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).

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4
votes
3answers
689 views

About the proof of Wajnryb's finite presentation of Mod(S)

I'm studying Farb and Margalit's A primer on mapping class groups and trying to understand Wajnryb's finite presentation of Mod(S). I understand that There exists a finite presentation, but I can't ...
19
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0answers
541 views

Polynomials with roots in convex position

Let $\mathcal P_n$ denote the set of all monic polynomials of degree $n$ with real or complex coefficients such that $P\in\mathcal P_n$ if for all $k\in\lbrace 0,1,\dots,n-2\rbrace$ the $n-k$ roots of ...
18
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4answers
1k views

Free splittings of one-relator groups

Roughly speaking, I want to know whether one-relator groups only have 'obvious' free splittings. Consider a one-relator group $G=F/\langle\langle r\rangle\rangle$, where $F$ is a free group. Is it ...
12
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2answers
2k views

Example of Noetherian group (every subgroup is finitely generated) that is not finitely presented

A Noetherian group (also sometimes called slender groups) is a group for which every subgroup is finitely generated. (Equivalently, it satisfies the ascending chain condition on subgroups). A ...
211
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29answers
138k views

Intuitive crutches for higher dimensional thinking

I once heard a joke (not a great one I'll admit...) about higher dimensional thinking that went as follows- An engineer, a physicist, and a mathematician are discussing how to visualise four ...
8
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1answer
285 views

Does every retraction of free groups arise from projection to a subset of a freely generating set?

Suppose $F_1$ and $F_2$ are free groups, and suppose $\alpha:F_1 \to F_2$ is a surjective homomorphism. Then, because $F_2$ is free, the homomorphism splits, and we get a subgroup $H$ of $F_1$ ...
8
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2answers
887 views

Can we make rigorous the 'obvious' characterisation of singular homology?

It is a well known and often touted fact that the singular homology groups 'count the k- dimensional holes' in a space (see: How does singular homology H_n capture the number of n-dimensional "...
7
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2answers
3k views

tangent sphere bundle over sphere

are there some general description about tangent sphere bundle over sphere? (it is a special $S^{n-1}$bundle over $S^n$) say for n=1,it is trivial,$S^0\times S^1$,for n=2,it is $SO(3)\cong \mathbb{R}...
5
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1answer
847 views

Do continuous maps give continuity in the 'topology' of Hausdorff distance?

I was reading this question: limiting behaviour of converging loops on a torus And I wanted to be able to give an argument along the lines of: "If your loops are converging in your torus, their ...
8
votes
2answers
361 views

limiting behaviour of converging loops on a torus

Suppose (Ln) is a sequence of loops in a torus S1 × S1 converging in the Hausdorff metric to some set L in the torus. Suppose also that for each loop Ln the projection map p:S1 × S1 -> S1 ...
7
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4answers
663 views

Do you know how to construct a compact hyperbolic 3-manifold with three or four totally geodesic boundary components?

Do you know how to construct a compact hyperbolic 3-manifold with three or four totally geodesic boundary components? The only constructions I could find have one boundary component. A reference ...
4
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2answers
2k views

Properties of the n-dimensional Stereographic Projection

Hello, I'm looking for an argument that the n-dimensional stereographic projection maps circles (intersections of affine two-dimensional subspaces with S^n) to circles in R^n. I've looked around and ...
27
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2answers
1k views

Does every finitely presentable group have a presentation that simultaneously minimizes the number of generators and number of relators?

This should probably be an easy question, but I don't know how to answer it: Suppose G is a finitely generated presentable group. Suppose a is the absolute minimum of the sizes of all generating sets ...
7
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1answer
694 views

Counting submanifolds of the plane

After thinking about this question and reading this one I am led to ask for an uncountable collection of homeomorphism types of boundaryless connected path-connected submanifolds of the plane. My ...
86
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1answer
8k views

The mathematical theory of Feynman integrals

It is well known that Feynman integrals are one of the tools that physicists have and mathematicians haven't, sadly. Arguably, they are the most important such tool. Briefly, the question I'd like to ...
50
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4answers
4k views

Drawing of the eight Thurston geometries?

Do you know of a picture, drawing, or other concise visual representation of the eight three-dimensional Thurston geometries? I am imagining something akin to the standard picture (of a sphere, plane,...
17
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1answer
791 views

Locus of equal area hyperbolic triangles

Henry Segerman and I recently considered the following question: Given a fixed area $A < \pi$ and two fixed points in the upper half-plane model for hyperbolic $2$-space, what is the locus of ...
5
votes
2answers
496 views

Is there a knotted torus in 4-sphere whose complement's fundamental group is infinite cyclic ?

I am reading the book 'surface in 4-space' about the unknotting conjecture (Page 97): a 2-knot (2-sphere in 4-sphere) is trival if and only if the fundamental group of the exterior is infinite cyclic. ...
5
votes
3answers
1k views

How big can the Hausdorff dimension of a function graph get?

This question is inspired by How kinky can a Jordan curve get? What is the least upper bound for the Hausdorff dimension of the graph of a real-valued, continuous function on an interval? Is the ...
5
votes
1answer
421 views

Automorphisms of $\pi_1$ induced by pseudo-Anosov maps

Suppose $X$ is an orientable surface with non-empty boundary and $f:X\to X$ is a pseudo-Anosov automorphism that acts identically on $H_1(X,\mathbf{Z})$. Let $x$ be a fixed point of $f$. For any $\...
9
votes
3answers
1k views

A Reference for Schubert's Theorem

Schubert's Theorem in Knot Theory says that any knot can be uniquely decomposed as the connected sum of prime knots. Unfortunately the original paper is in German. Does anyone know a good english ...
21
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2answers
3k views

Does this approach for the Poincaré conjecture work?

Several months ago a paper was posted at http://arxiv.org/abs/1001.4164 called "Another way of answering Henri Poincaré's fundamental question." The author gave a talk on it today at my institution. ...
9
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5answers
993 views

Möbius and projective 3-manifolds

A projective 3-manifold is a smooth manifold that admits an atlas with values in the real projective 3-space such that all transition maps are restrictions of projective transformations. A Möbius 3-...
16
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2answers
708 views

The kernel of the map from the handlebody group to Outer automorphisms of a free group

Let $K$ be a compact oriented 3-dimensional handlebody of genus $g$. The group $H_g$ of isotopy classes of diffeomorphisms of $K$ is called the handlebody group. (It embeds as a subgroup of the ...
2
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2answers
2k views

Group action, Fixed point set and Orbit Space

I want to know to what extent is the group action determined by its fixed point data and orbit data, i.e. if $G$ acts on $M$ in two ways with the same fixed point set and orbit space, on what ...
10
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3answers
615 views

Is every (finite) group action on R^n by diffeomorphisms conjugate to a linear action?

I want to know if every smooth (finite)group action on $\mathbb{R}^n$ is conjugate to some linear action.Thank you!
10
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1answer
464 views

Self-homomorphisms of surface groups

Let $X$ be a closed, orientable surface of genus at least 2, and let $\phi: \pi_1(X) \to \pi_1(X)$ be a surjective homomorphism. Is $\phi$ necessarily injective?
25
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5answers
2k views

Compactification theorem for differentiable manifolds ?

Just parallelling this question, that seemed not to admit an easy answer at all, let's "soft down" the category and ask the same thing in the case of $\mathcal{C}^{\infty}$-differentiable manifolds [...
5
votes
2answers
2k views

Classification of mapping tori

Assume $M$ is a topological space and $f\in \operatorname{Homeo}(M)$, then $f$ obviously plays a significant role in the torus bundle $$M_f = M\times I/\{(x,0)\sim (f(x),1)\mid x\in M\}.$$ Hence ...
13
votes
2answers
2k views

exotic differentiable structures on manifolds in dimensions 5 and 6

It's a result of low-dimensional topology that in dimensions 3 and lower, two manifolds are homeomorphic if and only if they are diffeomorphic. Milnor's 7-spheres give nice counterexamples to this ...
5
votes
2answers
188 views

A question about axes of symmetry in the plane.

Suppose J is a Jordan curve in the Euclidean plane E and that there are two perpendicular straight lines in E, each of which is an axis of symmetry of J. Does the intersection point of these two ...
12
votes
3answers
1k views

To what extent is convexity a local property?

A polyhedron is the intersection of a finite collection of halfspaces. These halfspaces are not assumed to be linear, i.e. their bounding hyperplanes are not assumed to contain the origin. The ...
11
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2answers
1k views

Isometry classification of spherical space forms

A spherical space form is a compact Riemannian manifold of constant sectional curvature $1$, or equivalently, the quotient of the unit sphere by a finite group of orthogonal transformations that have ...
18
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2answers
4k views

Turning pants inside-out (or backwards) while tied together

An entertaining topological party trick that I have seen performed is to turn your pants inside-out while having your feet tied together by a piece of string. For a demonstration, check out this ...
7
votes
2answers
617 views

Knot complement diffeomorphism groups and embedding spaces

I'm interested in the following collection of questions: Let $S^n_k = \sqcup_k S^n$ be a disjoint union of $k$ distinct $n$-dimensional spheres. Write $Emb(S_k^n, S^{n+2})$ for the space of ...
6
votes
1answer
527 views

Rational homotopy type of a complement

Let $X$ and $X'$ be smooth closed manifolds. Take closed subpolyhedra $D\subset X$ and $D'\subset X'$ (with respect to some triangulations) and let $f:X\to X'$ be a homotopy equivalence such that $f(D)...
4
votes
1answer
822 views

Contractability of Exotic R^4s

Notation: $\mathbf{R}^4$ is a smooth manifold with underlying topology $(\mathbb{R})^4$; ${\mathbb{R}}^4$ is the standard smooth structure. The two things I know best about $\mathbf{R}^4$ is that it ...
6
votes
1answer
714 views

Weight filtration for smooth analytic manifolds

In his ICM 2002 talk (Topology of singular algebraic varieties, available also on arXiv) B. Totaro says on p. 3 (of the arXiv version): "Using the method of Guillen and Navarro Aznar I was able to ...
3
votes
1answer
2k views

How to shown that the Tangent Bundle of a vector space is a Vector Bundle

Hello, I have the following question about the tangent bundle $T_M = \bigcup_{p \in M} \{p\} \times T_p M$ defined on a manifold $M$ of class $C^r$ modeled on a normed space $X$. My problem is ...
9
votes
1answer
716 views

Applications of Faber's conjecture

Faber's perfect pairing conjecture states that the tautological ring $R^*$ of the moduli space $\mathcal{M}_g$ of curves of genus $g$ behaves as if it were the rational cohomology of a closed, ...
11
votes
1answer
803 views

classification of smooth involutions of torus

Let $\mathbb{Z}_2=\{1,g\},T^2=\{(e^{i\theta_1},e^{i\theta_2})\}$ and place $T^2$ in $\mathbb{R}^3$ as the locus of the rotation of $2\pi$ rads of the circle$\{(y,z)|(y-2)^2+z^2=1\}$ around $z$ axis. ...
2
votes
1answer
647 views

isotropic deformation retract of Weinstein manifolds?

I found the following paragraph in the paper " Intro to symplectic field theory " which I don't understand what does it mean precisely? Suppose W is a symplectic (or Kahler) manifold. D, smooth ...
5
votes
1answer
464 views

Heegaard genus in hyperbolic 3-manifolds

It is well known that for a closed hyperbolic 3-manifold $M$ the rank of $\pi_1(M)$ is bounded above by some universal constant $K$ times the volume of $M$. Using similar methods, i.e. the thick-thin ...
13
votes
3answers
862 views

Rational homotopy theory of a punctured manifold

Let $M$ be a smooth simply connected manifold and let $N$ be $M$ minus a point. Is it possible to construct an explicit Sullivan model for $N$ (i.e. a commutative differential graded algebra (cdga) ...
17
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2answers
1k views

Involutions of $S^2$

are there some complete results on the involutions of 2 sphere? at least I have three involutions: (let $\mathbb{Z}_2=\{1,g\}$,and $S^2=\{(x,y,z)\in\mathbb{R}^3|x^2+y^2+z^2=1\}$) 1.$g(x,y,z)=(-x,-y,-...
3
votes
1answer
356 views

Singular, holonomy-free connections on Riemannian surfaces?

Consider principal connections on the frame bundle of a compact, connected, smooth, orientable Riemannian surface embedded in $\mathbb{R}^3$. On a disk $D$, it is apparent that you can construct a ...
18
votes
4answers
3k views

Why is complex projective space triangulable?

In an exercise in his algebraic topology book, Munkres asserts that $\mathbf{C}P^n$ is triangulable (i.e., there is a simplicial complex $K$ and a homeomorphism $|K| \rightarrow \mathbf{C}P^n$). Can ...
15
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10answers
2k views

Orbifold fundamental group in terms of loops?

In chapter 13 in his notes on 3-manifolds, Thurston defines the orbifold fundamental group to be the group of deck transformations of the universal cover of the orbifold. He also makes a statement "...
10
votes
2answers
593 views

Presentation of the monoid of surfaces

In the following every surface is assumed to be connected. I've read that the commutative monoid of homeomorphism classes of closed surfaces is generated by $P$ (projective plane) and $T$ (torus) ...
39
votes
0answers
2k views

Minimal volume of 4-manifolds

This question came up in a talk of Dieter Kotschick yesterday. The minimal volume of a manifold is the infimum of volumes of Riemannian metrics on the manifold with sectional curvatures bounded in ...