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).
4,016
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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 ...
- 397
-4
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55
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Are ultra-homogeneous spaces connected? [closed]
Given a métric space (M,d), M is ultrahomogeneous if and given any two isometric compact subsets A, B ⊂ M, every isometry f : A → B
extends to a global isometry $\hat{f}$
: M → M.
- 1
1
vote
1
answer
75
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 ...
6
votes
0
answers
124
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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 ...
- 61
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votes
0
answers
84
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Realizing a finite subgroup of $\mathrm{Homeo}^+(S_g)$ as a subgroup of $\mathrm{Isom}^+(S_g)$ for some suitably chosen hyperbolic metric on it
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 ...
- 29
0
votes
0
answers
111
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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 ...
- 139
5
votes
0
answers
133
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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 (...
- 3,347
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100
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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 ...
- 3,347
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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 (...
6
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1
answer
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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 $...
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3
votes
1
answer
133
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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
58
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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 $...
- 1,993
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votes
1
answer
289
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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 ...
- 129
4
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0
answers
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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 ...
- 141
3
votes
1
answer
113
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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 ...
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2
votes
1
answer
155
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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 $T_{1}M$ ...
- 73
2
votes
0
answers
50
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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 ...
- 503
1
vote
0
answers
55
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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 ...
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1
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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 ...
- 379
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votes
1
answer
356
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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 ...
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3
votes
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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-...
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2
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1
answer
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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_{...
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votes
1
answer
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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) ...
- 105k
5
votes
2
answers
456
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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 ...
- 105k
9
votes
1
answer
264
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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
99
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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 ...
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votes
2
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571
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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$ ...
- 397
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votes
1
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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 $\...
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votes
1
answer
147
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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 ...
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votes
1
answer
245
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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....
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votes
3
answers
841
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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 ...
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362
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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 ...
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2
votes
1
answer
150
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 ...
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votes
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146
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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
232
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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-...
- 379
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votes
0
answers
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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 ...
- 119
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votes
2
answers
166
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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$ ...
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1
answer
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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 ...
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0
answers
143
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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 ...
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9
votes
1
answer
354
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 ...
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2
votes
1
answer
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Entanglement, quadrics and $\mathbb{P}^2(\mathbb{C}^3)$ [closed]
First of all: I apologise in advance for if my question will be arid, wrong written or even nonsensical.
I was at a talking with a professor last week, and the question of "Entanglement and ...
- 147
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votes
0
answers
280
views
What are some of the big open problems in $4$-manifold theory?
I've recently been studying some Manifold Theory and got very interested in their topological as well as geometric properties. From my understanding of the current literature, most the big and ...
- 121
9
votes
1
answer
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Mapping class groups are finitely generated
Let $N$ be a compact smooth manifold. By "mapping class group" I will mean
$$\pi_0 \operatorname{Diff}(N)$$
i.e. the isotopy-classes of diffeomorphisms of $N$.
My presumption is that this ...
- 42.1k
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votes
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answers
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Is Heegaard-Floer homology the Lagrangian Floer homology of $M=\text{Sym}^g(\Sigma), L_0=\mathbb{T}_{\alpha},L_1=\mathbb{T}_{\beta}$?
Is Heegaard Floer homology the Lagrangian Floer homology of $M=\text{Sym}^g(\Sigma), L_0=\mathbb{T}_{\alpha},L_1=\mathbb{T}_{\beta}$? I am interested in the relationship between the theories
...
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votes
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answers
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"Next steps" after TQFT?
(Disclaimer: I'm rather nervous that this isn't appropriate for MathOverflow, but given the contents of my question I don't really know a better place to ask something like this.)
Recently, I've been ...
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votes
1
answer
134
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English version of a paper by Gusarov
I am looking for the english translation of the paper in russian Variations of knotted graphs, geometric technique of n-equivalence, St. Petersburg Math. J. 12-4 (2001) by Gusarov.
There is a .ps file ...
- 23
4
votes
1
answer
163
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Conformal map between flat and hyperbolic torus with a boundary
I am confused because I can define two very different complex structures on the torus with a puncture/boundary.
For my first construction, I can imagine removing a disk from a flat torus, inheriting ...
- 173
1
vote
1
answer
208
views
A torus bundle whose vertical tangent bundle is indecomposable
I would like to see an example of a fiber bundle $\mathbb{T}^2 \to X \xrightarrow{\pi} B$ whose fibers are tori $\mathbb{T}^2:=(S^1)^2$ and whose vertical tangent bundle $\ker(d\pi) \to X$ is not ...
- 768
3
votes
0
answers
96
views
Integrating over a fundamental domain in $\text{SL}(d,\mathbb R)$ under $\text{SL}(d,\mathbb Z)$ action and the choice of the fundamental domain
Let $\mathcal{F}$ be a fundamental domain in $\text{SL}(d,\mathbb R)$ under $\text{SL}(d,\mathbb Z)$ action. It is well known that there exists a unique $\text{SL}(d,\mathbb R)$-invariant probability ...
- 111
4
votes
0
answers
317
views
Does a contractible locally connected continuum have an fixed point property?
I'm surprised that I can't find any research on this topic. Maybe it's too obvious? Kinoshita proved that contractible continuum do not have FPP, but his example is not locally connected. Maybe if we ...
