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 ...

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6
votes
2answers
321 views

Simplicial replacements in smoothing theory

As far as I can tell, ever since Milnor's Microbundles and differentiable structures (1961) paper, whenever people talk about $Diff(\mathbb R^n)$ or $PL(\mathbb R^n)$ or $Homeo(\mathbb R^n)$, they ...
8
votes
2answers
193 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] ...
0
votes
1answer
84 views

Connecting two hypersurfaces in R^{n+1} by embedded curves

Let $M^n$ be a smooth closed embedded hupersurface in $\mathbb R^{n+1}$. Denote by $D$ the bounded connected component of $\mathbb R^{n+1}\backslash M$. We assume that $\mathbb R^{n+1}\backslash D$ is ...
0
votes
1answer
139 views

discrete subgroups of the isometries of a product

Suppose $X_1$ and $X_2$ are two nice metric spaces, e.g. two Riemannian manifolds, and let $G_i=Isom(X_i)$. Then $G_1\times G_2\subset Isom(X_1\times X_2)$. Suppose $X_1\times X_2$ is not compact and ...
0
votes
0answers
166 views

Fractional degree of a map?

Is there some natural notion of a fractional degree of a map? The degree of a map is a generalization of the winding number, and fractional winding numbers appear in the (mathematical physics) ...
3
votes
1answer
186 views

Symmetries of the standard probability space

The standard probability space $(I, \mathcal B, \lambda)$ consists of the interval $I = [0,1]$, its Borel $\sigma$-algebra $\mathcal B := \mathcal B(I)$ and Lebesgue measure $\lambda$. In ...
6
votes
0answers
236 views

Does a finitely generated aspherical group have an aspherical presentation with a finite generating set?

Let $G$ be a finitely generated group. Suppose $G$ has an aspherical presentation with a countably infinite generating set. Does $G$ have an aspherical presentation with a finite generating set? Here ...
1
vote
0answers
167 views

Toral decomposition

I have a couple of questions on the following theorem: Theorem. (Jaco, Shalen) Let $M$ be a compact, irreducible, orientable 3-manifold with incompressible boundary. There exists a collection ...
1
vote
0answers
131 views

Can a compact metrizable space be determined by its Hausdorff measures?

Suppose that $(X,d)$ is a compact metric space. Now suppose that $h:[0,a]\rightarrow[0,b]$ is a continuous function with $h(0)=0$ where if $x\leq y$, then $h(x)\leq h(y)$. Then define ...
6
votes
1answer
191 views

Characterisation of Q-rank 1

I'm looking for a reference and/or the original source for the following fact: An irreducible non-uniform lattice in a semisimple Lie group without compact factor has Q-rank 1 if and only if it does ...
16
votes
3answers
515 views

What is the state of the art for algorithmic knot simplification?

Question: Given a `hard' diagram of a knot, with over a hundred crossings, what is the best algorithm and software tool to simplify it? Will it also simplify virtual knot diagrams, tangle diagrams, ...
3
votes
1answer
264 views

On complex surfaces with Kodaira dimension 1

Let $S$ be a complex surface of Kodaira dimension $1$ and $\pi_{1}(S) \neq 1 $. What is known on possible diffeomorphism types of such $complex$ surfaces with a given fundamental group? Is it true ...
2
votes
1answer
216 views

Covering seifert manifolds

Let $M$ be a 3-manifold with boundary. If $M$ has an orientable finite cover that is a Seifert fiber space, then is $M$ also a Seifert fiber space?
0
votes
1answer
117 views

3-complexes not embeddable in 3-space

My question is about embeddability of 3-dimensional complexes in R^3. Do we have something like Kuratowski's theorem for complexes in 3-space which specifies a set of minors for non-embeddability?
11
votes
3answers
602 views

Is the class of n-dimensional manifolds essentially small?

Question: Consider the proper class of all $n$-dimensional smooth manifolds. If we take the equivalence classes where two manifolds are identified if there exists a diffeomorphism between them, is ...
3
votes
1answer
147 views

3-manifolds bounded by a non-compact Riemann surface of genus 0

Suppose $S$ is a non-compact Riemann surface in $\mathbb R^3$ that has no boundary and has genus zero (i.e. its fundamental group is generated only by its ends at infinity). A typical example would be ...
6
votes
3answers
556 views

Complex structure of the Teichmüller space in terms of Fenchel-Nielsen coordinates

The Teichmüller space $T_g$ of genus $g$ Riemann surfaces can be parameterized in terms of Fenchel-Nielsen coordinates, taking values in $\mathbb{R}^{3g-3}\times \mathbb{R}_+^{3g-3}$. The ...
3
votes
0answers
107 views

More questions about high-dimensional knot invariants

In a question yesterday I asked about the existence of algebraic invariants for embeddings of n-manifolds into n+2-spheres. The answers in the positive dimension all made certain assumptions about ...
7
votes
2answers
255 views

Existence of Morse functions on simply connected manifolds

Is it true that any simply connected closed manifold possesses a Morse functions that does not have critical points of index one? If the dimension is at least 5, this is a consequence of the results ...
9
votes
3answers
288 views

Invariants of high-dimensional knots

In the study of knots in three dimensions, it can be shown that the fundamental group together with a specification of a meridian and longitude form a complete invariant for knots. What is known about ...
1
vote
0answers
163 views

Topological razors (ball-like spaces)

Introduction Many admire the Euclidean space, and I am not an exception. I will try to catch the topological roundness of the $n$-ball in its greatest generality. I call the resulting axiomatized ...
3
votes
0answers
239 views

Representation variety vs. space of flat connections

The holonomy provides a bijection from the space of flat G-connections (modulo gauge equivalence) on a trivial G-bundle over M to a connected component of the representation variety ...
11
votes
2answers
378 views

Vector field on 3-sphere

Let $V$ be a vector field on $S^3$ such that its singularity points, namely the points at which the vector field vanishes, are only sinks or sources (i.e. the field is converging or diverging). Is ...
4
votes
0answers
119 views

Rigidity of secondary characteristic classes

For a representation $\rho:\pi_1M\rightarrow GL(n,C)$ and the associated flat $GL(n,C)$-bundle $E_\rho\rightarrow M$ one has the Cheeger-Chern-Simons classes $$\hat{c}_k(E_\rho)\in H^{2k-1}(M,R/Z)$$ ...
2
votes
0answers
155 views

Is $\mathbb T^\infty$ homeomorphic to an open subset in $\ell^2$?

More specifically, letting $\mathbb S$ be the set $\{\kern.5mm\exp({\rm i}\,t):0\le t<2\pi\kern.6mm\}$ with the induced topology from the complex plane, let $\mathbb T^{\kern.6mm\infty}$ be the ...
3
votes
0answers
108 views

Euclidean realisation of a polyhedral complex

Let us say that an Euclidean polyhedral manifold is a manifold that is glued from a finite number of Euclidean polyhedrons by identifying isometrically their co-dimension $1$ faces. Let us assume that ...
7
votes
4answers
195 views

Does Dehn filling always decrease Gromov norm?

As an immediate consequence of Proposition 6.5.2 of Thurston's notes, we have that, if $M$ is a compact 3-manifold with toric boundary and $\tilde M$ is obtained from $M$ via Dehn filling, then for ...
4
votes
1answer
188 views

Interesting families of groups as group extensions

Let me start this question with an example that hopefully makes clear what I am looking for: A discrete subgroup $G$ of the group of euclidean isometries of $\mathbb{R}^d$ is called a ...
7
votes
1answer
253 views

Generators for exact representations of 3-manifold groups

Does anyone have a list of matrix generators for a bunch of hyperbolic 3-manifold groups? I am testing an algorithm and am looking for a collection of test cases. I am looking for exact values of ...
0
votes
1answer
163 views

Coadjoint orbits and homogeneous symplectic $G$-manifolds

We know this important fact from A.A.Kirillov that : Every homogeneous symplectic $G$-manifold is locally isomorphic to an orbit in the coadjoint representation of the group $G$ or a central ...
7
votes
1answer
294 views

Analogues of the curve complex for Out(F)

Let $F$ be a finitely generated free group. Question: Is there an authoritative survey of analogues of the curve complex for $\mathrm{Out}(F)$? If not, as seems likely, would a passing expert be ...
3
votes
1answer
130 views

the space of noncrossing partitions of S^1

A non-crossing partition of the set $\mathbb{Z}_{mn}=\{ 1, 2,\dots,m n\}=\bigcup X_i$ is a disjoint union of sets such that if $ a < b \in X_1$ and $ c < d \in X_2$ then we can't have $a ...
10
votes
4answers
334 views

How to detect a simple closed curve from the element in the fundamental group?

(1) Given a fundamental group representation of a hyperbolic surface, i.e. $<a_j,b_j|\prod[a_j,b_j]=1>$, and given an element in this group, can we determine whether this element can be ...
9
votes
0answers
221 views

Provide a citation for the “spine lemma”?

I'm looking for a citable reference for the following (perhaps folkloric?) result on topological field theories. (There are obviously generalizations to other dimensions; I'm happy with just the ...
5
votes
1answer
130 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 ...
5
votes
1answer
108 views

Embedding of flat surfaces

Let $S$ be a orientable compact surface with a flat euclidean structure with conical singularities (cf. [T] for instance). Let also $\mathcal P$ be a polyhedral euclidean decomposition of $S$ (with ...
9
votes
0answers
162 views

Embedding of products of projective spaces into a projective space

Does anybody know estimates on the minimal dimension $k$ for which the product $P^n \times P^m$ can be smoothly embedded into $P^k$? I am interested in projective spaces over $\mathbb RR$ and $\mathbb ...
2
votes
1answer
143 views

isotopy classes of embeddings of the torus

Let's consider $S^1$-bundle $E$ over a 2-manifold $M$. How many isotopy classes of embeddings of the torus $\mathbb{T}^2$ in $E$? For each free homotopy classes $\gamma$ of mappings of the circle ...
8
votes
0answers
124 views

Naive Reidemeister-Schreier for $\mathbb Z$ quotients

I have a question about a "standard" variant of the Reidemeister-Schreier algorithm used by topologists when manipulating manifolds they either know or suspect are fibre-bundles over $S^1$. Say you ...
4
votes
3answers
245 views

Can the intersection of the boundaries of compact and convex sets be a single element?

Let $H_1,H_2,\dots,H_n$ be compact and convex sets in $\mathbb{R}^n$ such that $\bigcap_{j=1}^n H_j$ has non-empty interior and for each $i=1,2,\dots,n$ there exist at least one element $x \in H_i$ ...
7
votes
1answer
241 views

Diffeomorphism group of the 2-sphere with $C^0$ topology

What is known about the homeomorphism (or homotopy) type of the group of $C^\infty$ diffeomorphisms of $S^2$ equipped with $C^0$ topology? The group is the image of the inclusion ...
6
votes
2answers
226 views

Knotted projective planes and fake complex projective space

Paul Melvin gave a talk at Knots in Washington last year in which he asked whether the connected sum of an odd twist-spin of a classical knot and a standard cross-cap embedding of ${\mathbb R}P^2$ is ...
4
votes
0answers
211 views

Why does the plastic constant appear in the snub icosidodecadodecahedron?

The golden ratio, $$\phi =\frac{1+\sqrt{5}}{2}$$ appears (among other polyhedra) in the Platonic solids icosahedron and dodecahedron, and it's quite easy to see the significance of the discriminant ...
0
votes
0answers
102 views

K3 surface minus finite set

Let $S$ be a complex K3 surface, and $P\subset S$ a finite set of points in $S$. It is known that $$ H^i(S,\mathbb{Z})\cong H^i(S\setminus P,\mathbb{Z}) $$ for $0\le i \le 2$. Then the Euler ...
5
votes
1answer
123 views

Injective simplicial maps between Arc complexes

Let $A(S)$ denotes the Arc complex of a finite type hyperbolic surface $S$ with nonempty boundary. Let $\lambda:A(S)\rightarrow A(S)$ be a map such that on triangulations of $S$ i.e. on the top ...
0
votes
2answers
125 views

Convexity of a minimum function

I was reading a proof of $9g-9$ theorem which states that $9g-9$ length parameters are sufficient the parametrize the Teichmuller space of a closed surface of genus $g$. The proof uses the following ...
14
votes
3answers
542 views

fixed point property for maps of compacts

Definition. A topological space $X$ has the Fixed Point Property (FPP) if every continuous self-map $X\to X$ has a fixed point. Question. If $X$ and $Y$ are homotopy-equivalent compact metrizable ...
1
vote
2answers
273 views

Commutativity in the Fundamental Group and Knot Theory

Let $M$ be a connected $3$-manifold and let $\alpha$ and $\beta$ be elements in $\pi_1(M)$. Then $\alpha$ and $\beta$ can be represented by two knots $a$ and $b$ in $M$. We may further require that ...
5
votes
1answer
277 views

Universal covering map from $\mathcal{H}$ to $\mathbb{C}\setminus \mathbb{Z}\oplus i\mathbb{Z}$ (the countably punctured complex plane)

It's a consequence of the uniformization theorem for simply connected Riemann surfaces that the universal cover of $\mathbb{C}\setminus(\mathbb{Z}\oplus i\mathbb{Z})$ ($\mathbb{C}$ punctured at all ...
16
votes
3answers
370 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 ...