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).
4,133
questions
24
votes
1
answer
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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 ...
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 ...
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 $\...
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 ...
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 ...
20
votes
1
answer
2k
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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 ...
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 ...
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.
11
votes
1
answer
3k
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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 ...
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 $M \times R^1?$
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 ...
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 $\pi_1 (S^3 \setminus L)$. ...
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 ...
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 $S^1\rightarrow \mathbb{R}^2$ ...
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 ...
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 ...
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 ...
22
votes
0
answers
612
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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 ...
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 $\...
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 ...
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 ...
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 ...
3
votes
1
answer
1k
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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) ...
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 $\...
6
votes
3
answers
859
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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 ...
17
votes
4
answers
2k
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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 ...
8
votes
1
answer
5k
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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 ...
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 ...
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 ...
5
votes
3
answers
1k
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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 ...
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 ...
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 ...
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
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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 ...
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 $B$-...
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 ...
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 ...
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 ...
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 ...
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?
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 ...
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 ...
10
votes
3
answers
2k
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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 ...
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 ...
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 ...
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 ...
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 ...
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 ...