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 ...

learn more… | top users | synonyms (1)

2
votes
1answer
42 views

Looking for examples of large hyperbolic two-generator knots or 3-manifolds

We say a knot $K$ in $S^3$ is small if its exterior contains no closed properly embedded incompressible surfaces and we say $K$ is large otherwise. Does anyone know of an example of a large ...
0
votes
0answers
101 views

Canonical model on Kontsevich's moduli space

Motivation: Work of Eisenbud, Harris, and Mumford shows that $M_g$ is of general type when $g≥24$. What about Kontsevich's moduli space? Let $X$ be a smooth complex projective Calabi-Yau manifold, ...
9
votes
2answers
149 views

Triangulation with simplices of same volume

Let $M$ be a Riemannian smooth compact manifold. It is known that $M$ has a triangulation, for any dimension. But do we know if there exists a triangulation such that all simplices have same volume ? ...
6
votes
1answer
221 views

“structure group” for fibration

Regarding "fibration" as a homotopy analogue of "fiber bundle",I want to see parallel notions of "structure group" and "fiber change" in "fibration". Does it make sense to talk about "structure ...
3
votes
0answers
92 views

Can we use the “size” of smooth structure set to predict the information geometry or other topological information?

The "size" can mean the number of elements or the diameter of the set of smooth structures. Y. Shikata defined a distance function on it and proved that it is a distance. He then used it to prove that ...
-1
votes
0answers
43 views

Group of self-homeomorphisms of the projective line [closed]

let $S := \mathbb{P}^1(k)$ be the scheme of the projective line over a field $k$. My question is: what is the group of homeomorphisms of $Z(S) \mapsto Z(S)$, where $Z(S)$ denotes the Zarisky topology ...
3
votes
1answer
81 views

What is the Heegaard Floer Homology of a connect sum of $S^2 \times S^1$s?

There are many conjecturally-equivalent three-manifold Floer homologies, of which my understanding is the most-computable is Heegaard Floer homology. What is the (Heegaard) Floer homology of a ...
19
votes
2answers
524 views

Is there a smooth manifold which admits only rigid metrics?

Does there exist a (finite dimensional) smooth manifold $M$, such that every Riemannian metric on $M$ has no isometries except the identity? Of course, such a manifold must not admit a diffeomorphism ...
3
votes
1answer
170 views

Is there a degree one map from a product $B\times S^1 \to \#_n S^2 \times S^1$ for any n

For any $n \geq 1$, let $\Sigma_n$ denote the closed orientable surface of genus n. In http://arxiv.org/abs/1202.6302, the authors showed that for any $n$, there is a degree two, $\pi_1$-surjective, ...
5
votes
1answer
98 views

Are square tiled surfaces dense in the moduli space of translation surfaces?

I'm reading the survey "An introduction to Veech surfaces" by Pascal Hubert and Thomas Schmidt. At page 19 they state "In any fixed stratum, the set of square-tiled surfaces of that stratum is ...
5
votes
0answers
171 views

Does $\#_n S^2×S^1$ really admit a map of non-zero degree from $B×S^1$?

In Proposition 4 on page 6 of this paper, http://arxiv.org/abs/1202.6302, the authors claim to produce a degree 2 $\pi_1$-surjective map $f$ between $M=S^1 \times \Sigma_2$ and $N=\#_2 S^2 \times S^1$ ...
3
votes
1answer
167 views

Is the number of intervals you get by slicing the closed region under a nice graph continuous from below?

Let $I$ be a closed interval in $\mathbb{R}$, and let $f:I \to \mathbb{R}$ be a bounded function, smooth except at finitely many points (piece-wise smooth). There are also only finitely many critical ...
6
votes
1answer
123 views

Can a surface group act on a finite-valence simplicial tree?

Question. Let $S$ be a closed surface of genus $> 1$. Can $\pi_1(S)$ act faithfully and minimally on a simplicial tree of finite valence? Here "minimal" means that there is no invariant sub-tree. ...
9
votes
1answer
244 views

Are there non-smoothable homotopy/homology spheres?

A homotopy sphere is a topological $n$-manifold $M$ which is homotopy equivalent to $S^n$. A homology sphere is a topological $n$-manifold $M$ such that $H_i(M) \cong H_i(S^n)$ for all $i$. Note, by ...
23
votes
2answers
677 views

Why is it true that if two 4-manifolds are homeomorphic then their squares are diffeomorphic?

Near the top of the second page of this paper, it is claimed that if two 4-manifolds $X$ and $Y$ are homeomorphic, then their squares $X \times X$ and $Y \times Y$ are diffeomorphic. Why is this true? ...
1
vote
0answers
53 views

Laplace eigenvalue and floer theory

I'm trying to study the spectrum operator via Floer Theory. The idea is to consider the Rayleigh quotient as an operator on the Hilbert space $W^{1,2}$ and to study it's negative gradient flow. Any ...
0
votes
0answers
18 views

Extending a symplectomorphism of a surface with boundary, properly embedded in a 4-disk, to the ambient space

Let $S$ be a surface with boundary that is properly embedded, via a map $f$ say, inside a $4$-disk. When is the embedding isotopic to the identity? -- All embeddings and isotopes are in the symplectic ...
9
votes
1answer
230 views

Elements of infinite order in the topological mapping class group

Let $M$ be a closed topological manifold, and let $\operatorname{MCG}(M):=\operatorname{Homeo}(M)/\operatorname{Homeo}_0(M)$ denote the topological mapping class group of $M$ ...
8
votes
1answer
154 views

Are knots determined by their complements within a homotopy class?

Suppose $M$ is a closed 3-manifold and $K_1,K_2$ are two homotopic knots in $M$. That is, they are two embeddings $f_1,f_2\colon S^1 \to M$ such that there exists a homotopy $h\colon S^1 \times [0,1] ...
10
votes
1answer
146 views

Open (resp., closed) balls homeomorphic to open (resp., closed) discs on the plane

Let $\Sigma$ be a compact (smooth) surface, with a geodesic metric $d$ (compatible with the topology of $\Sigma$). Let $x \in \Sigma$, and suppose you have the following: for every $r<1$, the open ...
7
votes
0answers
159 views

Biholomorphic neighborhoods of the boundary of Stein domains

Let $(X_1,J_1)$ and $(X_2,J_2)$ be Stein domains with the same contact boundary $(Y,\xi)$. Under what conditions does there exist a biholomorphism between a neighborhood of their respective boundaries ...
5
votes
0answers
87 views

$H(M)$ necessarily highly non-integrable, i.e. forms contact structure?

Let$$M^{2n - 1} = \{z \in \textbf{C}^n : \textbf{h}(\textbf{z}, \textbf{z}) \equiv \textbf{z} \cdot \overline{\textbf{z}} = \textbf{1}\}$$be the unit sphere in $\textbf{C}^n$. Consider the ...
5
votes
0answers
94 views

Images of the $3$-dimensional solvable geometry

This is more of a mathematical art question. Most people are familiar with the mathematical movie "Not Knot", which explores a hyperbolic $3$-manifold. My question is: are there any images along the ...
4
votes
1answer
73 views

Locally nilpotent algebraic section of tangent bundle is complete?

Suppose $X$ is a smooth affine algebraic variety over $\mathbb{C}$ and let $V$ be an algebraic vector field (i.e. an algebraic section of the tangent bundle). If $V$ is locally nilpotent, meaning that ...
21
votes
1answer
290 views

Is the normal bundle of a torus trivial?

Question: Let $T^k \subseteq \mathbb{R}^n$, $ n > k$, be a smoothly embedded $k$-torus. Is its normal bundle trivial? What about the normal bundle of $S^k \subseteq \mathbb{R}^n$, $n > k$, the ...
7
votes
0answers
244 views

Can any smooth triangulation of a smooth manifold be blurred?

For the purposes of this question, let's say that a blurring of a smooth triangulation $T$ of a smooth manifold $X$ is a smooth homotopy $h\colon [0,1] \times X \to X$ such that ...
4
votes
0answers
125 views

Groups with infinitely many finite conjugacy classes

I've been coming across the condition "IMFCC: having infinitely many finite conjugacy classes" often in recent times and I was wondering if there is any serious difference between having "IMFCC" and ...
16
votes
1answer
394 views

Just how close can two manifolds be in the Gromov-Hausdorff distance?

Suppose that we have two compact Riemannian manifolds $(M,g)$ and $(N,h)$. Define the Gromov-Hausdorff distance between them in your favorite way, I'll use the infimum of all $\epsilon$ such that ...
5
votes
1answer
129 views

Primitive log-divergent graphs and convergence of Feynman amplitudes

To a connected graph $G$, quantum field theory attaches the integral $$ I_G=\int_{\sigma} \frac{\Omega_G}{\Psi_G^2} $$ where $N_G$ is the number of edges of the graph, $\sigma$ is the simplex of ...
30
votes
4answers
1k views

Why do people study representations of 3-manifold groups into $SL(n,\mathbb{C})$?

Varieties of representations and characters of $3$-manifold groups in $SL(2,\mathbb{C})$ have been intensively studied. They have provided tools to identify geometric structures on manifolds, and are ...
9
votes
0answers
154 views

Detecting a module for the free group algebra on a finite quotient

Let $F_2$ be the free group on two generators $x,y$ and let $R$ be the group algebra $\mathbf{Q}[F_2]$. Let $a,b,c$ be integers. Then define a right $R$-module $M = R / (ax + by + c) R$. I am ...
2
votes
0answers
83 views

relation between representations of homology class

Let X be a topological space, for its homology class [f], we can alway construct a simplicial complex K_f by gluing "cancelling boundary pairs" of f and an induced continuous map f' from K_f to X. ...
6
votes
0answers
75 views

Compatibility of spherical and hyperbolic geometry for fibred knots

Hyperbolic knots and links have a lovely peculiarity that you can always find a position for them in $S^3$ making two groups the same, one defined using the spherical geometry of $S^3$, and the other ...
4
votes
1answer
157 views

Fibered knots vs Heegaard genus

We call a knot $K$ in a 3-manifold $M$ fibered if $M\backslash K$ fibers over $S^1$ with fibers $\Sigma$ and such that $K$ is ambient isotopic to the boundary of the compactified fiber ...
7
votes
0answers
209 views

A “direct” proof that hyperbolic groups are not amenable

I am looking for a proof that a finitely generated hyperbolic groups is non-amenable [unless it is virtually cyclic] which is as "metric/combinatoric" as possible. Here are the two proofs I am aware ...
0
votes
2answers
93 views

obtaining circle bundle over torus by trefoil surgery

Does any integer surgery on a right or left trefoil knot give the $S^1$-bundle over $T^2$ with Euler number $1$?
0
votes
0answers
90 views

Mapping class groups acting on simple closed curves

Let $S_{g,d}$ be a genus $g$ compact Riemann surface with $d$ punctures. Let $\mathcal{M}_{g,d}$ be the moduli space of all such surfaces, i.e. genus $g$ compact Riemann surfaces with $d$ marked ...
2
votes
0answers
60 views

Singularities of Families of Differential Equations

Suppose you are given a family of differential equations with rational coefficients: $\partial_x^ny+a_1(x,x_1,\dots,x_m)\partial_x^{n-1}y+\dots+a_n(x,x_1,\dots, x_m)y=0$. At each point $X=(x_1,\dots, ...
8
votes
1answer
180 views

Homotopy type of diffeomorphism which are the identity on and near the boundary

Let $M$ be a compact manifold with boundary. Denote by $Diff(M), Diff_\partial(M)$ and $Diff_{U\partial}(M)$ the groups of diffeomorphisms of $M$ and the subgroups of the ones that are the identity on ...
4
votes
0answers
134 views

Questions on Thurston's metric on Teichmüller space

I'm reading the famous "Minimal stretch maps between hyperbolic surfaces" by William Thurston and I'm trying to understand the key theorem 8.1. I have many unclear points so I hope someone can help me ...
6
votes
3answers
254 views

For an arithmetic hyperbolic 3-manifold group, when is its trace field not its invariant trace field?

Edit: In my original post I failed to require the group to be a manifold group. The answer below from @BenLinowitz works in that case. I am really interested though in when the group is torsion-free, ...
4
votes
1answer
247 views

Who first considered constructibility of simplicial complexes?

A simplicial complex of dimension $d$ is called constructible if it is a simplex, or if it is the union of two constructible dimension-$d$ simplicial complexes along a dimension-$(d-1)$ intersection. ...
8
votes
1answer
119 views

Example where Calabi invariant is nontrivial?

Let $D^2$ denote the closed unit disk in $\mathbb{R}^2$. Let $\omega := dx \wedge dy$ denote the standard area form on $\mathbb{R}^2$ (and on $D^2$ by restriction). Let $\phi$ be a diffeomorphism of ...
6
votes
0answers
126 views

kernel of the mod $2$ Bockstein on the first cohomology group

Let $M$ be a path-connected finite $CW$-complex. Suppose the first integral homology group is $H_1(M;\mathbb{Z})= \mathbb{Z}_2^{\oplus r}\oplus A$ where $r\geq 1$ and $A$ is a finite abelian group of ...
3
votes
1answer
54 views

vector bundles induced by an action of a finite subgroup of $O(n)$

Let $M$ be a path-connected manifold. Let $G$ be a finite subgroup in $O(n)$ and suppose $G$ acts freely on $M$. Then we have an associated vector bundle $$ \xi(M,G): \mathbb{R}^n\longrightarrow ...
1
vote
0answers
107 views

Connecting homomorphism in generalized cohomology theory

I have some compact manifold with boundary $(M,\partial M)$, and there is a long exact sequence $$\cdots\to KO^{-1}(\partial M)\xrightarrow{\partial} KO^{0}(M,\partial M)\to KO^0(M)\to KO^0(\partial ...
1
vote
1answer
50 views

Existence of a continuous map of a disk with a given boundary image on a surface to its complement in $\mathbb{R}^3$

I am considering the following problem: given an embedded closed surface in $\mathbb{R}^3$ (unknotted) and a non-trivial simple closed curve on it, does there exist a continuous map of a disk to the ...
5
votes
0answers
89 views

cohomology ring of configuration spaces on $S^2$ and the projective plane

For a manifold $M$ and a positive integer $n$, the unordered configuration space $B(M,n)$ is the space consisting of all unordered collections of $n$ distinct points on $M$. Precisely, $$ ...
6
votes
1answer
134 views

non-orientability of vector bundles induced from a symmetric group action

Let $\Sigma_k$ be the symmetric group on $k$-letters. Let $M$ be a manifold with a free $\Sigma_k$-action. Then we can form a $k$-dimensional vector bundle $$ \xi:\mathbb{R}^k\longrightarrow ...
16
votes
2answers
778 views

What is an example of an orbifold which is not a topological manifold?

In Thurston's book The Geometry and Topology of Three-Manifolds it is proven that the underlying space of a two-dimensional orbifold is always a topological surface. Are there any easy examples of ...