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Questions tagged [geometric-topology]

The corpus of tools and results that arose from studying manifold theory using non-algebraic techniques, that is, as opposed to (algebraic-topology). The focus of the field tends to be on special objects/manifolds/complexes and the topological characterisation and classification thereof. A key example is the literature surrounding the Poincaré Conjecture.

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Question of deck transformation on double cover $\tilde{M}$ of non-orientable manifold $M$.

Suppose $M$ is a non-orientable connected manifold, $\tilde{M}$ is its orientable double cover. $\varphi$ is deck transformation of $\tilde{M}$ with $\varphi \not= id$. $\varphi$ is deck ...
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Integral homology group of a 3-torus cut out a donut

I know that the integral homology group of the manifold $M$ is given by $$ H_j(M,\mathbb{Z}) $$ I also have tried that $H_j(T^3,\mathbb{Z})$ is given by $$ H_0(T^3,\mathbb{Z})=\mathbb{Z}, $$ $$ H_1(...
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Proof of Kunneth theorem [on hold]

What are different ways to prove Kunneth theorem relating singular homology of product space $X * Y$ in terms of homology of $X$ and $Y$? or reference?I know some ways: use cell homology for cell ...
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Book recommendations “Computing Volume of some sets on hyperbolic surface with boundary”

I would like to learn how to compute the critical exponential of Fuchsian group and volume of some subset on the surface and its unit tangent bundle. I found that there is a book related to this, that ...
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What the boundary of the disc in Poincaré disk model exactly is by a geometric point of view?

Poincaré disk model is defined in a open disc, and the boundary of the disc represent something infinitely distant. But what the "something" exactly is? How to topologically or geometrically extend ...
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What is an orbifold with corners

Can one have a formal definition of orbifold with corners? note that it is not parallel to the definition of manifold with corners, as a manifold with boundary is already an orbifold(not with boundry)....
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Homology group of a triangulation(simplicial complex) space

Let $X$ be a connected oriented triangulation(polyhedron) space, i.e., homeomorphic to a geometric realization of an oriented simplicial complex $S$ with dimension $n$, and the boundary $\partial S$ ...
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Open tubular neighborhood of $2$-disk in $4$-ball and its exterior

I try to be familiar with the notion of smoothly sliceness of knots and disks. A $2$-disk $D$ is said to be a slice disk if it a smoothly and properly embedded in $B^4$. The boundary of $D$ in $S^3$ ...
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Tubular Neighborhood of Disk and Circle

I try to understand the notion of tubular neighborhood, but I cannot. Is there anyone who may explain for example (up to homeomorphism) what is the tubular neighborhood of circle and disk? For the ...
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founding equivalence classes of $D^n/S^{n-1}$ and this hoemomorphism with $S^n$ [duplicate]

Collapsing the unit sphere of the unit ball $S^{n-1} \subset D^n \subset \mathbb{R}^n$ to a point. How can I show that it ($D^n/S^{n-1}$) is homeomorphic to the sphere of one dimension higher ($S^n$)? ...
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Attaching map os two squares onto a sphere in isomorph to the union of sphere and a ellipsoid.

Consider the topological spaces $X = \partial ([0, 1]^3) \subset \mathbb{R}^3$ (the boundary of the unity cube in $\mathbb{R}^3$) and Y the unit sphere $\mathbb{S}^2 \subset \mathbb{R}^3$. Let $A \...
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I wondering how can I build a homeomorphism between two squares and a ellipsoid.

For example, if we have X equal the union on the squares $[0,1] \mbox{x}[0,1]\mbox{x}{0}$ and $[0,1] \mbox{x}[0,1]\mbox{x}{1}$ and $Y= (x^2)/4 + (y^2)/4 + (z^2)/1 = 1$. How can I construct a ...
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What do we know about Embedding problem of Covering spaces?

My question might be broad. Let $E$ be the covering space of $B$, what do we know about the embedding of $E$ (into some space)? I am looking for some general results or references. EDIT: After Lee ...
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Relationship between Betti number and Genus

I recently found out that Mathworld gave the same definition for both Betti number and Genus as: "the largest number of nonintersecting simple closed curves that can be drawn on the surface without ...
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Equivalent definition $\text{Cone}(K)$

Let $K\subseteq \mathbb{R}^{n} \times \{ 0 \}\subseteq \mathbb{R}^{n+1} $ and $v=(0,0,\cdots,1) \in \mathbb{R}^{n+1}$. For every $x \in K$, let $$Lx=\left \{ tv+(1-t)x \; | \ t \in [0,1] \right \...
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Why does a CW-complex consist of *open* cells?

Wikipedia states A CW complex is a Hausdorff space X together with a partition of X into open cells (of perhaps varying dimension) that satisfies two additional properties: For each n-...
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Complement of an unknotted oriented surface in $S^4$

Let $H$ be a genus $g$ handlebody embedded in $S^4$ and let $X = S^4 - N(\partial H)$ where $N(\partial H)$ is an open tubular neighborhood of the boundary of $H$. What is $X$? In the case where $g=...
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Using grid diagram to compute the Alexander polynomial

I have been reading the book 'Grid Homology for Knots and Links' (see https://web.math.princeton.edu/~petero/GridHomologyBook.pdf) - in Section 3.3 it provided a way to compute the Alexander ...
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Any compact orientable surface is a branched cover of a torus

Given a torus $T$, assume all the branch points are of index $2$, then by Riemann-Hurwitz theorem, the number of branch points is $2g-2$. Select $n:=2g-2$ points $\{x_i\}$ in $T$. Does that mean we ...
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Definition of an arrangement of the plane

... for a finite set $H$ of lines in the plane, the arrengement of $H$ is a partition of the plane into relatively open convex subsets, the faces of the arrangement. In this particular case,(*) the ...
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Can a connected compact subset in $\mathbb{R} ^3$ always be enclosed in a closed ball of minimum radius?

Given any connected compact set $A \subset \mathbb{R}^3$, does there exist a closed ball $B_r (x)$ of radius $r$ around $x\in A$ such that $r$ is minimum and $A\subset B_r (x)$ under usual topology ? ...
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Finding generators of a group from its action on a topological space

Summary I believe I've written a geometric group theory flavoured proof with a mistake in it, but I'm struggling to see why it might be wrong. I haven't found a counter example, but it also feels too ...
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Homology condition - bounding a disk in a handlebody?

Suppose that $\gamma_1,...,\gamma_n$ are a set of disjoint simple closed curves on a closed orientable surface $\Sigma$ that all bound disks in some handlebody $H$ with $\partial H = \Sigma$. Let $\...
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Complement of tubular neighborhood

Let $M$ be a closed, connected, orientable and embedded surface inside the unit 3-sphere $\mathbb{S}^3$ and consider a small tubular neighborhood $U$ of $M$: $$U = \{ x \in \mathbb{S}^3 : d(x, M) \...
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Show that all triangulations of a compact surface are equivalent

I'm having trouble with solving the following question: Let $T_1, T_2$ be two finite triangulations of a compact surface. Show that if $E_{T_1}\cap E_{T_2}$ is a finite set of points, where $E_{T_i}$ ...
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Manifold with a good exhaustion [duplicate]

Let $M$ a smooth manifold such that $M=\bigcup_{i=1}^{\infty} U_i$ for $U_i \subset U_{i+1}$, where $U_i$ is an open set of $M$ which is diffeomorphic to $\mathbb R^n$. Can we prove that $M$ is ...
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Why is this 3-manifold irreducible?

Let $M = \mathbb{R}\mathbb{P}^2 \times \mathbb{S}^1$. It is a prime 3-manifold, but it cannot be reducible, since the only reducible prime connected 3-manifolds are the $\mathbb{S}^2$-bundles over $\...
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Are weaves stable in higher dimensions?

A textile weave is stable object in three dimensions. A table cloth is an example. Here is an example of a weave from wikimedia: 1 - Are weaves (woven from one-dimensional threads) stable in higher ...
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Let $K$ be a simplicial complex (that need not be finite). Prove $|K|$ is Hausdorff.

As the title makes clear, I'm trying to solve a question which asks me to show the topological realisation of a simplicial complex is Hausdorff. The question reminds me that a subset of $|K|$ is open ...
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Sources for exploring Nil Geometry

I am first year graduate student and following the Thurston's book on 3-dimensional geometry and topology. Among all the 8 geometries of 3-manifold (from Thruston's classification), I heard from some ...
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Why a special ball in $S^3$ is unique?

I'm studying : An introduction to knot theory(by: W.B. Raymond Lickorish). To prove composition of two oriented knots is unique, Lickorish has written:"regarded $K_1$ and $K_2$ as being in distinct ...
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Is it possible to arrange the routes and the bus stops so that if one route is closed, it is still possible to get from any one stop to any other [closed]

A certain City has 10bus routes. Is it possible to arrange the routes and the bus stops so that if one route is closed, it is still possible to get from any one stop to any other, but if any two ...
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Are any two $n$-balls in $\mathbb{R}^n$ isotopic?

I have two related questions in the topological category. Let $B^n$ denote the closed unit $n$-ball, and let $S^{n-1}$ denote its boundary sphere. Is it true that there are exactly two isotopy ...
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Constructing 6-fold cover of $S^1 \vee S^1$ with deck transformation group $\cong S_6$

So i'm thinking that this will be a cover space of maximum possible symmetry. Will a "necklace" of 6 circles work? Any tips appreciated
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Proving that a closed, non-orientable surface cannot be embedded in $\mathbb{R}^{3}$ using Euler characteristic

So, though I am aware that it is conventional to use Alexander's duality theorem to prove that no closed, non-orientable surface $S$ can be embedded into $\mathbb{R}^{3}$; I was hoping perhaps to find ...
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Boundary of a Möbius loop in the fundamental polygon

Although it says here that, in the topology on the unit square representing the Möbius loop, the boundary consists of those points of the form $(a,a)$, I cannot find any justification as to why this ...
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Can a compact surface-with-boundary admit arbitrarily high genus subsurfaces-with-boundary?

More precisely, does there exist a compact surface-with-boundary $\Sigma$ with the following property? For every $g\geq 0$, there exists a subsurface-with-boundary $\Sigma'\subseteq\Sigma$, where the ...
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Proof of the inscribed rhombus conjecture

I ask to verify whether one of the aspects of a proof I have proposed regarding the inscribed rhombus conjecture (i.e. that every jordan curve $J$ inscribes a rhombus) is true. This proof is based on ...
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Covering space of a compact connected surface without boundary is a compact surface without boundary

Let $p: X' \rightarrow X$ be an n-sheeted cover of $X$. I proved that $X$ being compact implies that $X'$ is compact in the standard way. I started with an open cover of $X'$, projected it under $p$, ...
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1answer
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Surjective mapping of an orientable surface into $E^5$

When looking at this video: https://www.youtube.com/watch?v=AmgkSdhK4K8 I saw that the proof presented here relies upon the fact that, when a Mobius loop is mapped onto $R^3$ in such a way that the ...
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The self intersection of a Möbius loop for the inscribed rectangle theorem

https://www.youtube.com/watch?v=AmgkSdhK4K8 So, when seeing this proof of the inscribed rectangle problem, I noticed that the result used to prove the inscribed rectangle theorem was as follows: If a ...
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1answer
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The fundamental groups of 3-dimensional spherical space forms

Let $S^3/\Gamma_i\,(i=1,2)$ be a $3$-dimensional spherical space form, where $\Gamma_i \subset SO(4)$ is a finite subgroup acting freely on $S^3$. If $S^3/\Gamma_1$ is homotopy equivalent to $S^3/\...
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On the connected sum of a surface with a torus

I am studying the classification of Surfaces, and run into the notion of connected sum. We define it in terms of triangulations. I want to show the following. Let $S$ be a triangulated surface. I want ...
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Hyperbolic 3-manifolds of finite volume as link complements

This is an improved version of this question (sorry if I wasn't so clear there and sorry if this is well-known and I didn't find the reference). Let $N$ be a hyperbolic 3-manifold of finite volume ...
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1answer
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A compact set in $\Bbb R^n$ with smooth boundary is a manifold?

Can a compact set $\Omega \subset \Bbb R^n$ with smooth boundary be considered as a smooth manifold with boundary?(Smooth boundary probably means $\partial\Omega$ is a $n-1$-smooth manifold?) I think ...
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Vogel's Algorithm - Why can we read braid words from nested coherent Seifert surfaces?

I was reading this paper from R. Goldstein-Rose: http://math.uchicago.edu/~may/REU2017/REUPapers/GoldsteinRose.pdf In Figure 12 it was mentioned that if a Seifert surface is coherent and nested, then ...
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Understanding a comment by Thurston

In page 359 (right after Theorem 2.3) of the following paper Thurston, William P., Three dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Am. Math. Soc., New Ser. 6, 357-379 (...
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Is every hyperbolic 3-manifold of finite volume a link complement in some closed 3-manifold?

The question says all I need to know, but I will try to clarify it a little more. Let $M$ be a compact 3-manifold with nonempty torus boundary such that ${\rm int}(M)$ admits a complete hyperbolic ...
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1answer
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The effect of attaching the Möbius strip to the torus

We can attach a Möbius strip $M$ to a torus by using a homeomorphism between its boundary circle and $S^1 \times \{x_0\}$. Then the claim is that the inclusion map will send the generator of $H_1(S^1 \...
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Space homeomorphic to Mobius Strip? [closed]

I have a space $G$ of distinct pairs of points that are not ordered on $S^1$ with metric: $D = min(d(a,b) + d(a', b') + d(a, b') + d(a', b))$ Is $G$ homeomorphic to a mobius strip? This has been ...