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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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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 ...
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How to undo a type I Reidemeister move only using type II and type III moves?

I am pretty new to geometric topology and I am struggling to solve this problem. I have tried to approach this a lot of different ways and I will spare you the drawings. I am trying to undo this ...
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Example of a homeomorphism of a plane which fixes unit circle point wise

What would a non trivial example of a homeomorphism of a plane which fixes unit circle point wise. If I take \begin{equation} h(x,y)=\begin{cases} (x,y)& \text{$(x,y)\in S^1$}\\ (x+2,y+2) & \...
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How to characterize the group of homeomorphisms of unit disk in terms of the group of homeomorphisms of plane?

Let $G$ be the group of homeomorphisms of unit disk $(D)$ fixing boundary point wise and $P$ be the group of homeomorphisms of plane $\mathbb{R}^2$. Can we characterize $G$ in terms of $P$. Speaking ...
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Kirby calculus on E8 plumbing

I was trying to show that the 4-manifold described in Kirby diagram as a E8-plumbing (see the diagram below) has the same boundary as the 2-handlebody on the left-handed trefoil with surgery ...
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38 views

How can a torus be turned to a cylinder if the circumference of the outer ring is larger?

I’m not a math professional by any means, I’m just interested in math and this topic has been on my mind for a while, and I just couldn’t find an answer. Also, on the same vein, how can you make a ...
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63 views

Path components of $\mathbb{Z}/2\mathbb{Z}$-actions on a manifold

Let $M$ be a closed (compact and no boundary) topological manifold and $G=\mathbb{Z}/2\mathbb{Z}$ be the group of order two. Let $\mathcal{G}(M)$ be the space of continuous $G$-actions on $M$ (say, ...
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Orientation preserving homeomorphism on a torus

Let $Τ^2=S^1\times S^1$ be the standard torus, $f:T^2 \to T^2$ an orientation preserving homeomorphism that is the identity on a meridian $m$ and $A$ an $ε$-neighbourhood of $m$. My question is, if ...
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Are regular graphs always regular in a topological sense in a particular dimension?

Consider a large cloud of points (sites) in arbitrary dimensions. Now I introduce links between the sites, such that any site is connected to exactly two other sites (and there is no self-connections ...
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115 views

Characterization of $\mathbb{R}^n$?

Let $M$ be a smooth $n$-dimensional manifold with the property that any compact subset $K \subset M$ is contained in an $n$-dimensional smooth ball $K \subset B \subset M$. If $M$ is open, does it ...
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Proving an “inevitable intersection”

So, I need to prove that if a curve $C$ is homotopic to a point (with homotopy $H$ where deformation happens exclusively in the same number of dimensions as $C$), then all of the points within that ...
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Formal definition of euclidean space

Is there an agreed upon, precise definition of the euclidean space? Although I have searched in may places, I can't find anything that is rigorous. From what I have read, I have thought of defining it ...
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1answer
26 views

Proof regarding the genus of a topological space

Intuitively, I would think that if there exists a continuous mapping $f:X\rightarrow Y$ between 2 topological spaces $X$ and $Y$, then the genus of $Y$ must be smaller that the Genus of $X$ (as during ...
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How to prove that $R^n\setminus \{0\}$ is not contractible

If I must prove that $R^{n}\setminus\{0\}$ is not contractible, how may I do so formally. Using the intuitive notion of contractibility at a point as being that any surface homeomorphic to an $n$ ...
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1answer
36 views

Proof regarding n-connectedness

If one has to prove that $R^n - \{0\}$ is not $(n-1)$-connected, is it necessary to prove formally that there exists a non contractible $(n-1)$-sphere or can that simply be stated. If one must ...
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Is there any method to embed $K_p$ into a orientable surface?

It is known that $K_p$ can be embedded into genus $g=\lceil{\frac{(p-3)(p-4)}{12}}\rceil$ orientable surface. Do we know how to embed $K_p$ into the genus $g$ orientable surface?
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Do contractible homology manifolds have one end?

If $X$ is a space, then let us say that $\pi_0^\infty(X)$ is the set of equivalence classes of proper maps $[0,\infty) \to X$, modulo proper homotopy (the map $[0,1] \times [0,\infty) \to X$ should be ...
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1answer
50 views

Use Van Kampen theorem to find the fundamental group

I'm got stuck in calculating fundamental group of the plane ($\mathbb{R}^2$) minus finitely many points ($n$ points). I think the space is homotopic the wedge sum of n - circles ($S^1$). And its ...
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59 views

When a subspace of a space is totally disconnected

Let $X $ be a topological space and $Y $ a Hausdorff subspace of $X $ such that for every connected component $C $ of $X $ the set $C\cap Y $ is finite. How can we show that $Y $ is totally ...
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Curves on surfaces lifting to embedded curves in finite covers

Let $S$ be a orientable closed surface with genus $g \geq 1$ and let $\gamma \subset S$ be an immersed curve. Does there exist a finite cover of $S$ where $\gamma$ lifts to a curve that is homotopic ...
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Using the Gauss-Bonnet theorem to determine a surface

I'm using the Gauss-Bonnet theorem on the following exercise and I would like to see if what I've done is correct. Given a compact, connected surface $M$ with area $A(M)=1984707$ and Gauss curvature $...
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1answer
127 views

Request for applications of the Filling Theorem: geometric and algebraic Dehn functions are asymptotically equivalent

I am studying geometric group theory and metric spaces of bounded curvature. In particular, I was reading the seminal Gromov's Hyperbolic Groups and trying to filling the details. In the first page, ...
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A space obtained by $S^3$ by removing a Hopf link

If we remove a Hopf link from $3$-dimensional sphere $S^3$, can we obtain a space homotopy equivalent to (or deformation retract to) an annulus? If the answer yes, can we write it explicitly? EDIT: ...
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Which 3-manifolds can be cubulated?

I am trying to get a picture of what is currently known about cubulability of 3-manifolds, though cannot seem to find a good overview. I am personally most interested in compact 3-manifolds with ...
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Notation on fibre bundles

I came up this morning with the following question and after looking for it for a while on the internet i found this old question on math.stackexchange with no answers. Could anyone please give some ...
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1answer
32 views

Create a space between clouds

I have a dataset associated with labels. According to https://arxiv.org/pdf/1802.03426.pdf --> UMAP (Uniform Manifold Approximation and Projection) which is a novel manifold learning technique for ...
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131 views

Is a principal $\mathbb{Z}_2\ltimes PSU(4)$-bundle over a 3-manifold $M$ equivalent to an element in $H^1(M,\mathbb{Z}_2)\times H^2(M,\mathbb{Z}_4)$?

Given a 3-manifold $M$ and a principal $\mathbb{Z}_2\ltimes PSU(4)$-bundle $P$ over $M$ whose isomorphism class is represented by the homotopy class of a map $f:M\to B(\mathbb{Z}_2\ltimes PSU(4))$ ...
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1answer
29 views

What exactly this “elliptic” and “hyperbolic” mean on the picture describing Lobachevskian and Riemannian geometry?

Item 5 here has a figure calling Lobachevskian geometry hiperbolic and Riemann geometry elliptic and both figures have a perpendicular line. What does that perpendicular line and the both pictures ...
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Which Nonorientable 3 manifolds have torsion in $H_{1}$?

In thinking about closed, nonorientable 3 manifolds I've been interested in finding nonorientable loops which represent torsion classes in homology. More precisely, for a closed nonorientable 3 ...
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1answer
51 views

The second Betti number of a group

First I can't find the definition of second Betti number of a group. (Can you tell me any reference about this definition?) Also I don't know why $b_2(M)\ge b_2(G)$, where $M$ is a manifold with ...
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1answer
79 views

Is there an example of a manifold with fundamental group $\mathbb Z/3 \mathbb Z$?

I feel a little confused because I was told that there exist some manifolds with fundamental group $\mathbb Z/3 \mathbb Z$, but I can’t find an example, On the other hand, since any manifold $M$ has ...
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List the planes in $PG(3, 5)$ that pass through the point $(0: 0 :0 :1)$

I want to list all planes in $PG(3, 5)$ that pass through the point $(0: 0: 0: 1).$ My thought is to find the lines that not pass through this point first but I can not make sure how many different ...
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1answer
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How can you determine if a point is inside a parametric 2D manifold?

Asume I have an arbitrary, parametric, closed, orientable, surface; a sphere, ellipsoid, closed cylinder, weird general cone.... If you only have access to the parametrization, how can you determine ...
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1answer
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Definition of homotopy classes of simple closed curves

I don´t know whether this kind of questions should be asked here or not. I am trying to explain the concept of a homotopy class of simple closed curves (that is, the vertices of curve complexes) to ...
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1answer
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Group of orientation preserving homeomorphisms of circle $S^1$ acts transitively on the set of closed intervals of $S^1.$

The closed intervals here mean the arcs including endpoints on the circle. I tried to do it by taking the inverse image of those two closed intervals from $S^1$ to its covering space $\mathbb{R}$ and ...
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Definition of connected sum and orientation problem

I am reading Kosinski's book. To define the connected sum of $M_1^n$and $M_2^n$ (oriented and closed manifolds) we choose two embeddings of the disk $h_i:\mathbb{D}^n\to M_i$ such that $h_1$ preserves ...
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Representing a Multiple of the Fundamental Class of a Surface

Let us denote a connected, oriented surface of genus $g$ by $\Sigma_g$. It is easy to see that if we have a map of degree $1$ from $\Sigma_g$ to $\Sigma_h$ then $g\geq h$. Suppose now that $g\geq 2$ ...
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Locally compact in $\mathbb{R}^n$ and Hausdorff spaces

I have to show that (1) Show that $\mathbb{R}^n$ is locally compact. https://i.imgur.com/GL4PJj5.png I have provided a link with my answer since it's long and would take me forever to write in here....
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37 views

If V and W are linear subspaces of $R^n$, then V transversal W means just $V + W = R^n$

If V and W are linear subspaces of $R^n$, then V transversal W means just $V + W = R^n$. Could anyone give me a hint for this exercise please?
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1answer
44 views

How to show $\operatorname{Int}(z)$ is not empty?

For any closed smooth curve $z:[a,b]\to \Bbb C$, we define the interior $\operatorname{Int}(z)$ of $z$ as follows: Because $\operatorname{Im}(z)$ is bounded, we can find a circle $C$ such that $\...
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1answer
36 views

Proof for sphere eversion

Do you know where I can find (preferably online) a proof for the possibility of a sphere eversion?
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Any spin $M^3$, exists a natural induced $\text{Pin}^-$ structure on Poincare dual PD

It seems that "R. C. Kirby and L. R. Taylor, Pin structures on low-dimensional manifolds (1990)" shows given a spin structure on $M^3$, the submanifold $\text{PD}(a)$ can be given a natural ...
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1answer
79 views

Twisting the unit square n times before gluing( 2.1.6 in G&P).

The question is given below: I have made a Mobius band with a paper and twisted it 3-times but I could not describe what I see it may be a 3 knot shape, could anyone give me a hint for solving that ...
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1answer
32 views

Understanding noncyclic covers of knot complements

Let $K$ be a knot in $S^3$. I'm familiar with the process of taking the $n$-fold cyclic cover $X_n(K)$ of the knot complement $X(K) = S^3 - K$ and I know that this cyclic cover can be completed to a ...
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1answer
80 views

A difficulty in understanding the solution of problem 1.7.17 in Guillemin and Pollack.(p.47)

The question is given below: And here is exercise (16): And here is the solution to exercise(17) But I have difficulties in understanding the following parts of the solution: 1-Why the codomain of ...
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2answers
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Expansion of homeomorphism outside a disk

The following is an exercise in Bloch's Intro to Geometric Topology Let $B \subseteq \Bbb R^2$ be a set homeomorphic to the closed unit disk and $h :\partial B \to \partial B$, a homeomorphism. By ...
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1answer
49 views

Prove that $f$ is Morse function if an only if $det(H)^2 + \sum_{i=1}^k (\frac {\partial f}{\partial x_i})^2>0$(2)

I was trying to solve this question: Prove that $f$ is Morse function if an only if $\mathrm{det}(H)^2 + \sum_{i=1}^k (\frac {\partial f}{\partial x_i})^2>0.$ But while searching on this site I ...
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1answer
38 views

Show that all open sets in $\mathbb{R}^n$ are the same for metrics $d_1,d_2,d_\infty$

So I have shown that any $x,y\in\mathbb{R}^n$ that $d_\infty(x,y) \le d_2(x,y)\le d_1(x,y)\le nd_\infty(x,y)$, where $$d_\infty=\max\limits_{j\in\{1,\dots,n\}}\lvert x_j-y_j\rvert,$$ $$d_2(x,y)=\sqrt{...
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1answer
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Separation from family of closed and connected subsets

Let $X $ be a compact $T_0$ topological space and let $\{ A_i \}_{ i \in I }$ be a family of closed and connected subsets of $X $ and $ x\in X$ such that for each $i\in I $ there exists a closed and ...