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).
4,129
questions
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Is G(4,7) a Coxeter group
Let $G(4, 7)$ be an abstract group with the presentation
$$\langle a,b,c | a^2 = b^2 = c^2 = 1, (ab)^4 = (bc)^4 = (ca)^4 = 1, (acbc)^7 = (baca)^7 = (cbab)^7 = 1 \rangle $$
Richard Schwartz considered ...
6
votes
0
answers
141
views
Kirby diagram of Enriques surface (as the "(1/2) K3 surface")
Not to be confused with $E(1)\cong\mathbb{C}P^2\#9\overline{\mathbb{C}P^2}$, which is also known as a $\frac{1}{2}K3$ surface (in the sense that removing a neighbourhood of a regular torus fiber in $E(...
1
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0
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130
views
Higher dimensional Seifert surfaces and link numbers of higher knots
In 3-manifold topology, the notion of Seifert surface is well known. It is then used to define link numbers of knots.
Question: Consider embedding $N^n \rightarrow M^{2n+1}$ of n-dimensional manifold $...
2
votes
0
answers
52
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Tangential normal invariant isomorphism
Recently, I was reading the paper "Finite Group Actions on Kervaire Manifold" by Crowley, Hambolton. But I am having problem understanding a definition. Here it is,
In page 15-16 they are ...
3
votes
1
answer
294
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How many configurations of tubes are there?
Can $n$ disjoint lines in $\boldsymbol R^3$ be knotted? No... Let $X_n$ be the configuration space of $n$ disjoint lines in $\boldsymbol R^3$. It is not hard to see that $X_n$ is path connected: Let $...
11
votes
2
answers
1k
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When does a group act effectively and holomorphically on some Riemann surface?
Given a Riemann surface $X$, we have some fairly standard methods for identifying which groups $G$ admit an effective and holomorphic action $G \times X \to X$. For instance, some fairly elementary ...
1
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0
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27
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Unusual parameterization of the ring Dupin cyclide
I discovered the following by playing with the formulas given in the paper Sculptures in $S^3$ by Schleimer and Segerman.
First, define the following parameterization of the Clifford torus:
$$
p(\...
0
votes
0
answers
218
views
Pushforwards in vector bundles over a topological spaces
I have been reading the discussion from Pushforward and pullback..
I understand that it is quite straight forward to construct a pullback of a vector bundle. In the discussion it is clear that if we ...
0
votes
0
answers
33
views
Minimum diameter of spherically-inverted topological balls
Let $U$ be the closed unit ball in $\mathbb{R}^3$. Let $S$ be a round sphere whose center is in $U$ with radius at least $\delta_1 > 0$. Suppose $B$ is a closed topological ball of Euclidean ...
4
votes
1
answer
122
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Does the Alexandrov angle define convex functions along geodesics in CAT(0) spaces?
Let $X$ be a CAT(0) space and suppose $a,b,c,d\in X$ satisfy
$$
\max\{\angle_a(b,c),\angle_a(b,d)\}<\frac{\pi}{2}.
$$
Let $\gamma:[0,\ell]\to X$ be the geodesic with $\gamma(0)=c$ and $\gamma(\ell)...
2
votes
0
answers
129
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Explicit S-duality map
$\DeclareMathOperator{\Th}{Th}$
The Thom space of a closed manifold $M$ ($\Th(M)$) is the $S$-dual to $M_+ (=M \cup \{pt\})$. Let $M$ be embedded in $\mathbf R^{n+k}$. I found the duality map from $S^...
13
votes
1
answer
323
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Realizing integral homology classes on non-orientable manifolds by embedded orientable submanifolds
Let $M^m$ denote a compact, non-orientable smooth manifold and $\nu$ an integral homology class of dimension $n$. I am interested in understanding the representability of $\nu$ by embedded, orientable ...
1
vote
1
answer
169
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Question on ideal triangulation and geodesic lamination
Q1. Does a closed hyperbolic surface admit an ideal triangulation?
Here, an ideal triangulation of a surface means a partition of a surface by geodesics such that each component of the complement ...
6
votes
1
answer
237
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Does the inner automorphism group of the fundamental group of a closed aspherical manifold always have an element of infinite order?
Let $\pi_1$ be the fundamental group of a closed aspherical manifold of dimension $n$. In particular, $\pi_1$ is finitely presented, torsion-free and its cohomology is finitely generated and satisfies ...
5
votes
1
answer
274
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Realizing a finite subgroup of $\mathrm{Homeo}^+(S_g)$ as a subgroup of $\mathrm{Isom}^+(S_g)$
Let $G\leq \operatorname{Homeo}^+(S_g)$ be finite, where $S_g$ is a closed, connected, orientable surface of genus at least $2$. Then I have the following questions:
(1) Can $G$ always be realized as ...
0
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0
answers
139
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Weyl groups are Coxeter groups proof
I'm reading part of a proof that says that Weyl groups of apartments of buildings are Coxeter groups.
Let $\Delta$ be a building and let $\Sigma$ be a fixed apartment of $\Delta$. Let $C$ be a fixed ...
6
votes
0
answers
342
views
Why can't a Lie group act transitively on a finite volume hyperbolic manifold?
In the comments on the MathSE question "Is Seifert-Weber space homogeneous for a Lie group?",
it is claimed that if $ M $ is a manifold which admits a finite volume hyperbolic metric (...
4
votes
0
answers
133
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Is the total space of a $ U_1 $ principal bundle over a compact homogeneous space always itself homogeneous?
Let $ U_1 \to E \to B $ be a $ U_1 $ principal bundle. Suppose that $ B $ is homogenous (admits a transitive action by a Lie group) and compact. Then must it be the case that $ E $, the total space of ...
0
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0
answers
17
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Is the impression of an ideal boundary point (=end) the union of the impressions of the prime ends of the circle of prime ends associated to this end?
Let S be a compact orientable surface and U an open connected subset of S with finitely many ideal boundary points (or ends). U has a prime ends compactification which is a surface with boundary (...
8
votes
1
answer
343
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Given an embedded disk in $\mathbb{R}^n$, is there always another disk which intersects it nontrivially in a disk?
We call an open subset $D\subset X$ of a manifold $X$ an embedded disk, if there exists a homeomorphism $D\cong \mathbb{R}^n$.
The precise formulation of the question in the title is as follows:
Let $...
3
votes
1
answer
150
views
Embedding of half open half closed $n$-set in $n$-space
Let $n\geq 2$. Set $\Sigma= \{x\in \mathbb{R}^n: 1\leq |x|<2\}$. Assume $h:\Sigma
\rightarrow \mathbb{R}^n$ is continuous and injective.
Question: Must $h$ also be an embedding?
Some thoughts:
$h|...
2
votes
0
answers
64
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Decomposition length in the stable homeomorphism conjecture
Stable homeomorphism theorem (due to Brown--Gluck, Kirby, Quinn,...) states that any orientation preserving homeomorphism $f$ of $\mathbb R^n$ is stable, that is, it can be written as a superposition $...
11
votes
1
answer
316
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Embedded 2-tori in $S^1\times S^4$
I am interested in understanding the smooth isotopy class of embedded 2-tori in $S^1\times S^4$. Is it true that every two homotopic embedded 2-tori in $S^1\times S^4$ are smoothly isotopic? It would ...
4
votes
0
answers
92
views
KLO for operations over braids
KLO is a program that permits you to the do twistings, band operations over knots or Kirby diagram. However, I couldn't find a function on KLO that permits me to do the same thing over braids.
Is ...
3
votes
1
answer
126
views
Computer program for polyhedral manifolds
Suppose I have a 3-manifold obtained via face identifications of a polyhedron (e.g. the Poincaré sphere presented as a dodecahedron with opposite faces glued). Is there a program that exists for ...
2
votes
1
answer
196
views
Invariant measure of geodesic flow on unit tangent bundle of a modular surface
This is a paper written by Series "THE MODULAR SURFACE AND CONTINUED FRACTIONS".
I want to know about above construction natural invariant measure $\mu$ for the geodesic flow on $T_{1}M$ ...
2
votes
0
answers
76
views
Two different Bers embeddings
In An Introduction to Teichmüller spaces by Imayoshi and Taniguchi, they present in section 6.1.3 the Bers embedding as a map from Teichmüller space of a Riemann surface $X$ to the space of quadratic ...
1
vote
0
answers
67
views
Mapping class group interpretation of braid closure
Given a braid (diagram) $\beta\in B_n$, the associated closed braid is the knot/link formed by attaching the ends on which the strings lie. We can also, however, think of $\beta$ as being an element ...
5
votes
1
answer
132
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Properly embedded surfaces in handlebodies are compressible or boundary compressible?
I've read in a couple of different places (a paper and a blog) the following fact:
if $F$ is a surface, properly embedded in a three-dimensional handlebody of genus at least two, then $F$ is either ...
7
votes
1
answer
366
views
Exotic homeomorphisms of a cube
If $\varphi:\mathbb{R}\to\mathbb{R}$ is continuous, non-constant, non-decreasing, and differentiable a.e. with $\varphi'=0$ a.e., then the mapping
$$
\Phi(x,y)=(x+\varphi(x),y+\varphi(y))
$$
is a ...
3
votes
0
answers
90
views
Minimal set of geometric moves in various equivalence classes of triangulated geometries
I would like to get to know what is the minimal set of geometric changes "aka. moves" (topology preserving modifications / Pachner moves / bistellar moves) that can transform any 3-...
2
votes
1
answer
232
views
How does hyperelliptic involution act on the standard generators of the fundamental group of surfaces of genus g with n punctures?
Let $S_{g,n}$ be the surface of genus $g$ with $n$ punctures. We know that $\pi_1(S_{g,n})$ admits a presentation:
$$\left\langle~ \alpha_1,\beta_1,\dots, \alpha_{g},\beta_{g},\gamma_{1},\dots,\gamma_{...
3
votes
1
answer
252
views
Can such a set be simply connected?
$\newcommand\R{\mathbb R}$Let $U$ be an open subset of $\R^2$ such that the point $(0,0)$ is on the boundary of $U$. Let $f\colon[0,1]\to\R^2$ be the path that starts at $(0,0)$ and moves with a (say) ...
5
votes
2
answers
567
views
On the boundary of a simply connected set
Let $U$ be an open simply connected subset of $\mathbb R^2$. Let $x$ be a boundary point of $U$.
Does then there always exist a continuous function $f\colon[0,1]\to\mathbb R^2\setminus U$ such that $x ...
9
votes
1
answer
304
views
Torsion and nilpotence of framed manifolds under the Pontryagin-Thom map
The framed Pontryagin Thom construction produces a graded ring isomorphism from the framed cobordism ring $\Omega^{fr}_*$ to the stable ring of homotopy groups of spheres $\pi_*(\mathbb{S})$. I have a ...
1
vote
0
answers
104
views
An algebra with two multiplications, based on series-parallel diagrams?
Here is a commutative, unital, associative algebra $\mathcal{F}$ with two ways to multiply. The multiplications come from a construction with Boolean operations and series-parallel diagrams. I want ...
18
votes
2
answers
695
views
Integral homology classes that can be represented by immersed submanifolds but not embedded submanifolds
Let $M$ be an $m$-dimensional compact closed smooth manifold and $z\in H_n(M,\mathbb{Z})$ an $n$-dimensional integral homology class, with $m>n.$ Does there exist a pair of $M$ and $z$ so that $z$ ...
6
votes
1
answer
276
views
Reference for a property of Dehn twists
I was reading The symplectic Floer homology of a Dehn twist by P. Seidel, which you can find here.
In Lemma 3(ii) the following topological property of Dehn twists is stated without proof:
Let $\...
4
votes
1
answer
173
views
What are some properties of the leading eigenvalue of a product of inversions in mutually tangent spheres?
Let $S_1, \ldots, S_n$ be a collection of $n \geq 4$ pairwise tangent hyperspheres in $\mathbb{R}^{n-2}$ with disjoint interiors, and $\iota_i$ be the inversion in $S_i$. Viewing the conformal group ...
4
votes
1
answer
272
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Understanding $(\mathbb{Z}/3)^2 \times_{\mathbb{Z}/3} M$
I'm currently reading "Bordism of Elementary Abelian Groups via Inessential Brown-Peterson Homology" by Hanke (arXiv:1503.04563) and have come across some notation that I'm not familiar with....
16
votes
3
answers
884
views
Maximal degree of a map between orientable surfaces
Suppose that $M$ and $N$ are closed connected oriented surfaces. It is well-known that if $f \colon M \to N$ has degree $d > 0$, then $\chi(M) \le d \cdot \chi(N)$.
What is an elementary proof of ...
12
votes
0
answers
386
views
Is the Lipschitz structure on $\mathbb{S}^4$ unique?
Sullivan [S] proved that on any $n$-dimensional, $n\neq 4$, topological manifold there is a unique Lipschitz structure. Although it is generally accepted result, it seems that the proof lacks some ...
2
votes
1
answer
193
views
Subdivision of geometric simplicial complex
Let $\{v_0,v_1,\cdots,v_n\}$ be $n+1$ points in $\mathbb{R}^N$ which are geometrically independent. We define their convex hull to be a geometric simplex. Using this we can define geometric simplicial ...
0
votes
0
answers
158
views
Homeomorphism groups on manifolds and topological properties
Let $M$ be a compact $n$-dimensional manifold let $H(M)$ denote the homeomorphism group of $M$.
If $n=2$ then $H(M)$ enjoys nice properties such as being an ANR, is locally contractible, separable. ...
2
votes
2
answers
313
views
Twisted interval-bundles over a surface
I am trying to understand interval bundles over orientable surfaces. I know of course the basic examples: trivial interval bundles are just products. From what I understand, there is only one non-...
1
vote
1
answer
215
views
Reference request and prerequisites for understanding the Sphere Theorem and the Loop Theorem in 3-manifold theory
As part of my directed studies project, my advisor has suggested that I completely understand the proof of the Sphere Theorem and the Loop Theorem in 3-manifold theory and explain it to him. I have ...
2
votes
2
answers
184
views
Fibration of hyperbolic 3-manifold
A fibration of a manifold $\phi: M \to S^1$ gives rise to a short exact sequence
$$
1 \to \pi_1(N) \to \pi_1(M) = \mathbb{Z} \overset{f_\ast}{\to} 1
$$
where $N$ is the fiber.
I've heard that, if $M$ ...
2
votes
1
answer
314
views
What are the best definitions for smoothness of a 2D curve (real-valued function)?
Sounds like a trivial question, but could not find any answer other than the fact that there are many ways to define it. My problem is this: I look at different elevation maps,
some with sharp ...
1
vote
0
answers
156
views
Membership test of convex set
Let $K$ be a compact convex subset of $R^n$ which has some positive gaussian measure, say at least 1/2. For each nonzero vector $u \in R^n$, we
define another compact convex set $K * u$ in the ...
9
votes
1
answer
363
views
Morse theory on outer space via the lengths of finitely many conjugacy classes
Let $F_n$ be the free group on letters $\{x_1,\ldots,x_n\}$ and let $X_n$ be the (reduced) outer space of rank $n$. Points of $X_n$ thus correspond to pairs $(G,\mu)$, where $G$ is a finite connected ...