Questions tagged [surfaces]

A surface is a two-dimensional topological manifold. The term can also be used to describe a smooth surface, depending on the context.

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Identifying a curve on a closed surface of genus 4

The notation is the one used in the attached picture. Take a closed, orientable surface $\Sigma_4$ of genus $4$, obtained as the identification space of a polygon with $16$ sides in the usual way. The ...
Francesco Polizzi's user avatar
4 votes
1 answer
156 views

Euler class of vertical tangent bundle of the surface bundle over circle

Suppose $\Sigma$ is an oriented genus $g>1$ surface and $h:\Sigma\to \Sigma$ is a diffeomorphism preserving a point $p$. Let $M$ be the surface bundle over $S^1$ obtained by gluing $\Sigma\times I$ ...
Faniel's user avatar
  • 545
4 votes
1 answer
115 views

Topological type of complement of Heegaard curves in Heegaard surface $(\Sigma - \alpha - \beta)$

Suppose $(\Sigma, \alpha, \beta)$ is a genus-$g$ Heegaard diagram for a closed, oriented $3$-manifold $Y$, i.e. $\Sigma$ is an orientable genus-$g$ surface, and $(\alpha_1, \dots, \alpha_g)$ and $(\...
Matija Sreckovic's user avatar
2 votes
0 answers
59 views

Maximal orders and surface subgroups of even genus

Let $A$ be a quaternion algebra over a totally real number field $k$. Suppose that $A$ splits at exactly one real place of $A$. Let $\mathcal{O}$ be a maximal order in $A$. Then $\mathcal{O}$ contains ...
Jacques's user avatar
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2 votes
1 answer
94 views

Generically finite projection $\pi_L: X \to \mathbb{P}^2$ from plane $L$ and critical points

(In following we are working in "classical" complex setting: i.e. all involved schemes are considered to be varieties over $k=\mathbb{C}$) Let $X \subset \mathbb{P}^n$ be irreducible surface ...
user267839's user avatar
  • 4,994
7 votes
2 answers
246 views

Surjections from genus $n$ surface group to free group of rank $n$

Let $\Sigma_n$ be a genus $n$ surface, let $\mathcal{H}_n$ be a genus $n$ handle body, and let $F_n$ be a free group of rank $n$. Fix an identification of $\pi_1(\mathcal{H}_n)$ with $F_n$. I know ...
Annie's user avatar
  • 73
6 votes
2 answers
128 views

Generate $\mathrm{Mod}(S_g)$ by two Dehn twists

Let $S_g$ be a closed orientable surface of genus $g>1$. How can one prove that its mapping class group $\mathrm{Mod}(S_g)$ is not generated by two Dehn twists? A pair of simple closed curves in $...
Andrey Ryabichev's user avatar
0 votes
0 answers
27 views

Maximum for sum of distances on a constant curvature surface

(This is a modification of a previous question where my hopes were too general.) Let $X$ be a closed constant curvature surface and consider the function $f_n : X \times \cdots \times X \to \mathbb{R}$...
Sprotte's user avatar
  • 933
2 votes
0 answers
69 views

Leaves of bounded genus

Let $\mathcal{F}$ be a codimension one foliation in a closed $3$-manifold $M$. Does there exist an upper bound for the genus of the compact orientable leaves? That is, does there exist $G >0$ such ...
Eduardo Longa's user avatar
2 votes
0 answers
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Cubic lattice representation of a solid torus knot using the surface as a boundary

For physics simulation reasons, I would like to respresent a solid torus knot as a collection of integer points sat on a cubic lattice. If I were to do this using a sphere, I would do this by saying ...
Industrialactivity's user avatar
5 votes
0 answers
76 views

Intersections of geodesics in an "almost flat" plane

Let $g$ be a complete metric on $\mathbb{R}^2$, such that: Outside of a compact connected set $K\subset \mathbb{R}^2$, the curvature of $g$ vanishes. The integral of the Gaussian curvature in $K$ is ...
C M's user avatar
  • 381
4 votes
0 answers
129 views

Upper bound for the first eigenvalue of the Laplacian on surfaces with boundary

Let $\Sigma$ be a compact smooth surface with boundary. Define $$\Lambda(\Sigma) := \sup \{ \lambda_1(\Sigma,g) \operatorname{Area}(\Sigma,g) : g \text{ is a smooth Riemannian metric on $\Sigma$} \}$$ ...
Eduardo Longa's user avatar
9 votes
1 answer
201 views

Links and non-orientable surfaces

Let $\Sigma \subset \mathbb{R}^3$ be a compact embedded surface with boundary $\partial \Sigma$ and $i:\Sigma\setminus \partial\Sigma \to \mathbb{R}^3 \setminus \partial\Sigma$ the inclusion. Is the ...
mmen's user avatar
  • 413
4 votes
1 answer
166 views

Closed-form examples of CMC surfaces

Besides the trivial cases of cylinders and spheres, are there any other known examples of non-zero constant mean curvature surfaces which can be represented explicitly in a closed form? I am ...
Jiří Minarčík's user avatar
2 votes
0 answers
82 views

Vanishing of Goldman bracket requires simple-closed representative?

Let $\Sigma$ be a connected oriented surface, and $[-,-]\colon \Bbb Z\big[\widehat\pi(\Sigma)\big]\times Z\big[\widehat\pi(\Sigma)\big]\to Z\big[\widehat\pi(\Sigma)\big]$ be the Goldman Bracket. Note ...
Random's user avatar
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1 vote
0 answers
47 views

Uniqueness of graph embeddings on surfaces for highly connected graphs

A theorem of Whitney says that if $i_1$ and $i_2$ are two embeddings of a 3-connected planar graph into $S^2$, then there exists a homeomorphism $h : S^2 \to S^2$ such that $h \circ i_1 = i_2$. Is ...
Sprotte's user avatar
  • 933
1 vote
0 answers
63 views

Diffeomorphism induced by small perturbation

Consider the surface $S_{\epsilon}$ defined as: \begin{align} %S&=\{\vec x \in \mathbf{R}^3: x=0\}, \\ S_{\epsilon}&=\{\vec x \in \mathbf{R}^3:\epsilon (x^2 + y^2 + z^2 - 1) + x=0\}. \end{...
dennis's user avatar
  • 413
6 votes
1 answer
460 views

Topology change induced by small perturbation

Consider the surface $S_{\epsilon}$ defined as: \begin{align} %S&=\{\vec x \in \mathbf{R}^3: x=0\}, \\ S_{\epsilon}&=\{\vec x \in \mathbf{R}^3:f_{\epsilon}(\vec x)\equiv\epsilon ((x^2 + y^2 - ...
dennis's user avatar
  • 413
3 votes
1 answer
84 views

A closed curve can be homotopic to remove all intersections with a filling $\Gamma$ if it has zero geometric intersection numbers with $\Gamma$

Let $\Sigma$ be a compact oriented connected bordered surface other than the pair of pants. Let $\Gamma:=\{\gamma_i\}$ be a finite collection of simple closed curves on $\Sigma$ such that each ...
Random's user avatar
  • 793
2 votes
0 answers
117 views

Every surface of sufficiently large genus separates

Let $M^3$ be a smooth closed orientable manifold. Does there exist a non negative integer $g_0$ such that every closed orientable embedded surface $\Sigma \subset M$ of genus $g \geq g_0$ represents ...
Eduardo Longa's user avatar
4 votes
2 answers
81 views

Characterization of a non-trivial non-peripheral element of the free homotopy classes of a compact bordered surface

Let $\Sigma$ be a compact orientable connected $2$-manifold with a non-empty boundary. Let $\widehat \pi(\Sigma)$ denote the set of free homotopy classes of curves in $\Sigma$. We say $x\in \widehat \...
Random's user avatar
  • 793
4 votes
1 answer
104 views

Given $f$ from the cylinder $C$ to the interval constant on one boundary, is there a $r:C\to C$ constant on a boundary with $f\circ r = f$?

My question might be trivial, but my lack of knowledge of this particular subject has not enabled me to find the answer. What I want to know is the following. Let $I=[0,1]$ and $C=S^1\times I$ be the ...
Mathieu Baillif's user avatar
0 votes
1 answer
241 views

Why do we define the pants complex and the pants decomposition? [closed]

Why do we define the pants complex? I learned for the first time in A Presentation for the mapping class group of a closed orientable surface (by A. Hatcher and W. Thurston) that we have definition of ...
Usa's user avatar
  • 119
4 votes
0 answers
107 views

Earliest known proof of "Any degree one self-map of an orientable connected finite-type non-compact surface is homotopic to a homeomorphism"

I attended a talk where the speaker said the following is due to Nielsen. I searched here and there but couldn't find the corresponding paper, if any. So, what is the earliest known proof of the ...
Random's user avatar
  • 793
5 votes
0 answers
178 views

Groups of non-orientable genus 1 and 2

The non-orientable genus (aka crosscap-number) $\overline{\gamma}(G)$ of a finite group $G$ is the minimum non-orientable genus among all its connected Cayley graphs (and $0$ if $G$ has a planar ...
Kolja Knauer's user avatar
21 votes
2 answers
773 views

Fixed-point free diffeomorphisms of surfaces fixing no homology classes

One of my graduate students asked me the following question, and I can't seem to answer it. Let $\Sigma_g$ denote a compact oriented genus $g$ surface. For which $g$ does there exist an orientation-...
Robert's user avatar
  • 303
2 votes
1 answer
147 views

Intersection pairing on $H_1$ of surfaces

Let $X$ be a surface (real, not complex). With coefficients in a ring $R$, there is an intersection pairing $\langle -,- \rangle:H_1(X;\mathbb R) \otimes H_1(X;\mathbb R) \to R$. Is this pairing ...
Ethan Dlugie's user avatar
  • 1,045
2 votes
1 answer
111 views

Classification of degree one map between two closed orientable surfaces without using induction on the genus

A theorem of Edmonds (see Theorem 3.1. of "Deformation of Maps to Branched Coverings in Dimension Two") says that Theorem 1: A degree-one map between closed orientable surfaces is homotopic ...
Random's user avatar
  • 793
2 votes
1 answer
107 views

Isotopic homeomorphisms of surface induces same map on the space of ends

Let $\Sigma$ be a non-compact orientable connected two-manifold without boundary. Let $f,g\colon \Sigma\to \Sigma$ be two homeomorphisms. Suppose there is a homotopy $H\colon \Sigma\times [0,1]\to \...
Random's user avatar
  • 793
7 votes
2 answers
373 views

Kneser theorem about the Klein bottle

I know that in $1923$ H. Kneser showed that a continuous flow in a Klein bottle without singular points has a periodic trajectory. The original article is this, but does anyone know another old or new ...
Zaragosa's user avatar
  • 123
3 votes
1 answer
186 views

Area of a deformation of a closed surface

Let $(M^3,g)$ be a complete Riemannian manifold. Fix a two-sided immersion $\varphi : \Sigma^2 \to M$ from a closed surface into $M$, with unit normal $N$. Given $f \in C^\infty(\Sigma)$ and $\alpha : ...
Eduardo Longa's user avatar
12 votes
3 answers
956 views

How to generate all triangulations of an orientable surface?

$\newcommand{\comb}{\mathrm{comb}}$Consider an orientable surface $S$ with punctures and boundaries (each boundary having at least a marked point). A triangulation, up to orientation preserving ...
giulio bullsaver's user avatar
1 vote
0 answers
136 views

Doubly ruled surfaces in hyperbolic 3-space

A well-known theorem of classical surface theory states that the only doubly ruled surfaces in Euclidean 3-space are planes, 1-sheeted hyperboloids and hyperbolic paraboloids. There are a number of ...
Brendan Guilfoyle's user avatar
1 vote
0 answers
128 views

End space of non-compact 2-manifolds described with proper rays

I am wondering if the classification of noncompact surfaces given by Ian Richards can be stated with proper rays instead of nested sequences of connected open subsets with compact boundary. I asked ...
Carlos Adrián's user avatar
5 votes
1 answer
261 views

"Classification" of (orientable) 3-manifolds with genus-g-surface as their boundary

This is in some sense a generalization of the question I asked some time ago. I am very sorry if this question is too basic for MathOverflow, but I just started learning about some more detailed ...
G. Blaickner's user avatar
4 votes
1 answer
182 views

Inequivalent free $\Bbb Z/n\Bbb Z$-actions on orientable compact bordered surface

Let $S_{g,b}$ denote the orientable connected compact surface of genus $g$ with $b$ boundary components. A group homomorphism $\varphi\colon G\to \text{Homeo}^+(S_{g,b})$ is said to be free $G$-...
Random's user avatar
  • 793
1 vote
1 answer
139 views

Minimal surface enclosing balls

(This question is tangentially related to an earlier question I posed: Minimal surface enclosing two congruent balls.) Let $B_1,\ldots,B_k$ be unit-radius balls in $\mathbb{R}^3$, with pairwise ...
Joseph O'Rourke's user avatar
8 votes
1 answer
190 views

Minimal surface enclosing two congruent balls

Let $B_1$ and $B_2$ be two unit-radius balls in $\mathbb{R}^3$ whose centers are separated by a distance $d \ge 2$. Q. For sufficiently small $d$, is the minimal area surface enclosing $B_1$ and $B_2$...
Joseph O'Rourke's user avatar
2 votes
1 answer
102 views

Can we always find coordinates on a surface such that $K=K(u-v)$?

Let $(M^2,g)$ be a 2-dimensional Riemannian manifold. For any point $p\in M^2$ can we always find coordinates $(u,v)$ in a neighborhood $U$ of $p$ such that the Gaussian curvature is only a function ...
Daniel Castro's user avatar
4 votes
1 answer
297 views

A local isometric immersion from $\mathbb H^{n}$ into $\mathbb R^{2n-1}$

I found this local isometric immersion from $\mathbb H^{n}$ into $\mathbb R^{2n-1}$, given by Schur (1886) in Über die Deformation der Räume constanten Riemannschen Krümmungsmaasses as follows, $(1\...
Zaragosa's user avatar
  • 123
2 votes
0 answers
103 views

Restriction function as a Morse function

Let $\Sigma$ be a closed surface smoothly embedded in $\mathbb R^3$. For any Morse function $h:\mathbb R^3 \to \mathbb R$, can we isotope $\Sigma$ so that the restriction of $h$ on $\Sigma$ is also a ...
Zhiqiang's user avatar
  • 587
7 votes
1 answer
192 views

Factoring a projection of surface groups through a free group

Consider the surface group $S_g=\langle a_1,b_1,a_2,b_2,\dots,a_g,b_g \mid [a_1,b_1][a_2,b_2]\cdots[a_{g},b_{g}]=1\rangle$, which is the fundamental group of the closed orientable genus-$g$ surface. ...
Jeremy Brazas's user avatar
16 votes
3 answers
1k views

Does a random walk on a surface visit uniformly?

Let $S$ be a smooth compact closed surface embedded in $\mathbb{R}^3$ of genus $g$. Starting from a point $p$, define a random walk as taking discrete steps in a uniformly random direction, each step ...
Joseph O'Rourke's user avatar
6 votes
0 answers
156 views

Rigidity for convex surfaces in elliptic/hyperbolic space

From Alexandrov's work we know that any metric on the sphere with lower curvature bound $\kappa$ (in the sense of Alexandrov) can be realized as a closed convex surface (i.e. boundary of a compact ...
Pete's user avatar
  • 103
3 votes
1 answer
140 views

Finitely connected orientable surface

Let $(M,g)$ be a finitely connected orientable complete Riemannian surface, that is, $M$ is homeomorphic to a compact orientable surface $\Sigma$ minus $k \geq 1$ points. Do you have references or a ...
Eduardo Longa's user avatar
1 vote
0 answers
291 views

Question in the proof of Hilbert's theorem

I'm researching about Hilbert's theorem which says that there isn't isometric immersion of a complete surface with constant negative Gaussian curvature in $\mathbb{R}^3$. I'm taking as a reference the ...
Zaragosa's user avatar
  • 123
8 votes
2 answers
224 views

Degree one self-map of $\Bbb R^2\big\backslash \big\{(n,0):n\in \Bbb Z\big\}$ not homotopic to any self-homotopy equivalence

Consider the surface $\Sigma=\Bbb R^2\big\backslash \big\{(n,0):n\in \Bbb Z\big\}$. Does there exist a proper map $f\colon \Sigma\to \Sigma$ of degree $1$ and not homotopic to any self-homotopy ...
Someone's user avatar
  • 265
6 votes
1 answer
299 views

Are the “generalized Catalan numbers” of Dumitrescu–Mulase the "moments" of some "multivariate Wigner semicircle distribution"?

The classical Catalan numbers $$ C_n = \frac{1}{n+1} \binom{2n}{n}, $$ well-known for their numerous combinatorial interpretations (the second volume of Stanley's Enumerative Combinatorics famously ...
mlbaker's user avatar
  • 215
9 votes
0 answers
183 views

Algebraic context for Mednykh's formula?

Let $S$ be a closed orientable surface and let $G$ be a finite group, then Mednykh's formula says that $$ \sum_{V}d(V)^{\chi(S)} = |G|^{\chi(S) - 1} |\text{Hom}(\pi_1 S, G)| $$ where the sum is over ...
user101010's user avatar
  • 5,299
0 votes
0 answers
133 views

Can the infinite jungle gym surface be expressed by an exhaustion of compact surfaces with one boundary component?

Is it possible to write the infinite jungle gym surface as the increasing union of compact surfaces whose boundaries consist of only one simple closed curve? I believe that this is true. This surface ...
Fernando Oliveira's user avatar

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