Tagged Questions

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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Space of embeddings of circle in a surface

Let $S$ be a compact oriented surface of genus at least $2$ (possibly with boundary). Let $X$ be a connected component of the space of embeddings of $S^1$ into $S$. Question : what is the ...
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Realizing homology classes on surfaces with boundary by simple closed curves

Let $\Sigma$ be a compact oriented surface with boundary. Assume that the genus of $\Sigma$ is positive. We say that an element $h \in H_1(\Sigma)$ can be realized by a simple closed curve if there ...
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Explicit computation of the action of a Dehn twist on the fundamental group of a surface

Let $S$ be a compact orientable surface of genus $g$. Now let $p\in S$ and $\gamma$ a closed simple curve on $S$ disjoint from $p$. It is not very difficult to compute the action of a Dehn twist along ...
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Let $M$ be a 2-dimensional closed Riemmanian manifold diffeomorphic to $S^2$. S.B.Myers says "the cut-locus of every point $x\in M$ is a finite tree." How the set of point can be a tree? ...
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Costa's minimal surface and the structure of lungs

Seeing this image of Costa's minimal surface        (MathWorld image) made me wonder if the fine-grained structure of the human lung is somehow composed of pieces of ...
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Looking for software that computes intersection numbers (Heegaard Diagrams)

As a part of my research I am working with intersection matrices of Heegaard diagrams. Is there some software that could help me compute such matrices for some examples? Thanks.
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Dimension of the homology group with coefficients in $\mathbb{Z}/2\mathbb{Z}$

I asked this on math.stackexchange.com, but didn't get a single answer. Charles Weibel writes in his survey of homological algebra Riemann defined a surface $S$ to be $(n + 1)$-fold connected ...
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Contractibility of union of smooth rational curves in famillies

Let $\pi:\mathcal{X} \to S$ be a family of smooth surfaces containing a subfamily $\mathcal{Y} \subset \mathcal{X}$ proper, flat over $S$ parametrizing contractible curves in $\mathcal{X}$ satisfying ...
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Random metrics on compact orientable surfaces

Hello everyone, Let $S_g$ be a compact orientable surface of genus $g \geq 2$, and let $\mathcal{A}$ be the set of $\mathcal{C}^{\infty}$ Riemanniann metric on $S_g$ endowed with the topology of ...
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Fixing a proof of the systolic inequality for higher genus surfaces

I'm currently learning some stuffs about systolic inequalities. While reading the relevant sections (p329 to 340) in Berger's Panoramic View of Riemannian Geometry, I noticed a gap in one of the ...
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Covering spaces of surfaces

Let $\Sigma_g$ be a surface of genus $g\ge 2$, and let $\Sigma_k$ be an $m$-sheeted covering space of $\Sigma_g$. It is known that $k=m(g-1)+1$. An example of such a covering space is a regular ...
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Do the following set of Dehn twists generate the mapping class group?

If $S$ is the surface illustrated below, do the Dehn twists about the red curves generate the mapping class group $\operatorname{MCG}(S,\partial S)$?
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surfaces dominated by a product of curves

I would like to know for which projective smooth surfaces over a finite field there exists a dominant rational map from a product of curves to the surface
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Parallel translation on surfaces

Parallel translation of a vector along a geodesic in a surface is characterized by the following three properties: The vector being transported moves continuously. It has constant norm. It maintains ...
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Lagrangian Kleinian bottles

I remember some talks some time ago about proofs of nonexistence of Lagrangian Kleinian bottles in C^2 for the standard symplectic structure, mentioning that this were the only compact surface for ...
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Pursuit-Evasion on a Manifold

I know pursuit-evasion has been studied in many contexts, including on a manifold (e.g., Melikyan, "Geometry of Pursuit-Evasion Games on Two-Dimensional Manifolds"), but I have not seen this version: ...
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Morse theory in TOP and PL categories?

Apparently there are topological and piecewise linear versions of Morse theory. I would like to know of references that treat these topics. How is a Morse function defined for compact manifolds (with ...
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Classification of surfaces and the TOP, DIFF and PL categories for manifolds

A surface is simply a 2-manifold. The classification theorem for compact connected surfaces (with boundary) is commonly regarded in the categories TOP, DIFF and PL. Well known proofs (e.g. via ...
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Representing groups with two generators as graph automorphisms

Suppose we have a group $G$ which can be generated by two elements $x$, $y$. Call $H$, $K$, $L$ the subgroups of $G$ generated by $x$, $y$ and $y^{-1}x^{-1}$, respectively. With these data, we can ...
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What is parameterization of the trefoil knot surface in R³?

What is a parameterization, say (x(u,v),y(u,v),z(u,v)), of the trefoil knot surface in R³ whose cross-section can be circular or, in general, elliptic? Thanks!
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Naive definition of surface area doesn't work?

A first stab at a definition of surface area might go like this: Let S be a surface. Select finitely many points from S and make a bunch of triangles having these points as vertexes. Add up the ...
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Connected sum of surfaces

I'm looking for a detailed reference about connected sums. I'd like it to contain a proof that a connected sum of connected surfaces is independent - up to homeomorphism - of the various choices ...
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Contraction of curves on surfaces

Assume we have a surface $S$ (smooth if you want), and a map $f: S \to V$ that contracts a curve $C \subset S$. What condition would give a factoring of $f$ through a contraction $c: S \to V'$ ...
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Abelian subgroups of ball quotient

Let $X$ be a compact complex surface of general type which a ball quotient. Is it true that $\pi_{1}(X)$ can not contain ${\mathbb{Z}}^{2}$ as a subgroup? What kind of infinite abelian groups can ...
Let me first state the definitions : A not-nullhomotopic closed curve / loop $c$ on an orientable surface $X,c:[0,1]\to X$ is called simple closed curve is $c|[0,1)$ is injective and [ $c(0)=c(1) ] ;... 2answers 486 views Surface fitting with convexity requirement Hi all, Consider a cloud of points in 3D space (x,y,z). The data is well-behaved, once plotted the surface looks like some sort of spheroid. I assumed a form for the fitting function f(x,y,z) = c1 x^... 2answers 339 views Moving a canonical divisor on a normal surface away from the singular locus In a previous question Moving a Weil divisor on a normal surface away from a finite set of closed points I probably asked for too much. As J.C. Ottem pointed out, it is not always possible to move a ... 3answers 438 views Moving a Weil divisor on a normal surface away from a finite set of closed points Let$Y$be a normal surface and let$X$be a closed subscheme of codimension 2, i.e.,$X$is a finite set of closed points. Let$D$be a Weil divisor on$Y$. Question. Does there exist a Weil ... 0answers 153 views Is -(E,E) greater or equal to 2 for a minimal resolution I'm quite confused by the terminology minimal resolution and minimal model. Let$f:X\longrightarrow Y$be a minimal resolution of singularities, where$Y$is a normal surface. Let$E$be an ... 1answer 732 views intersection number I vaguely recall the following fact that I'd like to use in my research. It should be easy to see that this holds (if it does) but I can't seem to prove it. Let$p:X\longrightarrow S$be a (regular) ... 2answers 473 views The exceptional locus of a minimal resolution of singularities Let X be a surface. (A surface is an excellent integral normal separated 2-dimensional scheme.) Let$\psi:Y\longrightarrow X$be a minimal resolution of singularities and let$E$be an irreducible ... 1answer 599 views Hirzebruch surfaces I am sorry for too naive and stupid question, How can I express the 2nd Hirzebruch surface,$F_{2}$in terms of$SO(3)$. Can F_{2} be realizable as the total space of a bundle over$\mathbb{R}_{+}$... 2answers 697 views Names of certain surfaces Are there any generally used names for the following algebraic and nonalgebraic surfaces? Any references to literature where the surfaces are studied are also appreciated. Surface I. Implicit ... 4answers 554 views What is the quantity 2(handles)+crosscaps called? It is well-known that up to homeomorphism, the complete set of orientable surfaces is$\lbrace S_g : g=0,1,\dots \rbrace$, where$S_g$is the sphere with$g$handles. The complete set of non-... 2answers 348 views Can one obtain surfaces with interesting invariants as resolutions of singular surfaces? (Perhaps a not very well defined question) Let$(S_t)_t$be a (flat) family of compact complex surfaces. Assume the generic member is smooth while$S_0$has isolated singularities. As the simplest ... 2answers 1k views Simple curves on non-orientable surfaces. Given an element in the (first) homology group of a surface, I would like to know if it can be represented as a simple closed curve. For orientable surfaces, this is well-known, but I wasn't able to ... 5answers 1k views Is a rhombus rigid on a sphere or torus? And generalizations. If a rectangle is formed from rigid bars for edges and joints at vertices, then it is flexible in the plane: it can flex to a parallelogram. On any smooth surface with a metric, one can define a ... 0answers 298 views Abel-Jacobi map for regular fibered surfaces. Let$f:C\to S$be a regular fibered surface where$S=Spec(R)$,$R=dvr$. Assume$C$has smooth geometrically integral generic fibre$C_K$. We also assume the existence of a section$x\in C(S)$. Let$...
Let $C_1, \dots, C_n$ be a family of disjoint simple curves in a surface $\Sigma$. If $C$ is any simple curve in $\Sigma$, it turns out that we can map $C$ to a curve $C'$ (via a homeomorphism of \$\...