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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Existence of a cone neighbourhood in an open disc with a point in the border
I started reading C. P. Rourke and B. J. Sanderson’s “Introduction to Piecewise-Linear Topology” which is recommended by D. Rolfsen in “Knots and Links”, they provide the following definition:
1.1 A ...
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Finite subcover of an open cover with common boundary point
Let $S$ be a set of open sets. Suppose all open sets in $S$ have a common boundary point. That is, $\exists$ a point $p$ such that $\forall s\in S$, $p$ is a boundary point of $s$. It is known that $S$...
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Can the ambient isotopy connecting two smooth embeddings close to each other be taken close to the identity in the Whitney topology?
Let $M,N$ be smooth manifolds, of which $M$ is compact. It is a well known result that for any embedding $f:M\to N$ there is some neighborhood $\mathcal{W}$ of $f$ in the Whitney topology such that ...
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Does the generalized Schoenflies theorem generalize the Schoenflies theorem?
The object of study are topological embeddings $h\colon S^{n-1}\rightarrow S^n$, $n\ge2$. We have the
Jordan-Brouwer separation theorem: If $h\colon S^{n-1}\rightarrow S^n$ is an embedding, then $S^n-...
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Lens space $L^k(n) \equiv S^{2k-1}/\mathbb{Z}_n$: Stiefel-Whitney class and non/spin manifold
Define the lens space $L^k(n) \equiv S^{2k-1}/\mathbb{Z}_n$.
What is the property of Stiefel-Whitney class $w_1(TM)$ and $w_2(TM)$ for $M= L^k(n)$?
What is the spin or nonspin manifold property? Is ...
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Simple closed curve homotopic to an end
I am reading the paper Isomorphisms between big mapping class groups. On page 2, they defined the notion of a curve.
Definition: Let $S$ be a connected, orientable surface without
boundary. By a ...
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Can framings on plumbed manifolds be taken to be even?
The definition of "plumbed manifold" that I'm using in this context is the following - given a weighted tree $\Gamma$, build up a framed link $L(\Gamma)$ by chaining together two copies of ...
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Heegaard diagram on a surface
Textbook: Lectures on the topology of 3-manifolds by Nikolai Saveliev
In this book, there is a discussion about the Heegaard diagram of a closed orientable 3-manifold $M$. During the discussion it ...
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Stabilization of Heegaard splitting
So I'm trying to understand the stabilization operation which is to obtain a Heegaard splitting of closed orientable $3$-manifold $M$ of genus $g+1$ from genus $g$.
Given a Heegaard splitting $M = H_g\...
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Any closed orientable $3$-manifold admits a Heegaard splitting
Textbook: Lectures on the topology of 3-manifolds by Nikolai Saveliev
Theorem. Any closed orientable 3-manifold admits a Heegaard splitting.
Proof. Let $T$ be a triangulation of a closed orientable $...
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what does this mean "a periodic chain has nontrivial topology"
I'm reading an article about nonhermitian sytems it says that "a periodic chain has nontrivial topology". What does this mean?
Here is the article https://arxiv.org/pdf/1603.05312.pdf
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Expression of the "inverse" Hopf map (Hopf fiber)
The preimage of a point $p=(p_x,p_y,p_z)$ on the unit sphere $S^2$ by the Hopf map is the circle on $S^3$ with parameterization:
$$
\begin{array}{ccc}
\mathcal{C}_p \colon & (0,2\pi[ & \...
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Explain the general idea of topology for a sophomore student
I am a sophomore and still taking calculus 2 and 3. However, I asked several questions in class but the professor always answers me: you need to take a higher level of mathematics, topology in ...
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Pullback of Stiefel-Whitney class along vector bundle map
Let $\xi = (\pi: E \to B)$ be an $n$-dimensional vector bundle and let $u \in H^n(E, E_0; \mathbb{F}_2)$ be the mod $2$ fundamental class of $E$. (I.e. it is the class such that it is fiberwise the ...
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Construction of a smooth manifolds ($M,g$) from a given symmetry group
If a manifold $M^n$ is given then we know how to find a symmetry group(Automorphism group) of $M^n$ which will preserve some objects(like metric, connections etc) related to $M^n$.
But my query here ...
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What is filling graph $\Gamma$ embedded in a surface $S$?
I am studying graphs on surfaces. I found the following definition.
A graph $\Gamma$ embedded in a surface $S$ is filling if each connected
component of $|\Gamma|$ is diffeomorphic to a disc. Here, $|\...
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field related to "resolution-dependent" topology of manifold inside an ambient space
Let say we have a torus $T$ object floating in the $\mathbb{R}^3$ ambient space. For example the set is expressed as $T = \left\{(x, y, z) \in \mathbb{R}^3+ (\sqrt{x^2 + y^2} - R)^2 + z^2 = r^2\right\...
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Let K be a simplicial complex and σ a simplex of K. Show that the link lk(σ) is a subcomplex of K, if lk(σ) is not empty.
I wanna know if I choose the right way to do my proof? it's correct?
To show that the link lk(σ) is a subcomplex of K, if lk(σ) is not empty, we need to show two things:
Every simplex of lk(σ) is a ...
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Local Modification of Morse Functions
If $f_0 , f_1$ are two Morse functions defined on some smooth manifold $M$ with a common critical point $p\in M$, the same value $f_0 (p)=f_1 (p)$ and with the same stable discs $W^s_{f_1} (p)=W^s_{...
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A covering space of the mobius band
I was doing Hatcher exercise 1.3.21 recently and I failed to figure out the covering space of the space Y formed by attaching a mobius band to $\mathbb{RP^2}$. So I do some research on mathstackage ...
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Reference Request on Sullivan's paper
I am reading Sullivan's 1985 Non-wandering paper(for the paper, see https://www.math.stonybrook.edu/~bishop/classes/math627.S13/Sullivan-1985-Nonwandering.pdf). Section 3 in the paper says that a ...
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Proving that the cone over a knot is not locally flat
When introducing topologically slice knots (i.e. knots $K\subset S^3=\partial D^4$ which bound a locally flat disc in $D^4$) one explains the local flatness condition by noticing that without local ...
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Formula $D + B - S =1$ from The Geometry Center's "Outside In" YouTube Video
The popular YouTube video "Outside In" hosted by user ssgelm is a short film created by the University of Minnesota's Geometry Center to explain the intuition behind sphere eversion and ...
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Questions regarding basic definitions on CW complexes?
I am reading about CW complexes. My book uses the following definitions.
Definition: A cell complex $X$ is a Hausdorff space which is the union of disjoint subspaces $e_{\alpha} (\alpha \in \mathcal{A}...
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What does it mean for a fiber bundle to be local? equivariant theory
Let $\pi:P \rightarrow B$ be a fiber bundle with fiber $F$. What does it mean to say that locally, $\pi$ is of the form $U\times F\rightarrow U$?
Moreover, suppose, say, product fiber bundles with ...
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Minimum number of closed simple curves that separates a surface
I know from Jordan Curve Theorem that every simple closed curve separates the plane. For the torus, one curve isn't enough. Not even two of such curves necessarily separates this surface. What I would ...
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Continuous Map on Compact Oriented Manifold of Higher Degree
Let $M$ be a compact oriented manifold of dimension $\ge 3$.
Is there any known obstructions of $M$ for it to admit a continuous self-map of degree $>1$?
Note: the case when $\dim(M) = 2$ is known (...
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$\mathbb{C}P^2$ is not diffeomorphic to $\overline{\mathbb{C}P^2}$
I am working through 4-Manifolds and Kirby Calculus by Stipsicz and Gompf. At the beginning of Section 1.3, they have a list of exercises regarding $\mathbb{C}P^n$ and $\mathbb{R}P^n$. The part I ...
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Generalizing section-based operations on abstract polytopes
While writing some code to explore abstract polytopes, I've noticed that many intuitive polytope operations (like the Conway polyhedron operations) can be obtained by defining the output polytope $Q$ ...
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Classification of topologically constrained foliations on $X$ in low dimensions i.e. $2,3$
Take $X=(0,1)^n.$ Fix points $p,q$ s.t. $\text{dist}_n(p,q)=\sqrt{n}.$
Problem: Classify analytic regular foliations of $X$ with $(n-1)-$dim. leaves which are topologically $(0,\sqrt{n})\times S^{n-2}...
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How does the Liouville Current of a constant curvature hyperbolic surface depend on the metric?
I am currently studying hyperbolic metrics on surfaces right now, and want to understand the construction of the Liouville Current for a metric $\varphi$ on my (orientable, compact) surface $S$. In ...
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Components of Proper Embedded Connected Manifold
Let N be a connected properly embedded submanifold of M, where M is simply connected. Suppose N has codimension 1, then I want to prove that the complement of N has exactly two components.
I think I ...
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Minvol bounded below by characteristic numbers: Why?
For a closed smooth oriented manifold $M$, define
$$
\text{Minvol}(M) = \inf_{g: \substack{|K_g| \le 1\\g \text{ complete}}} \text{vol}_g(M),
$$
where $|K_g|$ here denotes $\sup_{p \in M}\sup_{\Pi \...
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Why, for Menger Sponge (3D generalization of Cantor Set and Sierpiński Triangle), there exists area surface? Shouldn't it be zero instead of infinite?
Imagine that you have a cube with an edge length of $a$. Let's call this cube $S_0$.
Now, imagine that you virtually sub-divide it into $27$ smaller cubes, each with an edge's length of $a/3$. Remove ...
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Non-existence of a homeomorphism between two sub-spaces of $\mathbb{R}^{2}$
Let $A_n = \{(x,y)\in\mathbb{R}^{2}:x^2+y^2=\frac{1}{n}\}$ , $n\geq 1$ and let $B = \{(x,0)\in\mathbb{R}^{2}:-1\leq x\leq 1\}$. Prove that:
$$
\bigcup_{n=1}^{3}A_n\cup B \mbox{ is not homeomorphic to }...
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Check proof that every manifold has a basis of precompact open sets.
I've written a prove of the theorem. I need someone to check if is correct and explain why Claim 1,3,7 are true.
Definition: Let $M$ be a topological manifold, Then the subset $U \subset M$ : $f(U)=...
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Prove that every manifold is paracompact
Following Lee's book on smooth manifolds.
I'm trying to understand the proof of the theorem (Every manifold is paracompact) and there are some topological claims that i don't understand. I've marked ...
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Equations for folded lemnisate-like curves
I'm trying to curve a set of lemnisate/infinity sign curves, such that the circles overlap along the z axis. Ie.
The equation for this curve is:
$x = a\cdot\sin(t)\cdot\cos(t)\\$
$y = a\cdot\sin(t)\...
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Using Laudenbach-Poénaru to justify Kirby diagrams
It is always referred as the "classical argument" that one only needs to specify the 1 and 2-handles in the Kirby diagram of a 4-manifold $X$ since by Laudenbach-Poénaru there is a unique ...
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Can any compact smooth $n$-dimensional submanifold of the open $n$-ball be deformed into a finite (n-1)-dimensional simplicial complex?
Please, forgive me if I say something silly. I would like to prove that for any smooth compact $n$-dimensional sub-manifold $M$ of the open $n$-ball $B$ there is a finite $(n-1)$-dimensional ...
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How many corners does an otherwise intrinsically flat surface need in order to be homeomorphic to a sphere?
I want to build a virtual world that feels like an unbounded flat plane but actually "connects back to itself" with the topology of a sphere. To do this, we can build the world out of ...
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Poincaré conjecture proof's precision relative the derivative number
First of all, this is a question from amateur in geometric topology.
Since most probably I won't be able to follow currently accepted proofs (they are lengthy and field-specific), I have to ask this ...
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Adding Extra Structures on Functor
Suppose $X$ is a real line with usual topology, and let $PT(X)$ be a category of all elements (points) of $X$ with mapping between points as a morphism. Let TOP be a category of all topological ...
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Serre fibrations between spaces of embeddings-Reference Request
Given topological manifolds $M,N$ of the same dimension, let $\operatorname{Emb}(M,N)$ denote the subspace of $\operatorname{Map}(M,N)$ consisting of the (topological) embeddings $M\to N$. Here $\...
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Existence of connected neighborhood in product via projection
Definition: Let $X$ be a space, a neighborhood of a subset $A$ of $X$ is a set $N$ containing an open set (in $X$) that contains $A$.
That is, a neighborhood need not be open.
Assume $Y'$ is a ...
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Local to global theorem for quasi isometry
I am looking for a local - to - global principle in quasi-isometry.
Suppose $Z$ and $X$ are proper, geodesic, hyperbolic metric spaces such there is a local quasi-isometry $f: Z \to X$. That it for ...
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Is there a complete set of topological invariants for smooth 3-manifolds?
Does there exist a complete set of topological invariants for smooth 3-manifolds? (If not for all smooth 3-manifolds, maybe for a certain class?)
By equipping the manifold with a metric, can some ...
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Extending continuous maps from spheres to Euclidean spaces
Fix $d\in\mathbb{N}$. Consider the following sets as topological spaces with the subspace topology from $\mathbb{R}^{d+1}$.
$$S^d = \{ (x_0,\ldots,x_d)\in\mathbb{R}^{d+1}\mid \sum x_i^2 = 1\}$$ $$ D^{...
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Inclusion of continuous functions with compact open topology into product topology is continuous
Let $X,Y$ be topological spaces and $C(X,Y)$ the set of continuous functions from $X$ to $Y$ equipped with the compact open topology. It has a subbase consisting of sets $$V(K,U):=\{f\in C(X,Y)\ |\ f(...
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Real Line Bundle Corresponding to a Double Cover
I am studying spin structures on $SO(n)$-bundles using some lecture notes. Right after defining the twist of a spin structure $(P,\psi,\rho)$ on $Q\xrightarrow{} X$ by a double cover $\pi:R\...
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