# 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 ...
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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 ...
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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 ...
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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 ...
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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 $x\in C$...
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### 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 ...
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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 ...
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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 ...
1 vote
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### 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 ...
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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 ...
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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 ...
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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 ...
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### 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 ...
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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 ...
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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 ...
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### 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 ...
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### 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 (...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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(...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...