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Results tagged with gt.geometric-topology
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user 1465
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).
60
votes
6
answers
7k
views
Torsion in homology or fundamental group of subsets of Euclidean 3-space
Here's a problem I've found entertaining.
Is it possible to find a subset of 3-dimensional Euclidean space such that its homology groups (integer coefficients) or one of its fundamental groups is not …
- 44.4k
32
votes
3
answers
2k
views
A Pachner complex for triangulated manifolds
A theorem of Pachner's states that if two triangulated PL-manifolds are PL-homeomorphic, the two triangulations are related via a finite sequence of moves, nowadays called "Pachner moves".
A Pachne …
- 44.4k
25
votes
2
answers
841
views
Which homotopy classes $S^3 \to S^2$ lift to embeddings $S^3 \to S^2 \times D^3$?
The question is, for a smooth embedding
$$f : S^3 \to S^2 \times D^3$$
one can compose the map $f$ with projection $\pi : S^2 \times D^3 \to S^2$, giving the map $\pi \circ f : S^3 \to S^2$.
Which hom …
- 44.4k
24
votes
0
answers
1k
views
Monoid structure of oriented manifolds with connect sum
Take the class of all compact, connected, boundaryless, smooth oriented $n$-dimensional manifolds, each taken up to orientation-preserving diffeomorphism. This is a commutative monoid with operation …
- 44.4k
24
votes
2
answers
4k
views
Isotopy extension theorems
I'm looking for the origins of the isotopy extension theorem in categories other than the smooth category.
Precisely, in the smooth category, the isotopy extension theorem says that if $f : [0,1] …
- 44.4k
22
votes
3
answers
1k
views
"Largest" finite-dimensional Lie subgroups of Diff(S^n), are they known?
The group $Diff(S^n)$ ($C^\infty$-smooth diffeomorphisms of the $n$-sphere) has many interesting subgroups. But one question I've never seen explored is what are its "big" finite-dimensional subgroup …
- 44.4k
22
votes
1
answer
1k
views
Word problem for fundamental group of submanifolds of the 4-sphere
Given any finitely-presented group $G$, there are a few equivalent techniques for constructing smooth/PL 4-manifolds $M$ such that $\pi_1 M$ is isomorphic to $G$. For most constructions of these 4-ma …
- 44.4k
16
votes
3
answers
1k
views
Looking for "large knot" examples
This question is about knots and links in the 3-sphere. I want to find an example of a "large" knot or link with some special properties. I'm looking for some fairly specific examples, but I'm also …
- 44.4k
15
votes
1
answer
514
views
Where is the Steenrod Realization problem at?
I'm wondering if there is a more modern reference out there for the Steenrod Realization Problem than the book of Connor and Floyd?
Realizing homology classes in a manifold via embedded submanifolds, …
- 44.4k
14
votes
0
answers
500
views
Structure of Gordian graph of knots
The Gordian graph of knots has the knot isotopy classes as it's vertices, and an edge whenever you can pass from one knot to a other via a "finger move", equivalently if for some diagram of the knot y …
- 44.4k
13
votes
2
answers
1k
views
Explicit embeddings of Cappell-Shaneson knots
In 1976 Cappell and Shaneson gave some examples of knots in homotopy 4-spheres and for some time these examples were considered as possible counter-examples to the smooth 4-dimensional Poincare conjec …
- 44.4k
13
votes
1
answer
1k
views
Mapping class groups of small Seifert-fibred 3-manifolds
Are computations of the mapping class groups of small Seifert-fibred 3-manifolds recorded in some convenient location?
For most Seifert manifolds working out the mapping class group is easy-enough ( …
- 44.4k
12
votes
1
answer
508
views
Lower bounds on dimensions of faithful representations of braid groups
Let $B_n$ be the braid group on $n$ strands. It's a theorem of Daan Krammer and Stephen Bigelow that there is a a faithful representation
$$B_n \to GL_{n \choose 2} \mathbb Z[t^{\pm}, q^{\pm}] $$
i …
- 44.4k
12
votes
3
answers
1k
views
Knot theory question: bridge number vs. min generators of fundamental group of complement
Given a knot in the 3-sphere in Bridge Position you can find a presentation for the fundamental group of the complement (a Wirthinger presentation) containing $n$ generators and $n-1$ relators, where …
- 44.4k
12
votes
2
answers
555
views
Curvature flows for PL closed curves in the plane?
I'm curious to what extent people have studied "curvature flows" on PL closed curves in the plane.
There's a paper by Gage and Hamilton from 1986 that describes the long-term behaviour of smooth cur …
- 44.4k