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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How can this horrible definition of "Quasi-Rectangle" be re-formulated to be more intelligible?

$$\renewcommand{\mangle}{\measuredangle}$$ Our goal is to either define, and/or characterize, a set of objects that visually resemble misshapen rectangles with slightly-rounded corners What ...
Theodore Shepard's user avatar
-1 votes
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Computing homology from finite closed cover [closed]

Consider a finite collection of closed $d$-balls $\{B_1,\ldots,B_n\}$ which cover a smooth $d$-manifold, $M=\bigcup_{i=1}^{n}B_i$. Suppose we wish to compute the (integral) (co)homology of $M$ from ...
rab's user avatar
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2 votes
1 answer
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Nulhomopty Lemma (Just for a small part)

I was reading the proof for null-homotopy lemma in Munkres's Topology book, and I cannot convince myself for the suggestion: One can replace $S^2$ by the one-point compactification $\mathbb{R}^2 \cup ...
Ulaş's user avatar
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-1 votes
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Polar coordinates generate the same σ-algebra as the usual Borel σ-algebra of R^n? [closed]

I want to prove the statement in the title but first some background information: I am studying multivariate probability theory. For a multivariate random variable, one often uses polar coordinates to ...
Victor's user avatar
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1 answer
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How to show infx∈D⟨p,x⟩>supy∈Ω⟨p,y⟩?

Suppose thet $\Omega \subset \mathbb R^n$ is closed convex, and $ D \subset \mathbb R^n$ is compact convex. If $\Omega \cap D = \varnothing$, then please show$\exists p \in \mathbb R^n$ with $inf_{x \...
Harry's user avatar
  • 23
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0 answers
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Open sets on a surface with locally connected boundary

Let $\Sigma$ be a surface and $\Omega$ be an open subset of $\Sigma$. Suppose that $\Omega$ is homeomorphic to the open unit disk $\mathbb{D}$ and is relatively compact in $\Sigma$. When $\Sigma$ is ...
Dilemian's user avatar
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Geometric intuition behind hyper-sphere volume recurrence relation

There is a recurrence relation for calculating the volume of a hyper-sphere and a logical explanation for why it holds. Is there a geometric intuition behind this to help me intuitively understand ...
Hank's user avatar
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Symmetric vector fields on the sphere

Let $C$ be a closed curve on $S^2$ (possibly self-intersecting) that is symmetric with respect to the origin. Assume a unit tangent vector field $F$ is defined on $C$ with $F(x)=F(-x)$ for all $x\in C$...
MathLearner's user avatar
2 votes
1 answer
59 views

A Weak Type of Convexity for Smooth Jordan Domains in $\mathbb{R}^2$?

Suppose that $D \subset \mathbb{R}^2$ is a closed Jordan domain with boundary curve $J$ and interior $\text{Int}(D)$. I'm sure there's already a term for this, but say that $D$ is $n$-star-convex if ...
John Samples's user avatar
1 vote
1 answer
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Is there any 4-manifold which is known to have a unique smooth structure?

A well-known problem in 4-manifold topology is the smooth Poincaré conjecture, which states whether any smooth manifold homeomorphic to S⁴ is actually diffeomorphic to S⁴, that is, whether S⁴ has a ...
homologic's user avatar
2 votes
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A 4-dimensional 3-handle attachment to $S^2 \times D^2$ to get $D^4$.

Given $Y= S^1\times D^2$ and attaching sphere of 3-handle as $S= S^2 \times \{pt\}$, then how to visualise that after attaching 3-handle we get $D^4$ as the resultant manifold? This is part of the ...
Prerak Deep's user avatar
1 vote
1 answer
58 views

Local homeomorphism and lifting

Assume $X,Y,X',Y'$ are nice spaces (not necessarily manifolds). If $\pi_{X}:X\rightarrow X'$ and $\pi_{Y}:Y\rightarrow Y'$ are local homeomorphisms, if $f':X'\rightarrow Y'$ is a continuous map, then ...
monoidaltransform's user avatar
1 vote
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Different statements about the peripheral system as a complete knot invariant

I am somewhat confused about the different flavours in which the statement "the peripheral (group) system is a complete knot invariant" usually comes, and I believe not all of them have ...
Minkowski's user avatar
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Peripheral subgroup determined up to conjugation

Let $K$ be an oriented knot in $S^3$, let $X_K := S^3 - \mathrm{int} \ N(K)$ be its knot exterior and let $i: \partial X_K \hookrightarrow X_K$ be the subspace inclusion. The peripheral subgroup is ...
Minkowski's user avatar
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5 votes
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What is the image of a smooth map?

Let $f: S^2 \to \mathbb{R}^n$ be a smooth map from the two-dimensional sphere to euclidean space. Let $X = \mathrm{Im}(f) \subset \mathbb{R}^n$ be the image topological space (note: the quotient ...
unknownymous's user avatar
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4 votes
1 answer
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Property of Lipschitz Domains

I have been working on a research problem of mine and came across the concept of Lipschitz domains. I am curious about whether it is possible to show that there always exists a bi-Lipschitz map from $\...
sudeep5221's user avatar
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If the cut locus of surface is a segment, the surface is the topologic sphere?

I drew a few examples and looked at them, seemly, if the cut locus of a surface is a (not closed) segment, then the surface is a topologic sphere. I google it, but fail to find efficient reference. So,...
Enhao Lan's user avatar
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2 answers
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Whether $\mathbb{T}\times [0,1]$ is diffeomorphic to $\mathbb{D}^2$?

Here $\mathbb{T}$ denotes the torus $\mathbb{R}\backslash \mathbb{Z}$, and $\mathbb{D}^2$ the closed unit ball in $\mathbb{R}^2$. Since $\mathbb{T}\times [0,1]$ and $\mathbb{D}^2$ are both smooth ...
Jiawen Zhang's user avatar
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1 answer
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Degree of the twist map $S^m\wedge S^n $ to itself.

Consider the map $S^m\times S^n \rightarrow S^n\times S^m$ which takes $(x,y)$ to $(y,x)$. This induces a map on the smash product $S^m\wedge S^n=S^{m+n}$. I am at a loss thinking what the degree of ...
aritracb's user avatar
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Semi-open triangles topology is not a product topology

In order to give an example of a topology of $\Bbb{R}^2$ which is not the product of a topology of $\Bbb{R}$, I have seen on a book the topology $\mathcal{T}$ of semi-open triangles induced by the ...
Superdivinidad's user avatar
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Is the product manifold $S^{1} \times \text{Bryant soliton on } R^{n-1}$ is non-collapsed?

I am puzzed that whether the product manifold $S^{1} \times \text{Bryant soliton on } R^{n-1}$ is non-collapsed, as my understanding of the definition non-collapsed and computation, I believe it is ...
Grantsome's user avatar
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1 answer
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Every point is umbilical, then plane or sphere. Proving from local to global.

I'm trying to fill in some details in Do Carmo's curves and surfaces proof on p.149, If all points of a connected surface $S$ are umbilical points, then $S$ is either contained in a sphere or in a ...
MathPhysForFun89's user avatar
1 vote
0 answers
70 views

Can an isotopy between diffeomorphisms be through diffeomorphisms?

Let $M$ be a smooth manifold (without boundary). Suppose $F:M \times I \rightarrow M$ is a smooth (say $C^{\infty}$) isotopy between two diffeomorphisms $f_0$ and $f_1$ of M in the sense that $\...
Skyskie's user avatar
  • 333
2 votes
0 answers
39 views

Bipartite intersection graph

Recently, I'm trying to learn geodesic currents on free groups through a paper by Kapovich and Lustig. Let $\langle, \rangle: \overline{CV}(F_N)\times Curr(F_N)\to\mathbb{R}$ be the intersection form ...
quuuuuin's user avatar
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1 answer
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Prove that a square without one inner point is not simply connected

I understand that this is a consequence of Brouwer's theorem. For a circle without an inner point, we have that the circle's mapping to itself does not contract, i.e. if $f = \mathrm{id}\colon S^1 \to ...
jhonnnnnny's user avatar
2 votes
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Computation of homology groups of Milnor's exotic sphere

Milnor's sphere $M$ is defined as the total space of the $S^3$ fiber bundle over $S^4$ with clutching map $f : S^3 \to SO(4)$ given by $u \mapsto (x \mapsto u^ixu^j)$, where $i, j$ are constants and $...
pnguyen0719 's user avatar
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Clique complex of expander graphs simply connected?

Given an expander graph family (an injective sequence of graphs with uniformly bounded vertex degree and a Cheeger constant/Laplacian spectral gap uniformly bounded away from zero). Can the ...
Florentin Münch's user avatar
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1 answer
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Simple questions on the parametrization of the surface $\left\{(x;y;z)\in\mathbb{R}\times\mathbb{R}^{+*}\times\mathbb{R}:x^2+z^2=1/y\right\}$

I am studying alone and I would like to have a feedback on my two first answers in "a)" and "b)" and get help on "c)". Question: We have the following parametric surface ...
OffHakhol's user avatar
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1 answer
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Prove that $\left\{ (x;y;z)\in\mathbb{R}\times\mathbb{R}^{+*} \times\mathbb{R}:x^2+z^2=1/y\right\}$ is a submanifold and justify his dimension

In introduction I would like to say that I just begin to study (alone) what is are submanifold so I need your help in order to be sure that my understanding of those concept is correct. Question: Let $...
OffHakhol's user avatar
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1 vote
0 answers
72 views

Do homeomorphisms of disks in a sphere extend?

Suppose $D_1$ and $D_2$ are two n-dimensional closed disks (topologically embedded) in the sphere $S^n$ ($n>=1$). Is there a homeomorphism $f:S^n \rightarrow S^n$ such that $f(D_1)=D_2$? Maybe I ...
Skyskie's user avatar
  • 333
0 votes
1 answer
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Is There a Conceptual Connection Between the 3D Winding Number and Ray Casting Algorithms?

The 3D winding number provides a numerical answer to whether a point is inside or outside a closed surface, with its definition arising from surface integration. In my recent journey through ...
K.R.Park's user avatar
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0 answers
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Fitting / drawing a closed curve of a given length within given dimensions

Fitting / drawing a closed curve of a given length within given dimensions. I'm trying to fit / draw a given length of a closed curve within given dimensions. Example: Curve length=941mm, width=3mm, ...
Rick T's user avatar
  • 457
0 votes
0 answers
30 views

Realizable subsets of the sphere at infinity

Suppose we have an arbitrary finite collection $X = (x_i)_{i = 1} ^ n$ of the boundary sphere $\partial D$ where $D$ represents the Poincare disk. Does there exist a (discrete?) subgroup $G$ of the ...
discretephenom's user avatar
0 votes
2 answers
134 views

Regarding intersecting surfaces of two surfaces

When I am plotting two quadratic surfaces $y^2/2 + z^2/2 - 16.7 z = -30$ and $-(x^2/2) + 16.7 z = 200$, then its intersecting orbit is found to be a single orbit (see attached image). But, when I ...
Arssat's user avatar
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0 answers
42 views

Limit set of subgroup of Hyperbolic Isometry

When working with subgroup of Klenian groups (or in general just asking for discrete subgroup forgetting about the dimension of Hyperbolic space) the classic definition of limit set $\Lambda(\Gamma)$ ...
Augusto Matteini's user avatar
3 votes
0 answers
73 views

What is the volume of the largest surface of revolution with constant positive curvature that can be embedded in the unit cube?

Consider a surface of revolution $S$ with constant positive curvature and an embedding $e:S \hookrightarrow X^3$ for $X^3=[0,1]^3$ with conjugate points $p,q$ anchored on $\partial X^3$ where $\...
John Zimmerman's user avatar
1 vote
2 answers
83 views

What is a transversal intersection? Can it be explained without tangent spaces and tangent bundles?

What is a transversal intersection? Can it be explained without tangent spaces and tangent bundles? Background: Transversal intersection was used to explain If the interior of two convex manifolds ...
SRobertJames's user avatar
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0 votes
1 answer
81 views

Can topological arguments determine the intersections of a plane and cube?

Determine the intersection of a plane $P$ and cube $U$ in $\mathbb R^3$ using topology. Note: This can be solved analytically (using brute force); this question asks for a topological argument (...
SRobertJames's user avatar
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0 votes
1 answer
60 views

If the interior of two convex manifolds intersect, what is the dimension of their intersection?

In $\mathbb R^n$, let $A,B$ be two convex manifolds of dimension $a,b$ respectively whose interiors intersect. It would seem that their intersection is a convex manifold of dimension $\min(a,b)$. Is ...
SRobertJames's user avatar
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0 votes
0 answers
49 views

Can a Lie group act transitively on a geometrically finite hyperbolic manifold?

Let $ M $ be a hyperbolic manifold. If $ M $ has finite volume (this includes all compact hyperbolic manifolds of course) then no Lie group can act transitively on $ M $. But what about geometrically ...
Ian Gershon Teixeira's user avatar
2 votes
2 answers
104 views

Why do $2g+1$ distinct closed curves separate a compact orientable surface of genus $g$?

The genus of a surface is the maximum number of pairwise disjoint simple closed curves that do not separate a surface. Why do $2g+1$ distinct closed curves separate a compact orientable surface of ...
Fernando Oliveira's user avatar
0 votes
2 answers
107 views

What is the Jones Polynomial for the Borromean Link?

I was looking up the Jones Polynomial for a project I’m working on and came up with this equation from the knot atlas: $$ -q^3-q^{-3}+3q^2+3q^{-2}-2q-2q^{-1}+4 $$ However, I know that when entering VL(...
ParabolicX's user avatar
9 votes
2 answers
2k views

What is the formal definition of a hole? [duplicate]

In topology, there is a definition of "number of holes" of a manifold, like a torus. However, I have never seen the definition of hole by itself. Intuitively, a hole is a region of space ...
user107952's user avatar
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3 votes
0 answers
49 views

For an explicit mapping $S^m\to S^6$ for $m<12$, how can we understand whether it is homotopy zero?

In my work I have mappings from $S^m$ to $S^6$, where $m\leq 11$. People know the homotopy group of these mappings. However, is there some algorithm that helps to recognize when mappings (we can ...
Timur's user avatar
  • 31
3 votes
1 answer
92 views

Existence of non-vanishing section of normal bundle (codimension $\geq 2$)

Suppose $M^n \hookrightarrow P^{n + k}$ is an isometric embedding of a manifold $M$ in an arbitrary ambient manifold $P^{n+k}$ for $k \geq 2$. Does there exist a global, non-vanishing section of the ...
JMK's user avatar
  • 795
0 votes
0 answers
40 views

Algorithm to recognize spherical abstract polytopes

A finite abstract polytope of rank 3 (an abstract polyhedron) consists of adjacency data for a collection of polygonal faces and their shared edges and vertices. This data is sufficient to uniquely ...
Karl's user avatar
  • 11.5k
2 votes
1 answer
74 views

Show that continuity-preserving maps are homeomorphisms

Let $X$ and $Y$ be topological spaces. Call $f:X \rightarrow Y$ continuity-preserving if, for all continuous functions $\gamma: \mathbb{R} \rightarrow X$, $f \circ \gamma$ is also continuous. I know ...
paad89's user avatar
  • 137
3 votes
0 answers
130 views

Examples using this long exact sequence of cohomology [duplicate]

I began reading this paper (also provided below), and I would really appreciate if someone could help me with the premises behind the first sentence. I have not been able to find the given long exact ...
June in Juneau's user avatar
3 votes
0 answers
39 views

Generalization of Euler's formula for graphs embedded on surfaces of higher genus

Question 1: Let $G$ be a graph embedded on an orientable surface of genus $g$, not necessarily cellularly. Prove that $v-e+f \geq 2-2 g$, where $v, e$ and $f$ denote respectively the number of ...
Pipnap's user avatar
  • 381
2 votes
1 answer
52 views

A prime orientable 3-manifold containing a nonseparating 2-sphere is homeomorphic to $S^2\times S^1 $.

I am reading the notes, trying to understand the proof of Proposition 3.5: I have no idea why $X$ is homeomorphic to $S^2\times S^1$ with the interior 3-ball removed. I think by the "obvious ...
Kat's user avatar
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