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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).
6
votes
0
answers
210
views
"A typical pair of two-dimensional surfaces in four dimensions will intersect at finitely ma...
The title quotes C.H. Taubes in the Princeton Companion to Mathematics (p.404).
I found this surprising despite the natural lower-dimensional analog
(a typical pair of loops in
$\mathbb{R}^2$ will in …
- 151k
1
vote
0
answers
217
views
Patterns in local winding number sequences
This is something of a followup to an earlier question
Can any sequence of consecutive integers be realized as winding numbers?, whose answer is Yes.
Now I would like to define a local winding number …
- 151k
8
votes
1
answer
717
views
Can any sequence of consecutive integers be realized as winding numbers?
For a closed plane curve $C$, define its sequence of winding numbers to
be the sorted list of the winding numbers of each of the distinct regions
of the plane demarcated by $C$.
For example, this curv …
- 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
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
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
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
2
votes
1
answer
101
views
Complexity of recognizing equivalent translation surfaces
"A translation surface is a union of polygons with pairs of parallel edges identified by translation, up to cut and paste equivalence."
I take that succinct (and not fully precise) definition fro …
- 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
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
10
votes
3
answers
1k
views
Which polygons have *simple* periodic billiard paths?
I know (or, rather, believe) that it remains unknown whether every polygon
has a periodic billiard path.
But Howard Masur proved in the 1980's that every rational polygon
(vertex angles rational mult …
- 151k
5
votes
0
answers
275
views
Is there a connection between |roots| $\rightarrow$ 1 and Gromov's waist theorem?
Recent questions showed that roots of a random polynomial tend to lie on the
unit circle ("Why do roots of polynomials tend to have absolute value close to 1?"; "Distribution of roots of complex polyn …
- 151k
5
votes
1
answer
482
views
Knot invariants in 3-manifolds that are not $\mathbb{R}^3$ or $S^3$ or $B^3$?
This is just a reference request; I have no sharp mathematical question.
Inspired by the $(3+)$-year old MO question,
In knot theory: Benefits of working in $S^3$ instead of $\mathbb{R}^3$?,
I would …
- 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
3
votes
1
answer
458
views
Surface curves equidistant from a simple closed geodesic
Let $S \subset \mathbb{R}^3$ be a surface embedded in $\mathbb{R}^3$,
let's say (to keep it simple) of genus zero.
Let $\gamma$ be a simple, closed, oriented geodesic on $S$.
Because $\gamma$ is orien …
- 151k