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).
4,134
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An auxiliary problem while constructing the system of Jordan sets on a plane
Let $\mathfrak{S}$ be a system of rectangles in $R^2$ of the form $[a,b]\times [c,d]$
where $a,b,c, d \in R$, $a<b$, $c<d$.
Let $\mathfrak{A}$ be a system of simple sets based on $\mathfrak{S}$.
...
5
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141
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What are the finite quotients of the braid group?
Are all known finite quotients of the braid group given by reducing the Burau or Lawrence-Krammer representations mod $p$ and evaluating at some element in $\mathbb{F}_p$? I recently saw a paper ...
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1
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209
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Submersion of real projective space into Euclidean space
The famous Whitney immersion theorem states that any real projective space $\mathbb{RP}^n$ can be immersed into $\mathbb{R}^{2n}$.
However, I haven't found information about the submersion ...
3
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Does there exist a finite-volume hyperbolic Coxeter polytope with these properties?
I searched for a finite-volume, hyperbolic Coxeter polytope of dimension $n \geq 4$ with the following properties $a$ and $b$.
$a$) It has exactly one ideal vertex;
$b$) if a bounded facet and an ...
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2
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Prove these are not surface groups
For $g,n \geq 1$, let $\Gamma_{g,n}$ be the group with the following presentation:
$$\langle \text{$a_1,b_1,\ldots,a_g,b_g$ $|$ $[a_1,b_1]^n [a_2,b_2] \cdots [a_g,b_g]=1$} \rangle.$$
For $n = 1$, ...
5
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134
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Representing some odd multiples of integral homology classes by embedded submanifolds
Consider an $m$-dimensional compact closed orientable smooth manifold $M$ and an $n$-dimensional integral homology class $[\Sigma]$ on $M$, with $1 \le n \le m-1$. Then does there exist an odd integer ...
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195
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Smooth compactification of complex varieties and uniqueness
Since I'm working in differential geometry, for the following I'm strictly interested in the smooth setting over $\mathbb{C}$ and its relation to the setting over $\mathbb{R}$.
Here are a few useful ...
13
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321
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Nonsmoothable 4-manifolds
Does there exist a closed connected nonsmoothable 4-manifold $M$ such that:
$\kappa(M)=0$ (Kirby-Siebenmann invariant vanishes, hence, there is no "classical" obstruction to smoothability) ...
13
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1
answer
472
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Impossibility of realizing codimension 1 homology classes by embedded non-orientable hypersurfaces
Suppose we have an $n+1$-dimensional compact closed oriented manifold $M$ and an $n$-dimensional integral homology class $[\Sigma]\in H_n(M,\mathbb{Z})$ on $M.$ Then is it true that $[\Sigma]$ mod $2$ ...
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183
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Reference for a folklore theorem about h-cobordisms
I've seen referenced here that if $M$ and $N$ are closed topological $n$-manifolds and $f: \mathbb{R}\times M \to \mathbb{R}\times N$ is a homeomorphism, then $M$ and $N$ are h-cobordant.
I know that ...
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80
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$L^p$-compression of metabelian groups
Question: Is there a metabelian group, so that for some $\epsilon >0$ and all $p \in [1, \infty[$ the [equivariant] compression exponent in [any] $L^p$-space is bounded by $1-\epsilon$ (bound does ...
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1
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185
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Partitions of unity with arbitrary Lip-constants
Lets make things simple. Suppose we have a compact metric space $(X,d)$ and then some Lipschitz partition of unity exists, say a collection $\mathcal{F}=\{f_n\}$ subordinate to some open cover $\...
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Lattice generated by parabolics
Let $G$ be a semisimple Lie group of split-rank one and let $\Gamma$ be a non-cocompact lattice which is torsion-free.
For the group $G=\mathrm{SL}_2(\mathbb{R})$ it then follows that $\Gamma$ is ...
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Double ($p$-fold) coverings of $B^4$ along ribbon/slice disks
I have two questions that seem to be related.
I wonder if there is a user-friendly algorithm (starting from ribbon/slice presentation of knots/disks) for the construction of double (in general $p$-...
12
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Isotopic diffeomorphisms of the sphere
Assume that $f:\mathbb{S}^n\to\mathbb{S}^n$ is a diffeomorphism and assume that there is an orientation preserving diffeomorphism $F:\mathbb{R}^{n+1}\to\mathbb{R}^{n+1}$ such that $F|_{\mathbb{S}^n}=f$...
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Stable homeomorphism theorem and the annulus theorem
Brown and Gluck [BG] proved in 1964 that the stable homeomorphism conjecture implies the annulus conjecture.
Is the proof of this implication difficult?
Is there any other place with the proof of ...
5
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159
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Volume of the Weeks manifold and of the 5.2 knot complement
Some computations show that the Weeks manifold and the 5.2 knot complement have the same trace field (which is $\mathbb{Q}[x]/(x^3-x+1)$) and the (hyperbolic) volume of the second is 3 times the ...
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2
answers
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Is a simple closed curve always a free boundary arc?
Is it possible to extract a neighborhood around any point on a simple closed curve such that the boundary of this neighborhood intersects the curve at only two points?
For a simple closed curve $\...
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185
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Local to global complexity of triangulations
Alright 3rd time's the charm - editing again to put all my cards on the table.
Consider a PL $n$-manifold $M$. Define the complexity $c(M)$ of $M$ to be the minimum number of $n$-simplices needed to ...
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A question about the existence of spin maps
Let $M, N$ be two smooth manifolds, not necessarily spin. My question is the following:
How can we construct a non-constant spin map $f:M\to N$ of degree zero?
Here spin map means that $f$ preserves ...
4
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2
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532
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Compactification of a product of manifolds
Let $M$ be a smooth manifold. We make the assumption that $M$ can be viewed as the interior of a compact manifold with boundary $\overline{M}$. In practice, for an explicit manifold, any ...
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Proof of homotopic essential simple close curves are isotopic
In the book by Benson Farb and Dan Margalit A primer on mapping class groups, Princeton Mathematical Series 49. Princeton, NJ: Princeton University Press (ISBN 978-0-691-14794-9/hbk; 978-1-400-83904-9/...
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2
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302
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Reference for a topological result
I am reading the short paper due to Erdös and Bollobás "On a Ramsey-Turán type problem", where they obtain a lower bound for the number of edges on an $n$-graph without $K_4$ as a subgraph ...
12
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3
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Road map to learn about $\operatorname{Out}{F_n}$
I'm a last year undergraduate student and I have taken a graduate course in geometric group theory.
I'd like to start reading some more advanced stuff in geometric group theory and in particular about ...
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340
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Relation between trivial tangent bundle $\Leftrightarrow$ certain characteristic classes of tangent bundle vanish [closed]
We know that
framing structure means the trivialization of tangent bundle of manifold $M$.
string structure means the trivialization of Stiefel-Whitney class $w_1$, $w_2$ and half of the first ...
6
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342
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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 (...
6
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1
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142
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Translation length on annular curve graphs
Question about curve stabilisers acting on annular curve graphs, plus context since I'm interested in being fact-checked.
Definition: let the group $G$ act by isometries on a metric space $(X,d)$. ...
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150
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The geometric models of generalised Baumslag-Solitar groups
I am trying to understand a construction in the paper "The large scale geometry of the higher Baumslag-Solitar groups", GAFA, Geometric and functional analysis 11, 1327–1343 (2001), ...
4
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Finding inverses of certain elements in the set of normal invariants of a smooth manifold
Let, $V$ denote the Stiefel manifold of 2-frames $V_{10,2}$ . Consider the the map $S_\text{diff} (V) \xrightarrow{\eta} N_\text{diff} (V) $ in the surgery exact sequence of a smooth manifold. . ...
2
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1
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312
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Realizable geometrically finite hyperbolic 3-manifolds with prescribed conformal boundaries
By Bers' simultaneous uniformization theorem, if $\Gamma$ is a Fuchsian group, then $\operatorname{QC}(\Gamma)\cong \mathcal{T}(S)\times\mathcal{T}(\overline{S})$ where $S = \Bbb H^2/\Gamma$. In ...
3
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What is known about the map $\text{Mod}_g^1 \rightarrow \text{Aut}(F_{2g})$?
Follow up question, edited in on 12/20 below:
Letting $\text{Mod}_g^1$ be the mapping class group of a surface with one boundary component (and basepoint on the boundary) and identify its fundamental ...
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273
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Generating 21-vertex 4-regular plane graphs with 8 faces of degree 3 and 15 faces of degree 4
Is there any way to generate all 4-regular plane graphs with 21 vertices, 8 faces of degree 3, and 15 faces of degree 4? If so, how many of these graphs are there and what are they?
13
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3
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Best known Margulis constants?
A Margulis number for a hyperbolic $n$-manifold $M=\mathbb{H}^n/\Gamma$ is a number $\epsilon>0$ such that for each $x\in\mathbb{H}^n$ the group generated by the elements in $\Gamma$ which move $x$ ...
22
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3
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Equilaterally triangulated surfaces with prescribed boundary
There is a problem in Richard Kenyon's list (Wayback Machine) which I would like to post here, because although I have thought about it from time to time, I have not been able to make the slightest ...
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131
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The James and Morse filtrations of homotopy groups
Denote by $JX$ the James construction on a path connected, well-pointed space $X$. This space is filtered by subspaces $X=J_1X\subseteq\dots\subseteq J_nX\subseteq\dots\subseteq JX$ and is the domain ...
23
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2
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2k
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Uniqueness of compactification of an end of a manifold
Let $M$ be an $n$-dimensional manifold (smooth or topological). I call $\bar{M}$ a compactification of $M$ if it is an $n$-dimensional compact manifold with boundary $\partial \bar{M}$, an $(n-1)$-...
4
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118
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Triangulating piecewise-linear manifolds
Question 1: Is this the mainstream definition of a PL-manifold?
Definition. A PL-manifold is a manifold with an atlas $(\varphi_i)_{i\in I}$ in which all transition maps $\varphi_j\circ\varphi_i^{-1}$ ...
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1
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175
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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 ...
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335
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Why is the Teichmüller space of a surface homeomorphic to a component of the $\mathrm{PSL} (2, \mathbb R)$ character variety of its fundamental group?
$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\PSL{PSL}$
I have a reference request for a proof for the following statement in the title:
The Teichmüller space $T_g$ of the surface $S_g$ of genus ...
1
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1
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169
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A question about Gromov-Lawson construction
We all know that if we consider the connected sum $S^n\# S^n$ of two spheres $S^n$ for $n\geq 3$, then by Gromov-Lawson construction(cf. Gromov, Mikhael; Lawson, H.Blaine Jun., The classification of ...
4
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1
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Induced fiber sequence and Eilenberg–MacLane space in Whitehead tower of $BO$
In Whitehead tower of $BO$, there is a induced fiber sequence:
1.
$$
Z_2 \to B SO \to BO \overset{w_1}{\rightarrow} B Z_2
$$
How does this map $\overset{w_1}{\rightarrow}$ from $BO$ to $B Z_2$?
...
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3
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689
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Jordan curve theorem for cylinders
Hello,
I would like to know if the following result is true:
Let $A,B$ be two embedded circles in $S^2$ which do not intersect and let $C$ be the $\textit{closed}$ region bounded by $A$ and $B$ (...
8
votes
2
answers
551
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Jordan curves in $\mathbb R^n$ and inscribed equilateral triangles
Inscribed square problem wants that we know "Does every Jordan curve admit an inscribed square?"
From my amateur viewpoint it seems that the concept of Jordan curve can be straightforwardly ...
3
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1
answer
450
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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 ...
4
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2
answers
457
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$0$-surgery of slice knots and contractible manifolds
We know that if we attach $4$-dimensional $2$-handle $D^2 \times D^2$ to $S^1 \times S^2$, then we produce a contractible $4$-manifold. In this case, $S^1 \times S^2$ is $0$-surgery on the unknot.
If ...
3
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0
answers
210
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Classifying spaces beyond CW complexes
We know that for a reasonable topological group $G$ (say a compact Lie group) admits a classifying space for $G$-bundles within the category of countable CW complexes. That means, there is a space $BG$...
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0
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141
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Minimal first Pontryagin class $p_1=1$?
From Hirzbuch theorem,
the signature of 4-manifold $\sigma = p_1/3$ with the first Pontryagin class $p_1$.
I know that the $\sigma=p_1/3 =1$ so $p_1=3$ for complex projective space $\mathbb{P}^2$.
Is ...
4
votes
1
answer
142
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Salvetti complex of dihedral Artin group
The Salvetti complex of a RAAG is well-known and it is fairly simple, since each complete graph gives rise to a tori. The case of Artin groups is wilder, since we do not have tori anymore. The ...
39
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1
answer
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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 ...
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1
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113
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Expansion of metric near boundary of 3 dimensional Poincaré-Einstein/hyperbolic manifolds
In Mazzeo-Alexakis, there is a brief discussion that if $(M^3,g) \sim \mathbb{H}^3/\Gamma$ (for $\Gamma$ convex cocompact), then the metric can be expanded near the topological boundary as
$$g = \frac{...