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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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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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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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Special case for Giraud's Theorem

I was wondering how Giraud's Theorem would work for spherical polygons. Do we know a proof?
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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?

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 ...
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Gluing $2n$-gons

I am confused about this how I would visualize/prove this. Given that we have a regular $2n$-gon that preserves orientation (the surface is orientable because the $2n$-gon's opposite sides are glued ...
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Showing $χ$ for a three manifold?

How would I show that the Euler number for $ (S^1 × S^1 × S^1) $ is $0$? Would it be different if we considered $S^2 × S^1 $ or just $S^3$? If so, how? Thanks for the help.
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Why is the fundamental group of a closed hyperbolic $n$-manifold is a (uniform) lattice in $SO(n,1)$?

Why is the fundamental group of a closed hyperbolic $n$-manifold is a (uniform) lattice in $SO(n,1)$? Why is $SO(n,1)$ the (orientation-preserving) isometry group of real hyperbolic $n$-space? Is ...
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Show that $( X \sqcup_f Y)/Y $ is homeomorphic to $X/A$.

Let $X$ be a topological space and let $A \subset X$ be a subset. We define the topological space $X/A$ to be the quotient space $X/\mathcal{R}$ where $\mathcal{R}$ is the equivalence relation defined ...
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Simply Connected Open Manifold being Contained in Simply Connected Closed Manifold

I am aware of the work of Siebenmann characterizing when an open manifold can imbedded in a compact manifold with boundary, but I am having trouble understanding the simpler case when the compact ...
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Explicit model of the $E_{n}$-operad in simplicial sets

The space of $k$ little $n$-disks, denoted $E_{n}(k)$, is usually constructed in the category of topological spaces as the space of $k$-tuples $(c_{d_{1},p_{1}},\dots, c_{d_{k},p_{k}})$ of disjoint $n$...
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A map to an aspherical space factorises?

Let $X$ be a compact smooth manifold with fundamental group $G$ and $Y$ be an aspherical space, then why does the map $f:X\to Y$ factorise to $X\to BG\to Y$?
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Non-Compact Topological Four-Manifolds

I know there exists a complete classification of non-compact surfaces and some work has been done on non-compact three-manifolds, but is anything known about non-compact topological four-manifolds?
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Topological Four-Manifolds that are not Simply Connected

I am familiar with the result of Freedman that closed, simply connected four manifolds may be characterized up to homeomorphism by their intersection forms. This result is basically an extension of $h$...
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Covering space of a $3$-manifold with infinite fundamental group

For a $3$-manifold $X$, s.t. $\pi_1(X)$ is infinite, how to see the universal covering space of $X$, $\tilde{X}$ is a non-compact $3$-manifold and $H_3(\tilde{X})=0$?
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Diffeotopy group, Mapping Class group, Isometry group

There are several closely related concepts on the symmetries or symmetry groups of the space. My apology, but some vague imprecise definitions may be as: Mapping class group (MCG) is an important ...
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Random geodesic on Bolza surface

On a standard flat torus, we can construct a random geodesic (i.e. a dense geodesic on torus) by choosing at the beginning an irrational direction from the point (0,0). My question is how can we ...
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Is this link L10a169?

Please consider the following link I am using SnapPy with the following code ...
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1answer
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Homotopically vs geometrically atoroidal

In the book Aschenbrenner, Matthias; Friedl, Stefan; Wilton, Henry, 3-manifold groups, EMS Series of Lectures in Mathematics. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-154-5/pbk). ...
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Winding semi-circle?

Let us say that $F : \Bbb R → S^1$ is defined by, $F(t) = exp(2πti) = cos(2πt) + isin(2πt)$ Say that I am given some open arc in $S^1$, defined by the set: $C(x) = \{y ∈ S^1 | d(x, y) < 0.01\}$ ...
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Compare Geometrical Shape and Subset (Transformation Invariant comparission)

I am working on CAD Automation projects, which requires comparison of two geometrical shape (Transformation invariant comparison), the expected outcome could be exact match or subset match. Please do ...
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Group of automorphisms of the disk fixing the boundary

I would like to know everything you know (group structure, dense or interesting subsets etc) about the group of diffeomorphisms $$\psi: \mathbb{D}^n \to \mathbb{D}^n$$ that such that $\psi|_{\partial ...
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1answer
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Computing homology of complement of an embedding

Let $I=[0,1]$ and $S^3$ the $3$-sphere. Assume we have injective maps $f_1,f_2:I\to S^3$ such that $\mathrm{Im}f_1\cap\mathrm{Im}f_2=\emptyset$. I have the following problem: Compute $H_*(S^3-\...
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1answer
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Equivalent Loops

If $a$ and $A$ are equivalent loops and $b$ and $B$ are equivalent loops, how could we show that $c$ and $C$ are equivalent loops given that $C = A * B$ ? Basically, we have to show that $[C] = [c]$.
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Homotopic Equivalence of the set of all continous maps from $X$ to $\Bbb R^n$

If we have the set of all continuous maps from $X$ to $\Bbb R^n$ denoted by $C(X,\Bbb R^n)$ and $\sim$ is the relation on this set, how could we prove that every two elements in $C(X,\Bbb R^n)$ is ...
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Some extension for connectedness

Let $X $ be a quasi compact topological space and $Y $ be a subspace of $X $. I am looking for some topological properties on $Y $ such that if $Y $ is a connected subspace of $X $, then $Y $ has ...
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About natural identifications in knot theory

Let's consider a knot $K$ in a general closed oriented 3-manifold $Y$. And for technical simplicity assume $K$ is rationally nullhomologous, ie, $[K]\in H_1(Y)$ is a torsion element. Now choose a ...
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Gale's Theorem Analog for smaller caps

Let $S^k$ denote the unit hypersphere in $\mathbb{R} ^{k+1}$. For a point $a\in S^{k}$ let $H_{\epsilon}(a):= {\{ x | \langle x,a \rangle > \epsilon}\}$. If $\epsilon =0$, then $H_{0}(a)$ is simply ...
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Square Torus homeomorphism

If we have a square torus on R$^2$ defined by the equivalence relation $(a_1, b_1)$ ~ $(a_2, b_2)$ if and only if $a_1 - a_2$ and $b_1 - b_2$ are integers, how would you show that they are ...
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1answer
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Proving different projective planes homeomorphic?

I am having major trouble showing that the version of the projective plane here (with a Mobius strip) is homeomorphic to the projective plane that is defined as the quotient of the sphere $S^2$ via ...
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Euler’s Characteristic for nonorientable surfaces?

I recently learned about this in a topology class but wanted to know how to apply the Euler Characteristic for surfaces such as projective planes and Klein bottles (nonorientable surfaces). I have ...
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1answer
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Decompose a unit ball into 3 convex disjoint parts parts with common boundary

I guess that for arbitrary $n\in\mathbb{N}$ it is impossible to decompose the open unit ball $B(0,1)$ of $R^n$ in four disjoint sets $A$,$B$,$C$,$D$, such that $A$,$B$,$C$ be convex open subsets and ...
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Non-standard examples of smash products and joins

While working with joins and smash products (wedge sums) of topological spaces, I noticed that all examples I know (and find in textbooks) are either involving discrete spaces or spheres, e.g. that $S^...
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Classification of contractible 4-manifolds

Is there a general homeomorphism classification of contractible topological 4-manifolds (possibly with boundary or noncompact)? In the compact case, any such manifold has a homology 3-sphere as its ...
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Lower bound of embedding dimension for finite CW-Complex of dimension $d$

Consider a finite CW-Complex $C$ of dimension $d$. Let $n$ be the smallest integer such that the complex embeds in $\mathbb{R}^n$. From Whitney it follows that $n \leq 2d$. How would you bound the ...
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On the proof of the Whitney trick (from Scorpan's book)

I'm trying to study a proof of the h-cobordism theorem from Scorpan's "The wild world of 4-manifolds". Given a handle decomposition for the cobordism, the Whitney trick is used to eliminate every pair ...
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Definition of a geometric simplicial complex

For $a_0,...,a_k$ affinely independent points in $\mathbb{R}^N$ for $N\ge k$ we define a $k$-simplex $\sigma$ to be $\sigma=\{\sum_{i=1}^kt_ia_i|t_0+...+t_k=1, t_i\ge 0\}$. A simplicial complex $K$ ...
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1answer
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Isotopy between homeomorphisms of open balls

Let $X$ and $Y$ be topological spaces. A homotopy from a continuous function $f: X \to Y$ to a continuous function $g: X \to Y$ is a continuous function $H : X \times [0, 1] \to Y$ such that $H(x, 0) =...
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Killing cohomology of surfaces in finite covers

Let $S$ be a closed orientable surface and $R$ a commutative ring. Given a nonzero element $\alpha \in H^1(S;R)$ is there a finite cover $p : \tilde{S} \to S$ such that $p^*(\alpha) = 0 \in H^1(\...
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1answer
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Obtaining the three torus via Dehn surgery

It is a well known theorem from the '60 (Lickorish-Wallace) that any closed orientable three dimensional smooth manifold can be obtained performing a sequence of integral Dehn surgeries along knots in ...
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1answer
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Extended mapping class group of $S^p \times S^q$

If I understand correctly, (1) the extended mapping class group of $S^2 \times S^1$ is $$ \mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2, $$ how do I understand the third generator that ...
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What shapes do these quotients represent? Do they have a name?

I am unable to visualize complex shapes in space. I can draw simpler shapes and explain which points correspond to which but can't for the life of me determine the name of the shape. I would like ...
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Reduced homology group $H_k(S^4 - N^4, \mathbb Z)=H^{4-k}(S^4 - N^4,\mathbb Z)=H_{k-1}(N^4, \mathbb Z)=\mathbb Z^2$?

Let $N^4$ be a 4-dimensional $D^2 \times T^2 = D^2 \times S^1 \times S^1$. (let us denote $\tilde H$ as the reduced homology or homology group) I know that $$ \tilde H_0(N^4,\mathbb{Z})=0, $$ $$ H_1(...
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1answer
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Number of minima in a ribbon disk?

I am asking this question mainly with the hope of finding a reference to (presumably well-trodden) topic. Let $K$ be a ribbon knot and define $I(K)$ to be the minimum over number of minima of all ...
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1answer
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Can a simple closed curve in a compact surface be dense?

I do not see an argument immediately that it cannot be, but it feels dubious. Does genus have anything to do with it?
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Self-Homeomorphism of an orientable surface with boundary

How can I show that a self-homeomorphism of an orientable surface with boundary that fixes identically a boundary component is orientation-preserving?
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What is the maximum number of spheres with same radius that can touch each other in n dimensions.

For 2 dimensions there can be three circles that can touch each other. In 3 dimensions there can be 4 spheres that can touch each other. What would be the number of hyper-spheres that can touch each ...
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A question on orientation in manifolds (with or without boundary)

I am currently reading Prasolov and Sossinsky's Knots, Links, Braids and 3-Manifolds. In their proof of the Dehn-Lickorish theorem there are some arguments that confuse me. They begin with a statement ...