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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).
3
votes
Fundamental group of the complement of a codimension two submanifold
To your first question, the answer is yes.
Take a $k$-component trivial link in $S^n$, i.e. the boring, linear embedding
$$\sqcup_k S^{n-2} \to S^n$$
that is the boundary of a linear embedding
$$\sqcu …
- 44.4k
20
votes
Accepted
Is there a 2 component link with full symmetry?
I think there is a non-hyperbolic link that does the job.
The link that I'm thinking of could be called the splice of two Bing doubles of a figure-8 knot. Another way to describe this link is to sta …
- 44.4k
2
votes
Handle decompositions subordinate to an open cover
If you call the standard $n$-simplex $\Delta^n$, i.e.
$$\Delta^n = \{ (x_0, x_1, \cdots, x_n) : x_i \geq 0 \forall i, \sum_i x_i = 1\}$$
then the function
$\phi : \Delta^n \to \mathbb R$
given by $\ph …
- 44.4k
5
votes
Equivalence of knotted spheres in $S^4$
$\DeclareMathOperator{\Diff}{Diff}$ The answer is "yes modulo some small potatoes".
There is one case where the answer is a simple no: if $K$ and $K'$ are mirror images of each other you can have a di …
- 35.5k
2
votes
Accepted
Numerical computation of the second Vassiliev invariant, and the permutation $(1 3 4 2)$
This is not a full answer to your question, but it gives some information.
The expression (1) is an integral version of the Polyak-Viro formula for the type-2 invariant, described here: https://academ …
- 44.4k
9
votes
Why/does 'low-dimension' topology end with dimension 4?
There are of course ways to construct difficult problems in high-dimensional manifold theory. One of the ways high-dimensional manifold theory differs strongly from low-dimensional manifold theory i …
- 44.4k
14
votes
Accepted
Isotopic diffeomorphisms of the sphere
This is Cerf's pseudoisotopy-implies-isotopy theorem.
Cerf's result is true in high dimensions, while it's independently known in a low-dimensional range. In dimension $2$ it goes back to the Earle–E …
- 82.7k
3
votes
Lie group framing and framed bordism
Lie group framing is a reference to the group action. A Lie group $G$ acts on the left of $G$ by the map
$$(g,h) \longmapsto gh.$$
Similarly there are actions on the right, and conjugation actions, e …
- 44.4k
9
votes
Generalization of the sphere theorem in dimension at least 4
I have a thread on the co-dimension $1$ generalization of Dehn's Lemma for 4-manifolds:
A problem/conjecture related to 4-manifolds that deserves a name. What name does it deserve?
My pair of papers w …
- 44.4k
5
votes
Accepted
How to get a presentation of the mapping class group of the $n$-punctured sphere
$\DeclareMathOperator\Diff{Diff}\DeclareMathOperator\Emb{Emb}\DeclareMathOperator\fix{fix}$There's a variety of ways to do this. If you take "mapping class group" to mean "isotopy classes of diffeomo …
- 44.4k
2
votes
Handle attachment information from Morse function and triangulation
Near a critical point $p$ of a genuine Morse function you have the local model
$$f(x) = f(p) + x_1^2 + \cdots + x_i^2 - x_{i+1}^2 - \cdots - x_n^2$$
Just below the height of $p$, your attaching sphere …
- 44.4k
13
votes
Is there a "knot theory" for graphs?
Yes, there's plenty of work on this. First of all, you have to define the notion of equivalence that you are interested in. Usually people only care about the graph up to handle-slide (turning the s …
- 2,821
89
votes
Accepted
Can every manifold be given an analytic structure?
(similar to Mariano's post)
Q1: no. There are topological manifolds that don't admit triangulations, let alone smooth structures. All smooth manifolds admit triangulations, this is a theorem of White …
- 4,713
6
votes
Accepted
Is there a geometric interpretation of the second derivative of the Alexander polynomial at ...
Given a knot in $S^3$, think of it as an embedding
$$f : S^1 \to S^3.$$
The configuration space of $5$ distinct points in $S^3$ is denoted $C_5(S^3)$, this is a $15$-dimensional manifold and it consis …
- 44.4k
4
votes
Accepted
Does every simply connected, orientable, non-compact, 3-manifold embed in $\mathbb{R}^3$?
When the manifold is the universal cover of a compact $3$-manifold $M$ (to begin with, lets say without boundary) then you construct the embedding by hands, using geometrization. In your question let …
- 44.4k