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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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
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2
answers
185
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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 $M$ ...
2
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1
answer
325
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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
156
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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 ...
9
votes
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363
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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 ...
2
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1
answer
251
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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 ...
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0
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428
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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 ...
9
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1
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336
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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 ...
6
votes
0
answers
191
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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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993
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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 ...
2
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1
answer
139
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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 ...
4
votes
1
answer
188
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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 ...
1
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1
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216
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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 ...
3
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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 ...
4
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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 ...
3
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1
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290
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A detail in Brown's proof of the generalized Schoenflies theorem
Consider a homeomorphic embedding $h:S^{n-1}\times [0,1]\rightarrow S^n$ and denote
$$S^{n-1}_t=h(S^{n-1}\times \{t\}).$$
The generalized Schoenflies theorem states the closure of each connected ...
8
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1
answer
205
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Integral homology classes of which no multiples admit embedded representatives with trivial normal bundle
Let $M$ be a closed smooth manifold of dimension $n$ and $z\in H_l(M,\mathbb{Z})$ a $k$-dimensional integral homology class. Theorem II.4 of Thom's classical 1954 paper states that for $l< n/2$ or $...
6
votes
1
answer
216
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How small need a perturbation be to not change the diffeomorphism type of a variety?
Let $f,g \in \mathbb{R}[x_0,\dots,x_k]$ be homogeneous polynomials and $X:=Z(f) \subset \mathbb{RP}^k$ be the projective variety defined by $f$.
Assume that $X$ is smooth and has codimension $1$.
Then ...
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159
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Identifying a curve on a closed surface of genus 4
The notation is the one used in the attached picture.
Take a closed, orientable surface $\Sigma_4$ of genus $4$, obtained as the identification space of a polygon with $16$ sides in the usual way. The ...
4
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0
answers
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Kernel of the geometric intersection form
Let $\Sigma$ be a closed surface and $\mathcal C$ the set of all free homotopy classes of closed (may be nonsimple) curves in $\Sigma$. Consider the geometric intersection form $i$ on $\mathbb Z\...
0
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1
answer
162
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Mappings of reducible 3 manifolds with boundary
In section 3 of his paper "Mappings of reducible 3 manifolds" McCullough, proves that every self-homeomorphism of a reducible 3 manifold can up to isotopy be written as a composition of ...
5
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1
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226
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Cancellation of elements in the Gromov boundary of a free group
Let $A$ be a finite set of free generators and their inverses and $F$ the free group generated by elements in $A$ (some call $A$ the alphabet of $F$). For each $g\in F$, use $\vert\,g\,\vert$ to ...
3
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0
answers
287
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"Maehara-style" proof of Jordan-Schoenflies theorem?
The highest upvoted answer to this old question Nice proof of the Jordan curve theorem? is a proof by Ryuji Maehara. I personally really liked/appreciated that Maehara's proof is
A) a fairly ...
2
votes
0
answers
134
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Characterization of the homotopy type of the $C^\infty$ topology
One of the subtle aspects of geometric topology is the interaction of function space topologies and homotopy theory. There are many reasonable topologies to put on the space of smooth embeddings $\...
10
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2
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273
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Reference request - Fibrations between spaces of embeddings
This is a cross-post of this question from MSE.
Given topological manifolds $M$ and $N$ of the same dimension, let $\operatorname{Emb}(M,N)$ denote the subspace of $\operatorname{Map}(M,N)$ ...
4
votes
1
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392
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4-manifold $M$ with intersection form of Leech lattice
Do we know any 4-manifold $M$ with intersection form of rank-24 Leech lattice?
Is $M$ smooth? Differentiable to which $n$-order?
Can $M$ be obtained by any surgery on 3 copies of E8 4-manifolds with ...
5
votes
1
answer
253
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0-surgery on a fibered hyperbolic ribbon knot
Does there exist a fibering hyperbolic ribbon knot such that the 0 surgery is exceptional? If so does there exist such an example where the result of 0-surgery is Seifert fibered?
I tried looking at ...
6
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1
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318
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Does $\overline{\mathbb{C}P^2 \setminus B^4}$ embed (smoothly/topologically) in $\mathbb R^5$?
Question: Does $\overline{\mathbb{CP}^2 \setminus B^4}$ (that is the closure of complex projective plane minus a 4-ball) embed (smoothly/topologically/piecewise linearly) in $\mathbb R^5$?
Background: ...
2
votes
1
answer
125
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Space of the trivial long knot in the thickened surface
Let $F$ be a compact oriented surface and $x_0\in F$ a basepoint. Consider the set $\mathcal E=Emb_0(I,F\times I)$ of embeddings $\sigma\colon I\hookrightarrow F\times I$, $\sigma(\partial I)=\{x_0\}\...
6
votes
0
answers
132
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S¹ action on a manifold which generates "non-torsion" loop in diffeomorphism group
I am interested in $S^1$-actions on smooth, closed, and oriented manifolds $M$. I suppose that the action has a fixed point (I also suppose $M$ is connected). Let $\operatorname{Diff}(M)$ denote the ...
5
votes
3
answers
204
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First usage of the terms pseudo-isotopy and concordance in manifold theory
I am hoping I can use the collective knowledge of the forum to piece together some history. I'm wondering where the terms pseudo-isotopy and concordance originated, in their modern forms as used in ...
0
votes
2
answers
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 ...
7
votes
2
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633
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A generic metric on $X\cup\mathbb Z$
$\newcommand\abs[1]{\lvert#1\rvert}$Let $(X,d_X)$ be a countable metric space such that $X\cap\mathbb Z=\{0\}$.
Problem. Is there a metric $d$ on the union $Y=X\cup\mathbb Z$ such that
$d(x,y)=d_X(x,...
3
votes
2
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446
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Finding a hyperbolic metric with geodesic boundary on a given Riemann surface
Let $X$ be a Riemann surface with analytic boundary. Assume that $X$ has negative Euler characteristic. Then there exists a conformal hyperbolic metric $X$ such that $\partial X$ consists of geodesics ...
6
votes
1
answer
395
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Abstract simplicial complexes - Reference for an elementary definition of mapping degree for simplicial maps?
I am interested to use the mapping degree for simplicial maps between (oriented) abstract simplicial complexes. What I mean by "elementary": My preference would be to use a definition of ...
4
votes
2
answers
335
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Knot theory in handlebodies of arbitrary genus
It is well known that not all graphs embed on the plane (e.g. the graph $K_{3,3}$). However, every finite graph embeds into a surface of some genus. One can think of this procedure as starting with a ...
2
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0
answers
229
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A Question about an article by Birman, Series
Birman and Series in their article GEODESICS WITH BOUNDED INTERSECTION NUMBER ON
SURFACES ARE SPARSELY DISTRIBUTED proved that the set of points on a hyperbolic surface (possibly with boundary) ...
4
votes
1
answer
208
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Euler class of vertical tangent bundle of the surface bundle over circle
Suppose $\Sigma$ is an oriented genus $g>1$ surface and $h:\Sigma\to \Sigma$ is a diffeomorphism preserving a point $p$. Let $M$ be the surface bundle over $S^1$ obtained by gluing $\Sigma\times I$ ...
8
votes
1
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420
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dichotomy in hyperbolic groups
Suppose $G$ is a word hyperbolic group i.e. every geodesic triangle in a cayley graph with respect to a finite generating set of $G$ is $\delta$-thin, for some $\delta>0$. There are various ...
1
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1
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165
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Bring's curve $\sum_{i=1}^5 x_i^k = 0$ for $k = 1,2,3$ and an analogue $\sum_{i=1}^6 y_i^k = 0$ for $k = 1,2,4,7$
Bring's curve or Bring's surface with genus 4 and $5!=120$ automorphisms can be given by the homogeneous equations,
$$x_1+x_2+x_3+x_4+x_5 = x_1^2+x_2^2+x_3^2+x_4^2+x_5^2 = \\x_1^3+x_2^3+x_3^3+x_4^3+...
2
votes
0
answers
141
views
Can distinct meridians commute in a knot group?
Suppose I have a knot $K$ in $S^3$. Given a diagram $D$ of $K$ I get the Wirtinger presentation $\langle x_1, \dots, x_a \mid r_1, \dots, r_c\rangle$ of its knot group $\pi(K) = \pi_1(S^3 \setminus K)$...
1
vote
1
answer
95
views
PL embedding of 3-manifold PL triangulation into 5-manifold PL triangulation
Are there any PL triangulations of closed orientable 3-manifolds that do not PL-embed into some PL triangulation of a closed 5-manifold?
3
votes
0
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178
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What's the meaning of this relation between volumes of $n$-balls and umbral calculus?
The volume of an $n$-ball of radius $1$ is $$V_{n}={\frac {\pi ^{n/2}}{\Gamma {\bigl (}{\tfrac {n}{2}}+1{\bigr )}}}.$$
The functional equation of Riemann zeta function is $${\displaystyle \pi ^{-{s \...
6
votes
1
answer
463
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A characterization of metric spaces, isometric to subspaces of Euclidean spaces
I am looking for the reference to the following (surely known) characterization of metric spaces that embed into $\mathbb R^n$:
Theorem. Let $n$ be positive integer number. A metric space $X$ is ...
4
votes
0
answers
151
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Examples of Lattices of Sp(n,1)
$Sp(n,1)$ is the isometry group of $n$-dimensional quaternionic hyperbolic space. It is written in literature that the group is an example of a hyperbolic groups. Can you suggest me any reference ...
2
votes
0
answers
94
views
Lifting homology classes to the unit tangent bundle, a la Johnson
Let $M$ be a oriented smooth closed 2-manifold, and let $\gamma$ be an oriented smooth simple closed curve in $M$.
In Spin structures and quadratic forms on surfaces, Johnson definines a standard way ...
8
votes
1
answer
303
views
The growth rate of a commutator set in a non-elementary group
Let $G$ be a non-elementary group generated by a finite set $S$. Here, a group is called non-elementary if it is not virtually abelian. Denote $S^{\le n}:=\{g\in G: |g|_S\le n\}$ for any $n\in \mathbb ...
3
votes
1
answer
139
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Cohomology of cocompact lattices in hyperbolic spaces
I have a (maybe too naive) hope that cocompact torsion-free arithmetic lattices in hyperbolic spaces $X \neq \mathbb{H}_\mathbb{R}^2$ are uniquely determined by their cohomology with coefficients in $\...
10
votes
3
answers
610
views
Doubles of 2-handlebodies
Let $X$ denote a $4$-manifold with boundary obtained by adding $k_1$ $1$-handles to $B^4$ and $k_2$ many $2$-handles to the resulting manifold i.e. $X$ is an arbitrary $4$-dimensional $2$-handlebody. ...
1
vote
1
answer
148
views
Tiling the hyperbolic plane by non-regular quadrilaterals
We add a bit to Which polygons tessellate the hyperbolic plane?.
Question: Are there hyperbolic quadrilaterals with all angles different (not necessarily irrational fractions of π) that tile the ...