while having your feet tied together by a piece of string. For a demonstration, check out this vide …
- 32.1k
17
votes
Applications of infinite graph theory
Here's a nice proof of the Cantor-Bernstein theorem in the language of infinite graphs.
Theorem. Let $G$ be an infinite graph with bipartition $(A,B)$. If $G$ has a matching saturating $A$ and a m …
- 32.1k
7
votes
Is it possible that every edge in a 1-planar drawing with minimum number of crossings is cro...
It is not possible that in an optimal drawing of a 1-planar graph, every edge is crossed. Here is a proof.
Suppose not and let $G$ be a smallest counterexample. I claim that $G$ is 2-connected. If …
- 32.1k
4
votes
Accepted
Least cardinality of a set of points in the plane
As Boris Bukh points out, three points suffice, but I'd like to point out that your question is related to this MO question.
Here is a summary of the information in the previous question. For the …
- 32.1k