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).
Questions tagged [gt.geometric-topology]
4142 questions
39
votes
4 answers
Thurston's "tinker toy" problem
In the article "On Being Thurstonized" by Benson Farb (located here), a particular result of Thurston is mentioned.
Namely, suppose a "tinker toy" $T$ is a contraption consisting of a multitude of rods. They can either be bolted to a table, or…
Rohil Prasad
- 1,591
31
votes
1 answer
Open immersions of open manifolds
For concreteness, I will work in the category of smooth manifolds, but my question makes sense in topological and PL category as well. Recall that a manifold $M$ is called open if every connected component of $M$ is non-compact.
Question. Is it…
Misha
- 30,995
27
votes
2 answers
Are non-PL manifolds CW-complexes?
Can every topological (not necessarily smooth or PL) manifold be given the structure of a CW complex?
I'm pretty sure that the answer is yes. However, I have not managed to find a reference for this.
A grad student
- 271
27
votes
4 answers
Examples of non-diffeomorphic smooth manifolds with diffeomorphic tangent bundle
Given a smooth manifold $M$, we define the differentiable structure on $TM$ in the usual way.
I would like to know examples of two smooth manifolds which are non-diffeomorphic, but their tangent bundles are.
Which is the smallest dimension in which…
Just a mathematician
- 1,317
22
votes
0 answers
Smooth thickenings of non-smoothable manifolds
It is known that any closed topological manifold is homotopy equivalent to an open smooth manifold.
Question 1. What can be said about the smallest
dimension of a smooth manifold
that is homotopy equivalent to a given closed topological…
Igor Belegradek
- 28,360
22
votes
1 answer
Pseudoisotopy in low dimensions
Recall that the space $P(M)$ of (smooth) pseudoisotopies of the compact manifold $M$ is defined as the space of all diffeomorphisms $M\times I\to M\times I$ that fix every point in $(M\times 0)\cup (\partial M \times I)$. I have worked on these…
Tom Goodwillie
- 54,421
21
votes
1 answer
If a manifold suspends to a sphere...
I have a topological manifold whose suspension is homeomorphic to the sphere $S^{k+1}$. Is it necessarily itself homeomorphic to $S^k$?
I know that this is not true if I replace "suspension" with "double suspension", because I found the helpfully…
James Cranch
- 3,034
20
votes
2 answers
Fibers of fibrations of a 3-manifold over $S^1$
Given a fiber bundle $S\hookrightarrow M \rightarrow S^1$ with $M$ (suppose compact closed connected and oriented) 3-manifold and $S$ a compact connected surface, it follows form the exact homotopy sequence that $\pi_1(S)\hookrightarrow \pi_1(M)$.…
Francesco Lin
- 1,027
20
votes
2 answers
What manifolds are boundaries of euclidian spaces ?
I would like to know if there are compact (n-1)-manifolds $N$ that are not spheres but such that there is a manifold with boundary $M$ which satisfies the following two properties:
$\partial M\cong N$
$M-\partial M\cong \mathbb{R}^n$
I am…
Geoffroy Horel
- 2,640
19
votes
1 answer
Are the Morse inequalities sharp for 5-manifolds
Given a compact smooth manifold $M$ denote by $b_i(M)$ the $i$-th Betti number and denote by $q_i(M)$ the minimal number of generators for $H_i(M)$. Let $f$ be a Morse function on $M$. The Morse inequalities say that the number of critical points of…
Stefan Friedl
- 2,587
19
votes
4 answers
4-dimensional h-cobordisms
I would like to know the state of the art concerning the following two questions.
1) Does there exist a smooth 4-dimensional h-cobordism (so between closed 3-manifolds) with non-vanishing Whitehead torsion ?
2) Does there exist a smooth…
Sylvain Courte
- 193
18
votes
1 answer
Homotopies of triangulations
I imagine this is pretty much standard, but surely someone here will be able to provide useful references...
Suppose $X$ is a topological space. Let me say that two triangulations $T$ and $T'$ of $X$, are homotopic if there is a triangulation on…
Mariano Suárez-Álvarez
- 46,795
18
votes
2 answers
Which immersed plane curves bound an immersed disk?
I am looking for a nice answer to the following question.
Which immersed plane curves bound an immersed disk?
Comments.
I am not sure what is a nice answer, but for sure I could make a stupid algorithm.
I am aware that there are plane curves…
Anton Petrunin
- 43,739
17
votes
1 answer
$(n-1)$-dimensional sphere in $S^n$ such that the closure of a component of complement is not contractible
Let $f:S^{n-1} \rightarrow S^n$ be a topological embedding and let $A_f$ and $B_f$ be the components of $S^n \setminus f(S^{n-1})$. If $\overline{A}_f$ and $\overline{B}_f$ are manifolds with boundary $f(S^{n-1})$, then the locally flat Schoenflies…
Sarah
- 291
16
votes
2 answers
Is there a 2 component link with full symmetry?
If you take a (labelled, oriented) 2 component link, it has a symmetry group which is a subgroup of the 16 element group $Z2 \times (Z2 \times Z2 \ltimes S_2)$ (mirror, reverse each component, swap the components). With Parsley, Cornish, and Mastin…
Jason Cantarella
- 707