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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Building a genus-$n$ torus from cubes

I wonder if this has been studied: What is the fewest number of unit cubes from which one can build an $n$-toroid? The cubes must be glued face-to-face, and the boundary of the resulting object ...
Joseph O'Rourke's user avatar
8 votes
2 answers
367 views

Hyperbolic structures on $S\times\mathbb{R}$

Let $S$ be a closed incompressible surface in a finite volume hyperbolic 3-manifold $M$ without cusps. Let $N$ be the cover of $M$ associated to $\pi_1(S) \subset \pi_1(M)$. The cover $N$ is ...
b b's user avatar
  • 1,591
10 votes
4 answers
1k views

Examples of acylindrical 3-manifolds

Let $C$ be the compact cylinder $S^1\times [0,1]$. A 3-manifold $M$ with incompressible boundary is called acylindrical if every map $(C,\partial C)\to (M,\partial M)$ that sends the components of $\...
HJRW's user avatar
  • 24k
5 votes
0 answers
270 views

deformed Gauss Bonnet formula?

I was reading about Gauss-Bonnet for circles, that integrating the curvature over the circle $\gamma(t)$ leads to the number of times $\gamma'(t)$ rotates around the origin. (The degree of the Gauss ...
john mangual's user avatar
  • 22.6k
5 votes
1 answer
282 views

Normal generation of Torelli

The only normal generators of the Torelli group of a closed surface I can find is Powell's 1977 paper (where the presentation is a bit complicated and given essentially without proof). Is there any ...
Igor Rivin's user avatar
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20 votes
1 answer
2k views

Can the Alexander horned sphere arise as a cell boundary in a finite CW-sphere?

Recently, I've been wondering to what extent certain types of pathologies can arise in finite CW complexes -- notice that I do not want to assume that I'm in the PL category or that the CW complexes ...
Patricia Hersh's user avatar
2 votes
2 answers
497 views

Multiple Dehn twists and minimal position

I have a question about a proof that I am reading in "A primer on Mapping Class Groups" by Farb and Margalit. Let $a$ be a simple closed curve in a compact surface $S$ (possibly with marked points ...
Sean's user avatar
  • 21
15 votes
5 answers
878 views

extension of surface homeomorphism

Can anyone give me a reference (or proof sketch) for the fact that there are psuedo-Anosov diffeomorphisms of closed hyperbolic surfaces which do not extend over any handlebody? Thanks.
michael freedman's user avatar
11 votes
1 answer
3k views

Cup products of connected sum

Hej, I am interested in the cohomology ring of the connected sum $M \# N$ of two oriented manifolds $M$ and $N$ in terms of the corresponding cohomology rings of $M$ and $N$. Mayer-Vietoris shows ...
Viktor's user avatar
  • 111
5 votes
0 answers
211 views

Universal exotic $R^4$

Let $R$ be a universal exotic $R^4.$ Is there an open orientable 3-manifold $M$ such that there is a smooth embedding of $R$ into $M \times R^1?$
cccmjj's user avatar
  • 51
9 votes
2 answers
702 views

Can we determine which monodromy of surface gives a fibered knot?

A fibered knot is a knot with a homeomorhism on compact surface with one boundary component. On the contrary, for a given homeomorohism on a compact surface with one boundary component, is there any ...
hopflink's user avatar
  • 537
2 votes
0 answers
370 views

link group of the trivial $n$ component link

Currently I am reading Milnor's paper "Link Groups". In the paper he defines the link group, for every $n$ component link $L$, as a certain quotient of the fundamental group $\pi_1 (S^3 \setminus L)$. ...
W. Politarczyk's user avatar
3 votes
1 answer
1k views

What does the 'V' in 'V-manifold' stand for?

The story of how the name 'orbifold' came about is pretty well-documented, but I can't find any explanation as to why Satake originally named orbifolds 'V-manifolds'. The 'manifold' part is clear ...
user avatar
5 votes
2 answers
583 views

Reference request: 2-dimensional Schonflies theorem

Does anyone know a reference for the 2-dimensional version of the Schoenflies theorem? To be precise, I'd like a reference for the fact that every continuous, 1-1 map $S^1\rightarrow \mathbb{R}^2$ ...
Dan Ramras's user avatar
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8 votes
1 answer
826 views

TQFT and Mapping Class Groups

It is well known how could we get a representation of the mapping class group of a surface S (which I assume compact, connected and orientable), given a TQFT. My question is: is there any reference ...
Ramses's user avatar
  • 229
4 votes
1 answer
340 views

interval exchange maps and surfaces

I apologize if the question is a well-known theorem, but I'm just starting to learn about laminations, so I don't know much. The question is roughly, if interval exchange maps have an underlying ...
Mircea's user avatar
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4 votes
0 answers
573 views

Topological version of two results in smooth Morse theory

Morse theory is generally presented in the DIFF category. However, there is a version of Morse theory in TOP (see the post Morse theory in TOP and PL categories? for references). It is well known ...
Victor's user avatar
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22 votes
0 answers
612 views

Smooth thickenings of non-smoothable manifolds

It is known that any closed topological manifold is homotopy equivalent to an open smooth manifold. Question 1. What can be said about the smallest dimension of a smooth manifold that is homotopy ...
Igor Belegradek's user avatar
0 votes
0 answers
118 views

uniform properness of lifts of uniform proper maps

Let $(X,d)$ be a unique geodesic metric space and $Y\subseteq X$ a subset, equipped with the induced path metric $d'$. We say that the inclusion $i\colon Y\hookrightarrow X$ is uniformly proper if $\...
Damiano Lupi's user avatar
8 votes
1 answer
1k views

Rational homology spheres and knots

It is known that the boundary of a branched double cover of the four ball branched over a surface bounded by a knot in is a rational homology $3$-sphere. Two questions come to my mind simply out of ...
Juan OS's user avatar
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20 votes
2 answers
1k views

Characteristic classes for block bundles

Block bundles are PL analogs of vector bundles, see e.g. Rourke-Sanderson's article in Bulletin of AMS, 1966. There is a classifying space $B\widetilde{PL}_q$ for rank $q\ $ block bundles, which is ...
Igor Belegradek's user avatar
3 votes
3 answers
345 views

When is a three-manifold deck transformation group solvable?

Suppose that $\pi:Y \to Y'$ is a regular covering of closed, connected, orientable three-manifolds and let $G$ be the deck transformation group. Furthermore, suppose that $Y$ is a rational homology ...
Tye Lidman's user avatar
3 votes
1 answer
1k views

Homeomorphism classification of 4-manifolds

Question 1. Let $X_i$ be an infinite family of closed, orientable, smooth 4-manifolds with the following properties: a) $\pi_1(X_i) = \mathbb{Z}\times \mathbb{Z_{2}}$ for any $i = 1, 2, \cdots $ b) ...
user23802's user avatar
1 vote
1 answer
137 views

Example of dynamical system $M$ such that $M \rightarrow \mathbf{R}\backslash M$ is not locally trivial?

Let $\Phi: M \times \mathbf{R} \rightarrow M$ be a smooth dynamical system having no periodic orbits, i.e. such that the canonical map $\pi:M \rightarrow \mathbf{R}\backslash M$ is a principal $\...
Nicolas Schmidt's user avatar
6 votes
3 answers
859 views

Reference request: embedded Morse theory

For references of embedded Morse theory, or so-called relative morse theory of a pair, I have found R.W. Sharpe's "Total Absolute Curvature and Embedded Morse numbers" totally not helpful. For further ...
JHM's user avatar
  • 2,254
17 votes
4 answers
2k views

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 ...
Victor's user avatar
  • 2,076
8 votes
1 answer
5k views

Manifolds are paracompact

By Definition, smooth manifolds are assumed to be Hausdorff and to satisfy the second countability axiom. I have heard (but never seen written) that these assumptions imply paracompactness (and thus ...
ThiKu's user avatar
  • 10.3k
10 votes
3 answers
830 views

Surface Eversions: Generalizing from Sphere and Torus Eversions

In 1958, Smale proved that a $2$-sphere can be "turned inside out", and throughout the 60s, 70s, and 80s, explicit constructions such as Thurston Corrugations, and Minimax eversions were developed to ...
Samuel Reid's user avatar
  • 1,401
39 votes
1 answer
6k views

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 ...
Victor's user avatar
  • 2,076
5 votes
3 answers
1k views

Matrices generating non-discrete subgroups of SL(2,R)

Jorgensen's inequality $\mid \left(Tr\left(A\right)\right)^2-4\mid+\mid Tr\left[A,B\right]-2\mid\ge 1$ gives a necessary condition for two matrices A,B to generate a discrete subgroup of SL(2,R). Are ...
ThiKu's user avatar
  • 10.3k
25 votes
1 answer
697 views

Diffeomorphisms of finite order not in the image of a circle action

Does there exist a closed smooth manifold $M$ and a diffeomorphism $f\colon M \to M$ such that: $f$ is isotopic to the identity, $f$ is of finite order, $f^n=ID$, and $f$ is not contained in the ...
Jarek Kędra's user avatar
  • 1,772
1 vote
2 answers
382 views

Mapping class between coverings of Riemann surfaces

Let $X$ be a closed Riemann surfaces of genus $g$ and let $p_1:Y_1 \rightarrow X$ and $p_2:Y_2 \rightarrow X$ be two $K$-sheeted, connected, unramified coverings of the Riemann surface. By the theorem ...
berl13's user avatar
  • 471
59 votes
7 answers
7k views

Status of PL topology

I posted this question on math stackexchange but received no answers. Since I know there are more people knowledgeable in geometric and piecewise-linear (PL) topology here, I'm reposting the question. ...
2 votes
1 answer
2k views

Difference between $S^2$-bundles over $S^2$ and $CP^2\sharp CP^2$

I want to distinguish $S^2$-bundles over $S^2$ from $CP^2\sharp CP^2$. As you know, a $S^2$-bundle over $S^2$ is $S^2\times S^2$ or $M= S^3\times_{S^1}S^2$ where $M$ is diffeomorphic to a ...
Hee Kwon Lee's user avatar
  • 1,070
5 votes
1 answer
447 views

Does the fundamental group of a surface have rigid subgroups?

Suppose $\Gamma$ is the fundamental group of a closed, oriented surface $S$. Let $B$ be a finitely generated, infinite index subgroup of $\Gamma$, and let $\Gamma_B$ be the compact core of the $B$-...
Lee Mosher's user avatar
  • 15.3k
23 votes
7 answers
9k views

Introduction to Floer Theory?

Can anyone suggest a good overview/introduction of the Floer machine for a beginner? (Someone pointed out some intriguing connections to surface mapping class groups, which might be enough incentive ...
2 votes
0 answers
209 views

Does tetrahedral complex in R^3 with "2-spherical" boundary have to be a 3d ball?

Let $S$ be a finite set of tetrahedrons in $\mathbb{R}^3$. Let $S$ be tetrahedral complex in a sense that if two tetrahedrons intersect, the intersection is a face of both. In what follows we view ...
IL.'s user avatar
  • 43
6 votes
0 answers
196 views

Surgering locally flat tori in 4-manifolds

Is there a locally flat torus in some not smoothable topological 4-manifold such that surgering on it produces a smoothable 4-manifold? Surgering means removing a tubular neighborhood and reattaching ...
Daniele Zuddas's user avatar
6 votes
1 answer
503 views

Periodic automorphisms of free groups

Hi, I am troubled with the following question: Does there exist a finite order automorphism of a free group, $f\in Aut(F)$, such that it fixes no non trivial conjugacy class and no non trivial ...
Rizos's user avatar
  • 63
12 votes
1 answer
795 views

Handlebody decomposition of an open 4-manifold

Let $M$ be the fake $CP^2$ (namely the closed topological 4-manifold which is homotopy equivalent but not homeomorphic to the complex projective plan). It is well-known that $M$ admits no smooth ...
Daniele Zuddas's user avatar
20 votes
2 answers
1k views

Manifolds with homeomorphic interiors

Suppose that two compact topological manifolds with boundary have homeomorphic interiors. Can we conclude that the two manifolds are homeomorphic? What happens in the smooth category?
Daniele Zuddas's user avatar
5 votes
0 answers
172 views

BKS pairing in the SU(2) Chern-Simons theory

I know that usually, the way to compare the Hilbert spaces arising from $SU(2)$ Chern-Simons theory with different Kähler polarizations is via the Hitchin connection. However, it should be possible, I ...
Blake's user avatar
  • 1,025
2 votes
2 answers
663 views

Sufficient conditions for a 3D tetrahedral complex to be homeomorphic to a 3D ball

Dear experts, let $T$ be finite tetrahedral complex in flat 3-dimensional euclidean space. Additionally, let $T$ be 'homogeneous' in a sense that each simplex in $T$ is a face of some tetrahedron ...
IL.'s user avatar
  • 43
10 votes
3 answers
2k views

Applications of knot theory to biology/pharmacology

What are the applications of knot theory to biology/pharmacology? I guess there should be some, since proteins are quite long and some of their properties are probably related to whether they are ...
8 votes
1 answer
507 views

How to compute the Monopole Floer Homology for Surface $\times S^1$ ?

We know that Monopole Floer homology of a 3-manifold $M$ depends on a spin-c structure. My question is that if $M$ is $F\times S^1$ ($F$ is a surface of genus larger than 1) then how can we compute ...
juliuslin's user avatar
10 votes
2 answers
495 views

Equivariant cohomology of the complement to the arrangement $\bigcup_{i\neq j}\vec x_i = \vec x_j$?

$\DeclareMathOperator\SO{SO}\DeclareMathOperator\SU{SU}\DeclareMathOperator\Conf{Conf}$Let $V=\mathbb{R}^d$ be a $d$-dimensional (Euclidean) vector space over real numbers. Let $G=\SO(V)$ be the ...
Anton Khoroshkin's user avatar
6 votes
1 answer
2k views

Connections on line bundles over the torus

If I understand correctly, every line bundle $L$ over the (2-dim) torus can be obtained from a quotient of $\mathbb{R}^2 \times \mathbb{C}$ by a $\mathbb{Z}^2$ lattice action. Different line bundles ...
Blake's user avatar
  • 1,025
3 votes
1 answer
450 views

When is the Freudenthal compactification an ANR?

Let $X$ be a locally compact metric ANR (or, if preferred, a locally compact simplicial complex). If needed, assume that $X$ has finitely many ends or is of finite dimension. My question is: What are ...
Michał Kukieła's user avatar
5 votes
1 answer
231 views

Finite index subgroups of the mapping class group with geometric meaning

I have got a question that is perhaps not precise in a mathematical sense. Is there a classification of all coverings of the moduli space of Riemann surfaces which are moduli spaces themselves, that ...
berl13's user avatar
  • 471
9 votes
5 answers
501 views

A terminological question concerning orbifolds.

The notion of orbifold is quite well established by now. I would like to ask how one should call a point of an orbifold with non-trivial stabilizer? Should one call this a singular point? Of something ...

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