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).

Filter by
Sorted by
Tagged with
3 votes
0 answers
290 views

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 ...
Shijie Gu's user avatar
  • 1,936
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(...
rab's user avatar
  • 139
1 vote
0 answers
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 $...
0x11111's user avatar
  • 473
2 votes
0 answers
52 views

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 ...
Sagnik Biswas ma20d013's user avatar
3 votes
1 answer
294 views

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 $...
seldom seen's user avatar
11 votes
2 answers
1k views

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 ...
Matthew Niemiro's user avatar
1 vote
0 answers
27 views

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(\...
Stéphane Laurent's user avatar
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 ...
Nash's user avatar
  • 47
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 ...
maxematician's user avatar
4 votes
1 answer
122 views

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)...
DavidHume's user avatar
  • 741
2 votes
0 answers
129 views

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^...
Sagnik Biswas ma20d013's user avatar
13 votes
1 answer
323 views

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 ...
Zhenhua Liu's user avatar
1 vote
1 answer
169 views

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 ...
one potato two potato's user avatar
6 votes
1 answer
237 views

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 ...
user513804's user avatar
5 votes
1 answer
274 views

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 ...
Rajesh Dey's user avatar
0 votes
0 answers
139 views

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 ...
Anonmath101's user avatar
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 (...
Ian Gershon Teixeira's user avatar
4 votes
0 answers
133 views

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 ...
Ian Gershon Teixeira's user avatar
0 votes
0 answers
17 views

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 (...
Fernando Oliveira's user avatar
8 votes
1 answer
343 views

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 $...
Tashi Walde's user avatar
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|...
monoidaltransform's user avatar
2 votes
0 answers
64 views

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 $...
Dmitrii Korshunov's user avatar
11 votes
1 answer
316 views

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 ...
Dmitrii Ivanov's user avatar
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 ...
Ivan So's user avatar
  • 141
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 ...
mrburch's user avatar
  • 155
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 $T_{1}M$ ...
user473085's user avatar
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 ...
Jacques's user avatar
  • 563
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 ...
user2357's user avatar
5 votes
1 answer
132 views

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 ...
luthien's user avatar
  • 379
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 ...
Piotr Hajlasz's user avatar
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-...
Kregnach's user avatar
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_{...
Rajesh Dey's user avatar
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) ...
Iosif Pinelis's user avatar
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 ...
Iosif Pinelis's user avatar
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 ...
João Lobo Fernandes's user avatar
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 ...
David Richter's user avatar
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 $z$ ...
Zhenhua Liu's user avatar
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 $\...
Don's user avatar
  • 163
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 ...
Sami Douba's user avatar
4 votes
1 answer
272 views

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....
Noah B's user avatar
  • 403
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 ...
Andrey Ryabichev's user avatar
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 ...
Piotr Hajlasz's user avatar
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 ...
Prateep's user avatar
  • 141
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. ...
Some Person's user avatar
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-...
luthien's user avatar
  • 379
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 ...
ZSMJ's user avatar
  • 131
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 $M$ ...
return true's user avatar
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 ...
Vincent Granville's user avatar
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 ...
Sandra's user avatar
  • 11
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 ...
Sarah's user avatar
  • 93

1 2
3
4 5
83