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).

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Convex surfaces with transverse boundary (contact geometry)

Suppose I have a compact surface $\Sigma$ in a contact 3-manifold, where the boundary $\partial\Sigma$ is transverse to the contact structure. Am I able to perturb $\Sigma$ rel boundary so that it is ...
8
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63 views

Find a 4-manifold bounding a 3-torus with any abelian representation in SL_2

Fix $x,y,z\in \mathbb{C}^*$ and let $M=S^1\times S^1\times S^1$ with $\rho:\pi_1(M)\to \operatorname{SL}_2(\mathbb{C})$ mapping the three generators to diagonal matrices with entries $(x,x^{-1})$, $(y,...
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1answer
134 views

Cobordism monopole Floer homology

From the famous book: Monopole and three manifold, Kronheimer and Mrowka(https://www.maths.ed.ac.uk/~v1ranick/papers/kronmrowka.pdf). It is known that: Let $Y$ be a closed oriented $3$ manifold, ...
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1answer
143 views

Computing $\pi_2$ of the complement of a 2-knot or spaces with aspherical splittings?

As far as I know, it is an open question if the complement of a ribbon disk $D^2 \subset B^4$ is aspherical. In reading "Some remarks on a problem of J.H.C Whitehead" by Howie, it is noted that there ...
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136 views

Cobordism theory of some weird space

Let $G=SU(3)$ and $N=SO(3)$, then $G/N= SU(3)/SO(3)$ = a 5-dimensional Wu manifold $W$. The $W$ is a homogeneous space (also a quotient space), but not a group. Previously, I am aware of the ...
3
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1answer
154 views

Books on foliations

I am looking for resources (books, notes, lecture video, etc. anything will do although printed material in English is preferable) on foliations which satisfy some or all of the following constraints. ...
6
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1answer
143 views

Mean curvature flow and knot theory

I am wondering if the mean curvature flow of one-dimensional submanifolds of $\mathbb{R}^3$ is understood well enough to give some perspective on (and hopefully a proof of) something like the Fary-...
5
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1answer
244 views

Whitney sum formula for topological Pontryagin classes

Is there a Whitney sum formula for topological rational Pontryagin classes? I thought the answer is yes, but now I cannot find a reference. Is it even true? The PL case would also be of interest.
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2answers
76 views

Transverse invariant measures to vector fields

Given a smooth, non-vanishing vector field on a compact manifold, when does the 1-dimensional foliation given by its integral curves admit a transverse invariant measure? I've seen examples of higher-...
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28answers
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Proofs where higher dimension or cardinality actually enabled much simpler proof?

I am very interested in proofs that become shorter and simpler by going to higher dimension in $\mathbb R^n$, or higher cardinality. By "higher" I mean that the proof is using higher dimension or ...
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157 views

Image of the mapping class group of surfaces into automorphism group?

Let $S_{g,p}^n$ be a compact oriented surface of genus $g$ with $p$ punctures and $n$ boundary components, and $\operatorname{Mod}(S)$ and $\operatorname{PMod}(S)$ be the mapping class group and the ...
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1answer
98 views

Maximally symmetric hyperbolic 3-manifolds with finite volume

In physics, standard cosmology is build with simple maximally symmetric 3-manifolds (spacelike time-slices of constant curvature, e.g. $S^3$ or less popular the hyperbolic space $H^3$). Since $S^3$ ...
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108 views

When a free action gives rise to a $G$-principal bundle

When a free action gives rise to a $G$-principal bundle Let a (topological) group $G$ act freely on a (topological) space $X$. Assume that $G \backslash X$ is Hausdorff. (equivalently the image of ...
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59 views

Implicit function theorem for PL maps

Let $K$ be a PL triangulation of a closed manifold and $f: K\to\mathbb R^k$ be a PL map. Equivalently, $f$ is a map that becomes linear on every simplex after subdividing. Suppose $v$ is a vertex in ...
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101 views

Why does the inverse Alexander polynomial appear in the MMR conjecture?

In an attempt to better understand why the inverse Alexander polynomial appears in the MMR conjecture, I was reading the paper [1] of Bar-Natan and Garoufalidis giving their proof of the conjecture ...
6
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1answer
383 views

Solutions of PDE under changing topology

Let suppose we have a PDE on a manifold. I'm interested in the following question. How does the space of solutions of this PDE change when the topology of the manifold changes? For example in 2D we ...
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145 views

Convergence of Fuchsian groups and existence of suitable homeomorphisms

Let $(\Gamma_n)_n$ ($\subset PSL(2,\mathbb{R})$) be a sequence of discrete groups, if we say that $(\Gamma_n)_n$ converges to a group $\Gamma$ this means that there exist isomorphisms $\tau_n:\Gamma\...
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203 views

Retracting off a compact set

Let $K$ be a compact set in $\mathbb{R}^n$ and let $U$ be a bounded open set that contains $K$. You may assume both are connected. Can we always find an open $V$ such that $K\subset V\subset\...
4
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1answer
203 views

Mirzakhani's hyperbolic method generalized to moduli space of stable maps

I've been learning about Mirzakhani's use of hyperbolic geometry to compute Weil-Petersson volumes of moduli space of curves, and the application to proving Virasoro constraints for a point. Why have ...
4
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2answers
191 views

Is it possible to connect every compact set?

Let $X$ be a "nice" space: metrizable, connected, locally path connected perhaps. Let $K\subset X$ be a compact set. Is there a always a compact connected $L\subset X$ such that $K\subset L$? This ...
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0answers
46 views

Spaces that are comparable with their compacts

This is an outgrowth of this question. For a (metrizable) space $X$ consider the following increasingly strong properties: (i) For every compact $K\subset X$ there is a map $f:X\to X$ such that $K\...
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50 views

Spherical space-form as the boundary of an Euclidean ball

Let $M^n$ be a smooth compact manifold such that the boundary $\partial M$ is diffeomorphic to a spherical space-form $S^{n-1}/\Gamma$, where $\Gamma \subset O(n)$ is a finite subgroup acting freely ...
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166 views

Retracting to a bigger compact

Consider the topological spaces $X$ with the following property: For every compact $K\subseteq X$ there is a compact set $L$ such that $K\subseteq L\subseteq X$ and $L$ is a retract of $X$. Let ...
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1answer
193 views

Powers of the Euler class, torsion free subgroup of Homeo($S^1$)

For any subgroup $G$ of $\text{Homeo}(S^1)$, we have the Euler class $\chi$ in the group cohomology $H^2(G;\mathbb{Z})$. One can think of this class as the pullback of the generator of $H^2(\mathrm{B}\...
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1answer
112 views

Unknotting algorithm in higher dimensions?

Suppose we are given a 2-knot (say by a movie). Is there an algorithm to tell if it is unknotted ? I suppose that it could matter if I say "topologically" or "smoothly" here since those could be ...
3
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2answers
120 views

Homology torsion in the double branched cover of a tangle?

Let $T$ be a locally unknotted $2$-tangle in $B^3$ and $\Sigma(T)$ be its double branched cover. Can $H_1(\Sigma(T))$ have a non-trivial torsion? (Obviously, not for rational tangles.)
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147 views

cohomology classes of complex submanifolds

I was wondering if there were restrictions in what the cohomology classes corresponding to complex submanifolds of a complex manifold could be. For example, say $T^4$ is regarded as a complex ...
8
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1answer
227 views

Mostow Rigidity Theorem and reconstruction from fundamental group

The Mostow Rigidity Theorem is phrased in terms of a relationship between isometries and isomorphisms of fundamental groups, which raises an obvious question. Given the fundamental group of a complete ...
6
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1answer
181 views

An orientable surface that cannot be embedded into $\Bbb R^3$? [duplicate]

I previously asked this question on MSE, without success. By Whitney's embedding theorem, every 2-dimensional manifold (aka. a surface) can be embedded into $\Bbb R^4$. Now, Wikipedia states in this ...
4
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1answer
89 views

Is a cosparse action on a CAT(0) cube complex an essential action?

Let $X$ be a CAT(0) cube complex. (From Sageev and Wise's Cores for Quasiconvex actions) A group $G$ acts cosparsely on a CAT(0)-cube complex $X$ if there exists a compact space $K$ and finitely many ...
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29 views

relative transformation of coordinates on a flat surface [closed]

I have a few coordinates that form a triangle. I have a relative point to that triangle. if the coordinates get translated to a new triangle I want to calculate the new relative point. How do I do ...
4
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2answers
121 views

When existence of loxodromic, WPD elements implies an action is acylindrical

Definitions Say that $(X,d)$ is a $\delta$-hyperbolic space and that $G$ is a finitely generated group acting on $X$ by isometries. Recall that an action of $G$ on $X$ is called acylindrical if the ...
5
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1answer
173 views

Quasi-isometric embedding of graphs in non-compact riemannian surfaces

Given a complete riemannian surface $(S,m)$, where $S$ is homeomorphic to $\mathbb{R}^2$, I would like to find a weighted graph $G$ (which means a graph with real non-negative weights on the edges), ...
2
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0answers
67 views

Riemann mapping theorem with boundaries and corners

I was reading this paper by Hollands and Yazadjiev, where on page 760, they claim that since $\hat{M}$ is an orientable simply connected analytic $2$-manifold with boundaries and corners, we may map ...
2
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1answer
73 views

Upper bounds on the genus of the surface produced by Seifert's algorithm

Let $K$ be a knot with genus $g$. Seifert's algorithm produces a surface of genus $k$ whose boundary is $K$. In general $k$ may be larger than $g$, but are there any bounds on how much larger it can ...
8
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1answer
186 views

Is a gluing of homeomorphic Mazur manifolds diffeomorphic to $S^4$?

A recent paper proves the existence of homeomorphic but not diffeomorphic Mazur manifolds (see also examples of exotic pairs of contractible Stein manifolds). Let's call them $M_1$ and $M_2$. If we ...
11
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2answers
614 views

Two Dehn fillings yielding a lens space?

Let $M$ be an oriented $3$-manifold with $\partial M$ torus. Suppose that two different Dehn fillings $M(r)$ and $M(r')$ are (oriented) homeomorphic to a lens space $L(p,q).$ Does that imply that $M$ ...
6
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0answers
79 views

Existence of codimension 1 topological foliations

One of the many famous theorems proven by William Thurston is that a closed connected smooth manifold $M$ admits a codimension 1 smooth foliation if and only if $\chi(M)=0$: W.P. Thurston, Existence ...
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0answers
111 views

Find a manifold with boundary of a geodesic ball being a torus [closed]

I would like to find the answers to the following questions: a. Find a complete $3$-dimensional Riemannian manifold $M$ and a point $p\in M$, such that the boundary of the open geodesic ball $B(p,1)...
7
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1answer
257 views

Map which is null-homotopic on compacts

This is the missing ingredient towards answering my previous question. Let $M$ and $N$ be path connected locally compact, locally contractible metric spaces (you may assume that they are manifolds). ...
2
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1answer
120 views

vector field on the torus [closed]

Is the following statement true? Let $T$ be diffeomorphic to the solid torus. Let $v$ be a vector field such that $v$ and $curl(v)$ are both tangent to $\partial T$ everywhere and $|v|$ is constant ...
9
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2answers
426 views

Is limit of null-homotopic maps null-homotopic?

The question is motivated by my failed comment to this one. Let $M$ and $N$ be path connected locally compact, locally contractible metric spaces (you may assume that they are manifolds). Let $\...
3
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1answer
101 views

Rigid Brunnian links for $n \geq 4$

Brunnian links consist of $n$ linked un-knot components such that the cutting of any component leaves all components unconnected. The most famous example is the three-component Borromean rings (or ...
7
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1answer
267 views

Non-density of continuous functions to interior in set of all continuous functions

Let $M$ be an $m$-dimensional manifold and $N$ be an $n$-dimensional manifold. Suppose also that the topology on $N$ can be described by a metric. Thus, the set $C(M,N)$ can be endowed with the ...
6
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0answers
80 views

References for the connectivity of complements of smooth submanifolds

Suppose $M$ is a smooth $m$-manifold and $Z$ a codimension-$d$ closed submanifold. Neither $M$ nor $Z$ has a boundary and neither is assumed to be compact. I believe that the inclusion $M \setminus Z \...
6
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1answer
181 views

Explicit fundamental domain for the action of $\operatorname{O}(n,1)(\mathbb{Z})$ on $\operatorname{O}(n,1)(\mathbb{R})$

Minkowski computed explicit fundamental domains for the action of $\operatorname{SL}_n(\mathbb{Z})$ on $\operatorname{SL}_n(\mathbb{R})/\operatorname{SO}_n(\mathbb{R})$ for each $n \leq 6$. In the ...
10
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0answers
150 views

Embedding 2-complexes null homotopically into 2-complexes

Whitehead's conjecture states that if $L$ is an aspherical 2-complex and $K$ is a subcomplex of $L$, then $K$ is also aspherical. It is known by work of Howie and Luft that if the Whitehead ...
6
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0answers
157 views

The last step in Ahlfors' and Sario's proof of the triangulability of surfaces

In their book, Riemann Surfaces, Ahlfors and Sario write, at the bottom of pg. 109 to the top of pg. 110, "Consider the sequences $\{V_n\}$ and $\{W_n\}$ introduced by Lemma 46B. We will show that ...
2
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1answer
201 views

Moduli, Teichmüller spaces and mapping class group of a sphere with four punctures

In the complex analytic setting, it is easy to see that the moduli space of a sphere with four punctures is $\mathcal{M}=\mathbb{CP}^1 / { 0,1,\infty }$, since I can use a Moebius transformation to ...
4
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0answers
87 views

Any cobordism invariant made of “characteristic classes”, on unorientable manifolds, must be a mod 2 class?

For any cobordism invariant (or simply bordism invariant) quantity $\omega$ that satisfy the conditions: $\omega$ can be fully decomposed from the cup product of characteristic classes (such as ...

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