# Questions tagged [gt.geometric-topology]

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).

4,227
questions

16
votes

0
answers

166
views

### The congruence subgroup property for mapping class groups and a conjecture of Grothendieck

This question is about a link between an open question in low-dimensional topology and a conjecture of Grothendieck, proved by Mochizuki. Let's start by stating them.
Recall that a subgroup $K$ of a ...

6
votes

1
answer

211
views

### Analytic continuation gives a covering space (and not just a local homeomorphism)

Let $\mathcal{G}$ be the space of germs of holomorphic functions defined on open subsets of $\mathbb{C}$, topologized in the usual way. There is a natural map $p\colon \mathcal{G} \rightarrow \mathbb{...

5
votes

0
answers

104
views

### Can every orientation preserving homeomorphism of a manifold isotoped to be identity on a locally flat embedded disk?

Let $M$ be a connected (topological) oriented $m$-manifold (say without boundary), and
let $\operatorname{Homeo}^+(M)$ be the group of orientation preserving homeomorphisms $M \to M$.
Is it true that ...

3
votes

0
answers

77
views

### Finiteness of non-orientable 3-manifolds with the same orientable two-fold cover

Given a compact, orientable and boundary incompressible 3-manifold $M$. Suppose that either $M$ is closed, or $\partial M$ consists of tori.
For which non-orientable 3-manifolds $N$, the orientable ...

0
votes

0
answers

54
views

### Approximating curves using only line sections and arcs

Given a curve $C$, I need to construct another curve $C'$ approximating $C$ under the following constraints: (1) $C'$ needs to be smooth, (2) $C'$ is composed of only line sections and arcs, (3) $C'$ ...

8
votes

0
answers

109
views

### Space of thick ending laminations

Let $\Sigma$ be a compact closed connected oriented surface of genus $g>1$. Klarreich proved that the space of ending laminations $\mathcal{EL}(\Sigma)$ is the ideal boundary of the curve complex $...

9
votes

1
answer

359
views

### Cohomological gap in arithmetic groups

$\DeclareMathOperator\SL{SL}$For the sake of this question, let's say that a group $G$ of finite cohomological dimension $n$ has a cohomological gap if for some $0 < i < n$ there is no subgroup $...

8
votes

1
answer

334
views

### Can I endow the following 3-manifold with a hyperbolic metric?

Consider the following three-dimensional topology. Start with $S^3$ and drill out four unlinked tori as shown in the picture. Then, fill in the gaps with the same tori but with their longitudes and ...

5
votes

3
answers

234
views

### On a metrized $n$-dimensional manifold $X$, does every $x \in X$ have a small ball $B_\delta(x)$ that is homeomorphic to $\mathbb R^n$?

Suppose that $X$ is an $n$-dimensional topological manifold that is also metrizable, and hence equipped with some metric that induces the topology.
For every point $x \in X$, let $B_\delta(x)$ be the ...

4
votes

1
answer

236
views

### Seifert surfaces of fibered knots

Given a fibered knot $K\subseteq S^3$, does every genus-minimizing Seifert surface appear as the fiber of a bundle $S^3\setminus K\to S^1$?

2
votes

0
answers

80
views

### Is isoperimetric hypersurface unique up to homeomorphism?

Is there a Riemannian structure on $\mathbb{R}^n $with two non homeomorphic compact hypersurfaces $M,N$ such that both satisfy the isoperimetric inequality. I precisely meanthe following:
$$\...

3
votes

1
answer

87
views

### Representation of Lie groups inducing a quasi-isometric embedding of their symmetric spaces

Let $G_{1}$ and $G_{2}$ be connected semisimple real Lie groups with no compact factors and finite center and let $K_{1}$ and $K_{2}$ denote some fixed choice of their maximal compact subgroups, ...

5
votes

2
answers

202
views

### $\mathbb{CP}(2)$ from gluing boundary of 4-ball

Many manifolds can be obtained from gluing the boundary of a ball. For example, $\mathbb{RP}(2)$ is obtained from gluing the two edges of a bi-gon (2-ball). Or, lens spaces are obtained from a 3-cell ...

4
votes

1
answer

227
views

### The complement of a properly embedded annulus in a handlebody is a handlebody

Suppose $H_g$ is a genus $g$, orientable handlebody. Suppose $A\subset H_g$ is properly embedded ($\partial H_g \cap A = \partial A$, a transverse intersection), non-$\partial$-parallel annulus. It ...

4
votes

0
answers

145
views

### Is there a notion of "locally flat" for CW complexes?

A submanifold $X^n\subset Y^m$ is locally flat if each point $x\in X$ has a neighborhood $U\subset Y$ so that $(U,U\cap X)\simeq (\Bbb R^m, \Bbb R^n)$ with the standard embedding $\Bbb R^n\...

3
votes

1
answer

155
views

### How to properly define a slice knot (or a locally flat disk)?

A knot $K\subset\Bbb S^3=\partial \Bbb D^4$ is said to be (topolopgically) slice if there is a locally flat disk $D\subset\Bbb D^4$ with $\partial D=D\cap \Bbb S^3=K$. As far as I understand, locally ...

7
votes

1
answer

418
views

### What are the covering spaces of $\mathbb{Q}$?

Let $X = \mathbb{Q}$, topologized as a subset of the real line. Is there a reasonable description of the covering spaces of $X$?
Here is something more precise. One way of constructing covers $p: \...

7
votes

1
answer

220
views

### When is the action of a mapping class group on the set of punctures realized by a finite subgroup of mapping classes?

I have a curious question about a natural sequence, which I haven't seen answered in the literature.
Let $\Sigma$ be an oriented surface of genus $g$ without boundary with a set $\mathcal P_n$ of $n$ ...

5
votes

0
answers

91
views

### For spaces $U$ and discrete sets $I,J$, are maps $f\colon U \times I \rightarrow U \times J$ commuting with the projection to $U$ covering spaces?

Let $U$ be a topological space, let $I$ and $J$ be discrete sets, and let $f\colon U \times I \rightarrow U \times J$ be a continuous map that commutes with the projection onto the first factor. In ...

6
votes

1
answer

196
views

### A stable splitting of linear surjections

Some computations I've been doing in Weiss calculus predict the following stable splitting of the space of linear surjections: $\Sigma^\infty_+ \mathrm{Sur}(\mathbb{R}^n,\mathbb{R}^{m_1+m_2})$
as the ...

2
votes

0
answers

67
views

### Uniqueness of symplectic mapping cylinder?

Given a Weinstein manifold $(X,\omega ,\phi )$ together with an exact symplectomorphism $\mu\colon X\to X$ we can form the symplectic mapping cylinder $$
M(\mu)\colon = X\times\mathbb{C}/((x,z)\sim (\...

4
votes

1
answer

197
views

### Equivalence of knotted spheres in $S^4$

Say we have two smoothly embedded spheres $K, K' \subset S^4$ that are equivalent in the sense that there is a diffeomorphism of pairs $(S^4, K)$ and $(S^4, K')$. Does it follow that they are ...

4
votes

1
answer

86
views

### $\partial$-incompressibility of a surface obtained when attaching a 2-handle to an irreducible 3-manifold produces a reducible 3-manifold

This question arises in my previous question.
Let $M$ be a compact, orientable, irreducible 3-manifold with incompressible boundary. Let $\alpha\subseteq \partial M$ be a simple closed curve, which is ...

5
votes

1
answer

349
views

### Proving the Cork Theorem

I am reading Kirby's paper paper "Akbulut's corks and h-cobordisms of smooth simply connected 4-manifolds" and I have a question about how to actually prove the cork theorem from the results ...

5
votes

1
answer

209
views

### Is it possible to fill a boundary component of an irreducible 3-manifold using a handlebody so that the resulting manifold is still irreducible?

Let $M$ be a compact, orientable, irreducible 3-manifold with boundary (possibly more than one component). Let $S\subseteq\partial M$ be one of its boundary components, which is an orientable surface ...

3
votes

1
answer

123
views

### Does the fundamental group of a compact 3-manifold induce full profinite topology on the fundamental group of its boundary?

Let $M^3$ be a compact, orientable, irreducible 3-manifold with incompressible boundary. Let $S\subseteq\partial M$ be one of its boundary components. Does $\pi_1(M)$ induce the full profinite ...

4
votes

0
answers

77
views

### Conditional convergence of sums over infinite sets

As undergraduates, we learn that conditional convergence of infinite series is highly sensitive to the order structure on $\mathbb N$: if $\sum_{n=0}^\infty x_n$ conditionally converges and $x \in \...

2
votes

1
answer

128
views

### Is there a Dehn-like presentation of a knot quandle?

The knot group can be presented using either a Wirtinger presentation (with generators corresponding to arcs of the knot diagram) or a Dehn presentation (with generators corresponding to regions of ...

11
votes

1
answer

314
views

### What is the minimal genus of a surface acted on by the symmetric group $S_n$?

For $G$ a finite group, it is easy to construct a (connected, orientable) surface with a faithful action of $G$. E.g.: take a disjoint union of $G$ many spheres, and add a 1-handle for every edge in ...

1
vote

0
answers

58
views

### Map from simplex to itself that preserves sub-simplices: revisited

Here it is proved that, if $f$ is a continuous map from an $n$-simplex $\Delta$ to itself, that maps each sub-simplex of $\Delta$ to itself, then $f$ must be onto $\Delta$ (surjective).
I would like ...

4
votes

1
answer

288
views

### Connectedness of Milnor fiber

Let $Q$ be a homogeneous polynomial in $n$ variables. Then it defines a locally trivial fiber bundle projection $Q:{\mathbb C}^n- Q^{-1}(0)\to {\mathbb C}-\{0\}$ (called Milnor fibration). Under what ...

2
votes

0
answers

45
views

### Fractional Dehn Twist coefficient of monodromy of rational open book

Given an open book $(S,h)$, the fractional Dehn twist coefficient $c(h)$ in some sense measures the difference between $h$ and its Thurston representative $g$. More specifically, one can consider the ...

4
votes

1
answer

237
views

### Intersection pairing on non-compact surface

Let $S$ be a smooth oriented connected $2$-manifold. We have an algebraic intersection pairing $\omega\colon H_1(S) \times H_1(S) \rightarrow \mathbb{Z}$. If $S$ is compact, then this is ...

4
votes

1
answer

137
views

### "Almost embedding" the complete 2-dimensional complex $\mathcal K_7^2$ into $\Bbb R^4$

Let $\mathcal K_7^2$ be the complete 2-dimensional simplicial complex on seven vertices, i.e. it has all $7\choose 2$ edges and all $7\choose 3$ 2-simplices (and no higher-dimensional simplices).
I ...

6
votes

1
answer

189
views

### Are finitely generated subgroups of $\text{GL}_n(\mathbb{Q})$ virtually special?

This might be a silly question--but are there any examples of finitely generated subgroups of $\text{GL}_n(\mathbb{Q})$ that are known to not be virtually special?

13
votes

0
answers

282
views

### When can we extend a diffeomorphism from a surface to its neighborhood as identity?

Let $M$ be a closed and simply-connected 4-manifold and let $f: M^4 \to M^4$ be a diffeomorphism such that $f^*: H^*(M;\mathbb{Z})\to H^*(M;\mathbb{Z})$ is the identity map. Moreover, let $\Sigma \...

13
votes

1
answer

483
views

### Low dimensional homotopy groups of $\operatorname{Top}(4)$

$\DeclareMathOperator\Top{Top}$I would like to compute $\pi_3\Top(4)$ and $\pi_4\Top(4)$. It is known that $\Top(4)/O(4) \rightarrow \Top/O$ is 5-connected and
$$
\pi_k(\Top/O) =
\begin{cases}
...

9
votes

0
answers

239
views

### Sheaf cohomology of non-paracompact manifolds (e.g. the long line)

I have long heard that manifolds are "affine". If we allow non-paracompact manifolds, then this seems to fail, since as explained in Dmitri Pavlov's answer, the Serre–Swan theorem fails. I ...

7
votes

2
answers

530
views

### Does there exist a Dehn filling of an irreducible 3-manifold with toroidal boundaries which is still irreducible?

Let $M$ be a compact, orientable, irreducible 3-manifold with incompressible toroidal boundary (there might be more than one boundary component). Is it always possible to choose appropriate slopes on ...

3
votes

1
answer

169
views

### Lengths of generators of surface group

Let $\Sigma$ be a closed genus $g\geq 2$ Riemann surface, which we equip with its unique constant curvature $-1$ hyperbolic metric. Let $\pi_1(\Sigma)$ be its fundamental group with respect to some ...

12
votes

2
answers

290
views

### Property P and R for general 3-manifolds

Let $Y$ be a closed oriented $3$-manifold and $K$ be a knot in $Y$. We say $K$ is the unknot if $K$ is contained in a local $3$-ball in $Y$ and is unknotted therein.
Generalized Property R:
If a Dehn ...

2
votes

0
answers

100
views

### Generalized Stokes's theorem and the relationship of volume to surface area of objects of arbitrary genus

In this post I saw that it could be explained with the Generalized Stokes's theorem why the derivative of the area of a circle is equal to the boundary of the circle (the circumference):
$$C=\frac{d}{...

17
votes

0
answers

575
views

### Must the number of smooth structures be countable or continuum?

Let $M$ be a manifold. Must the number of non-diffeomorphic smooth structures on $M$ be either countable or continuum? Could it be something in between when the continuum hypothesis fails?
Edit:
By ...

2
votes

0
answers

95
views

### Distributions of random walks on boundaries of balls in hyperbolic metric spaces

Suppose $G$ is a finitely-generated non-elementary hyperbolic group and consider a symmetric random walk on the Cayley graph $\text{Cay}(G,S)$ with generating set $S$. Denote the set of points $B_{\...

2
votes

1
answer

135
views

### Extending diffeomorphisms between surfaces

Suppose we have two smooth compact oriented surfaces $M_1$ and $M_2$ with boundary,both of them have genus $g$, and there are orientation preserving diffeomorphisms $\psi_1, \psi_2, \cdots, \psi_n$, ...

1
vote

0
answers

48
views

### Connected pre-images spanning $n$-cubes under dimension reducing maps

Let $I^n = [0,1]^n$ be the $n$-dimensional hypercube. For a continuous function $f: I^n \to \mathbb{R}^m$ with $m < n$, we're interested in the existence of points $p \in \mathbb{R}^m$ whose ...

5
votes

1
answer

92
views

### When do two measured foliations on a surface define a Riemann surface structure?

Let $S$ be smooth surface of finite type, i.e. it has genus g and n punctures (assume $S$ to have negative Euler characteristic). We know by Hubbard-Masur theorem that given a measured foliation $(F,\...

4
votes

0
answers

156
views

### Question on the construction of transversely oriented foliation on a sutured 3-manifold

The question is based on the proof of the main theorem of Gabai's paper on Foliations and the topology of 3-manifolds which is the following:
Theorem 5.1. Suppose $M$ is connected, and $(M,\gamma)$ ...

8
votes

1
answer

208
views

### Bilinear forms on abelian $p$-groups: Images of weak metabolizers in the Frattini quotient

Suppose $G$ is a finite abelian $p$-group for $p$ an odd prime and $b : G \times G \to \mathbf{Q}/\mathbf{Z}$ is a non-degenerate symmetric bilinear form. A subgroup $H \leq G$ is a weak metabolizer ...

3
votes

1
answer

417
views

### Detecting a PL sphere and decompositions

Question 1. A PL $n$-sphere is a pure simplicial complex that is a triangulation of the $n$-sphere. I'm looking for a computer algorithm that gives a Yes/No answer to whether a given simplicial ...