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Results tagged with gt.geometric-topology
Search options questions only
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user 6094
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).
65
votes
4
answers
4k
views
Tying knots with reflecting lightrays
Let a lightray bounce around inside a cube whose faces
are (internal) mirrors.
If its slopes are rational, it will eventually form a cycle.
For example, starting with a point $p_0$ in the interior
of …
49
votes
3
answers
8k
views
Thurston's 24 questions: All settled?
Thurston's 1982 article on three-dimensional manifolds1 ends with $24$ "open questions":
$\cdots$
Two naive questions from an outsider:
(1) Have all $24$ now been resolved?
(2) If so, w …
- 151k
45
votes
1
answer
2k
views
Pach's "Animals": What if the genus is positive?
Janos Pach asked a deep question 23 years ago (1988) that remains unsolved today:
Can every animal—a topological ball in $\mathbb{R^3}$ composed of unit cubes glued face-to-face—be reduced to a si …
- 151k
41
votes
1
answer
6k
views
Not all manifolds can be triangulated: In which dimensions?
I know that Ciprian Manolescu has settled the triangulation conjecture in the negative: Not all manifolds can be triangulated. I've only read secondary literature on this result, which did not detail …
- 151k
37
votes
2
answers
5k
views
Kervaire invariant: Why dimension 126 especially difficult?
Is there any resource that might help non-experts gains some understanding of why
the Kervaire invariant problem remains open now only in dimension $126$? ($126 =2^7-2=2^{j+1}-2$;
whether $\theta_j=\t …
- 151k
26
votes
5
answers
2k
views
Complexity of random knot with vertices on sphere
Connect $n$ random points on a sphere in a cycle of
segments between succesive points:
I would like to know the growth rate, with respect to $n$, of the crossing number
(the minimal number of crossin …
- 151k
24
votes
1
answer
2k
views
Building a genus-$n$ torus from cubes
I wonder if this has been studied:
What is the fewest number of unit cubes
from which one can build an $n$-toroid?
The cubes must be glued face-to-face,
and the boundary of the resulting objec …
- 151k
24
votes
2
answers
2k
views
Is every rational realized as the Euler characteristic of some manifold or orbifold?
Let me first ask the question for two-dimensional compact, connected manifolds and orbifolds.
Then, if the answer is No, one can remove various conditions on the dimension,
and allow non-compact examp …
- 151k
21
votes
1
answer
1k
views
Homeomorphism historically: When did it reach its modern formulation?
Q. When did the notion of homeomorphism reach its
modern formulation as a bicontinuous bijection, i.e., a
continuous bijection
between topological spaces whose inverse is also continuous?
…
- 151k
17
votes
1
answer
525
views
Is $\partial X$ a sphere for $X$ a complete CAT$(0)$ space?
Let $X$ be a complete CAT$(0)$ metric space, and $\partial X$ its boundary.
One way to define $\partial X$ is as the equivalence class of geodesic rays
$\gamma(t), \gamma'(t)$
that remain within a co …
- 151k
15
votes
5
answers
3k
views
Generalization of winding number to higher dimensions
Is there a natural geometric generalization of the winding number to higher dimensions?
I know it primarily as an important and useful index for closed, plane curves
(e.g., the Jordan Curve Theor …
- 151k
13
votes
3
answers
1k
views
Random Reidemeister moves to unknot
Suppose one has a link diagram of the unknot, and applies random Reidemeister moves
until the unknot is reached.
Surely it requires an exponential number of moves, exponential in, say, the crossing nu …
- 151k
12
votes
1
answer
368
views
Why do convex polytope options constrict with dimension, rather than expand?
There are an infinite number of regular polygons in the plane,
five regular polyhedra,
six regular polytopes in $\mathbb{R}^4$,
and then three regular polytopes in every dimension $d > 4$.
There are …
- 151k
11
votes
4
answers
1k
views
Distance between two knots
Are there some well-studied functions defining natural distance measures between two knots? One can imagine a function that counts, say, the minimum number of
moves, each of which passes one strand of …
- 151k
10
votes
2
answers
2k
views
Do all combinatorially distinct fundamental polygons correspond to surfaces?
The topology of a closed surface can be constructed
by identifying edges of a fundamental polygon of an
even number $2n$ of edges.
Labeling the edges and using $\pm 1$ exponents to indicate
direction, …
- 151k