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 ...

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1answer
32 views

Jordan curves in compact subset in $\mathbb R^3$ [closed]

Always exist a curve in compact and convex subset of $\mathbb{R}^3$?
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63 views

Surgery presentation for an abstract open book decomposition

Suppose $(\Sigma,\phi)$ is an abstract open book whose monodromy is expressed as a product of Dehn twists about boundary-parallel curves. Is there a standard way to produce a surgery presentation of ...
3
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1answer
40 views

Topological embeddings of a $n$-ball $B^n$ in $\Re^n$: is the image always an open of $\Re^n$?

I've an elementary doubt about topology and I just can't find the answer (nor I'm able to have an opinion): is it true or false that if $S$ is a subspace of $\Re^n$ homeomorphic to the open ball $B^n$,...
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33 views

Intuition of transversality equation

I am studying Differential topology from Guillemin / Pollack and unfortunately i cannot understand ıntuition of Transversality equation Let $f$ be a smooth map between smooth manifolds $X$ and $Y$ ...
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2answers
98 views

Prove there is a line that cuts the area in half

Suppose you are given two compact, convex sets A and B in the plane, prove there exists a line such that the area of A and B is simultaneously divided in half. Can you help me with this proof? What I ...
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1answer
56 views

Cell-decomposition

My professor said that this (picture) is an example for a non-cell-decomposition because there is missing an vertex at the end of the edge. He also claimed that we have $2$-faces, $3$-edges and $2$-...
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1answer
95 views

Calculating Berry phase for parallel transport of vector around closed path on sphere.

I am following the text here which explains that we are transporting a vector $\mathbf{v}$ around a closed circular loop (C) on a sphere under the constraints $\mathbf{e}_3\cdot\mathbf{v}=0$ and $\...
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1answer
45 views

Space as a network. Is it possible to model topological properties of Euclidean (or non-Euclidean) space using a discrete network of points?

I've read Wolfram's book 'A New Kind of Science', which was very interesting and introduced me to the topic of cellular automata. But he also introduces the idea to model the physical space by a ...
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0answers
24 views

Notation in H-spaces from Homotopy point of view

This book doesn't have a notation index. What do m|e, and rel(e,e) likely mean? Many thanks!
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0answers
36 views

How many topologically inequivalent loops are there on genus g surface?

Suppose we had a 2D surface with g holes in it, and suppose a child drew closed loops on that surface. How many topologically distinct loops can be drawn on the surface? Two loops are equivalent if ...
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3answers
181 views

uses of Riemannian geometry for questions not related to Riemannian geometry

Poincaré conjecture (whose formulation does not make use of notions from Riemannian geometry: "Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere") was eventually proved using ...
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167 views

Reference request: Tubular neighborhood theorem for non-closed manifolds via the exponential map.

Let $M\subset N$ be a submanifold. (Both $M$ and $N$ have no boundary, but $M\subset N$ need not to be closed as a subspace and none of them need to be compact) Choose a Riemannian metric on $N$ and ...
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1answer
290 views

Topological idea of orientability of manifold

While reading Poincare Duality a new idea of orientability of manifold came in my mind.I dont know wheather this idea is new or not, or even true or false. My idea is following... A $n$-dim manifold $...
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1answer
67 views

decomposing a function into embedding and projection

I have a simple question. If $f:\mathbb{S}^{2}\rightarrow\mathbb{R}$ is a non-constant continuous function, can we represent it as a composition $f=p\varphi$, where $\varphi:\mathbb{S}^{2}\rightarrow\...
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0answers
36 views

Euler Integral of a self-overlapping tube with a cusp singularity

I am studying in depth the following paper on Euler calculus applied to target enumeration: https://www.math.upenn.edu/~ghrist/preprints/eulerenumerationpart1.pdf Within this paper there is an ...
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1answer
95 views

Two disjoint real projective planes in real projective space?

Let $\mathbb{R}\mathbb{P}^3$ be the real projective three-space. It is clear that any two hyperplanes in $\mathbb{R}\mathbb{P}^3$ intersect. But I wonder whether one could embed two copies of the real ...
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1answer
58 views

Specifying an arbitrary point on a manifold

It is known that any arbitrary point x on the sphere $\mathbb{S}^2$ can be parametrised by the spherical coordinates $$\bf{x}=r(\theta,\phi)=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta),\quad 0\...
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0answers
42 views

Covering maps between Seifert fibered manifolds

Let $M$ and $\widetilde{M}$ be two Seifert fibered three manifolds. Suppose that there exists a covering projection $p: \widetilde{M} \to M$ preserving the Seifert structure. What is the relation ...
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23 views

Surfaces obtained by $\gamma$-reduction

$\mathcal{C}$ will denote the collection of all connected compact (not necessarily orientable) smooth 2-dimensional surfaces-without-boundary embedded in $M$ ( here $M$ is a complete Riemannian 3-...
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2answers
58 views

Do two embeddings of a Euclidean space into a higher dimensional one only differ by a diffeomorphism?

Let $d\le n$ and $$f,g\colon\mathbb{R}^d\hookrightarrow\mathbb{R}^n$$ be two smooth embeddings. Is there a diffeomorphism $$\phi\colon\mathbb{R}^n\rightarrow \mathbb{R}^n,$$ such that $$f=\phi\circ g$$...
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0answers
49 views

Singularity in Ricci flow vs Ricci soliton

In the paper "The formation of singularity in Ricci flow" Hamilton studied systematically the possible singularities of the flow.My question is why it is important to classify Ricci solitons in order ...
2
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1answer
69 views

Probability via Geometry, applications and examples

People, I am going to teach a lesson including something about Probability via Geometry and, if you have, I would like to know some references or materials (or even some good ideas) that can help me. ...
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1answer
69 views

Is the signature of inverse images of diffeomorphic submanifolds (along a homotopy equivalence) the same?

Suppose it is given an orientation preserving homotopy equivalence $h:N\to M$ between closed oriented connected manifolds. Let $X$ and $Y$ be diffeomorphic submanifolds of $M$, and assume $h$ to be ...
4
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1answer
91 views

Casson handles neighborhoods are representable by $D^2$-bundles over $S^2$.

On 250 page of Scorpan's book Wild world of 4-manifolds. there is a construction of an exotic $\mathbb{R}^4$. It starts from taking manifold $M = \mathbb{C}P^2 \# 9 \overline{\mathbb{C}P}^2$ and ...
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37 views

on branched covers

Consider a branched cover $f:M\to N$ with branch set $A\subset M$ and $B\subset N$. In Rolfsen's book Knots and Links, it is assumed as part of the definition of a branched cover that the dimension ...
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0answers
45 views

cohomology ring of cross-section space of fibre-bundles

Given an $m$-dimensional manifold $M$, let $TM$ be the tangent bundle of $M$ and $SM$ be the $m$-sphere bundle over $M$ obtained by fibre-wise one point compactification of $TM$. Let $\Gamma(SM)$ be ...
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0answers
35 views

What is an intuitive explanation of the Hopf fibration and the twisted Hopf fibration? [duplicate]

As the question suggests, what is an intuitive explanation of the Hopf fibration and the twisted Hopf fibration? I not a topologist by trade and I find the concept kind of hard to understand... Thanks ...
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1answer
70 views

Embedding 3-manifolds in Euclidean space

By the Whitney Embedding theorem, every 3-manifold can be embedded in $\mathbb{R}^6$. It's my understand that it's an interesting problem to see which 3-manifolds embed in $\mathbb{R}^4$; some do and ...
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0answers
49 views

incompressible one - sided surfaces

Suppose we are given an orientable 3 - manifold M and an embedded closed and one - sided surface S with normal bundle N. It is well known that $\delta N$ is an orientable subsurface covering S. ...
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1answer
96 views

How can I write Klein bottle as an adjunction space?

I want to find the homology groups of the Klein bottle by Mayer-Vietoris. For this I want to describe the klein bottle as an adjunction space. I think it can written as a pushout $S^1\cup_f D^2$ but I ...
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2answers
81 views

homology group of adjunction space

I start to study homology theory and i want to understand homology groups of adjunction space In this picture i can't see $V$ deformation retracts to $X$ neither intuitively nor explicitly help ...
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0answers
38 views

isotopy equivalence of maps

In the book Encyclopaedia of Mathematics, Vol. 6, Question: I do not understand the part Does this mean $F_1\circ f_0=f_1: X\to Y$ or as subsets of $Y$, $$ \{y\in F_1(f_0(X))\}=\{y\in f_1(X)\}? $...
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0answers
74 views

Rigorous definition of an embedded connected sum.

Can someone point me to a rigorous definition of a connected sum of two smooth embeddings? I know about the usual construction, the problem is that I can't find a proof that this construction is well ...
2
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1answer
42 views

Computation with Mayer-Vietoris and fundamental classes

Let $M$ be an $n$-dimensional closed oriented connected manifold and suppose that $\bar{U},\bar{V}\subset M$ are $n$-dimensional manifolds with boundary so that $M=\bar{U}\cup \bar{V}$ and $\bar{U}\...
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0answers
84 views

Understanding the “shape” of a singular Riemann surface

Consider the singular Riemann surface given by the following expression: $$z^d w^d-z^d-w^d+t=0\ ,$$ where $t$ is a parameter in $(-1,1)$ and $d$ is a positive integer greater than 2. For $t\neq0$ the ...
4
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1answer
50 views

isotopy equivalence between manifolds

The definition below is from Encyclopaedia of Mathematics: Volume 6. Question: For any $n\geq 1$, is the $n$-dimensional closed cube $$[0,1]^n=[0,1]\times [0,1]\times\cdots \times[0,1]$$ isotopy ...
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0answers
48 views

Gluing 3 dimensional tetrahedra with orientation reversing edge

I am not sure how to proceed on exercise 3.2.3 in Thurston's book "Three Dimensional Geometry and Topology". The wording is as follows: "In a gluing of three dimensional simplices, each edge enters ...
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46 views

equivalent characterizations of manifolds such that configuration spaces are homotopy equivalent

Let $M$ and $N$ be manifolds. If $M$ is homeomorphic to $N$, then the $k$-th configuration spaces $F(M,k)$ is homeomorphic to $F(N,k)$. If $M$ is homotopy equivalent to $N$, then the $k$-th ...
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1answer
58 views

An intutive proof of 'replacing two-caps by a handle'

I am trying to understand a statement given in Polchinski Vol.1 - a torus with cross-cap can be obtained either as (g,b,c) = (0,0,3) or as (1,0,1), trading two cross-caps for a handle. Here, g is ...
2
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1answer
118 views

The first Kirby move and $\mathbb{C}P^2$

A surgery on a link in $S^3$ can be regared as 2-hadnle attachements to $D^4$ (resulting 4-manifold $W$ whose boundary is the result of the surgery). I would like to know how the first Kirby move (...
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0answers
67 views

Isotopy: Definition

An isotopy is a homotopy from one embedding of a manifold $M$ in $N$ to another such that at every time, it is an embedding. In this definition, I am wondering why $M$ and $N$ are required to be ...
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0answers
31 views

Is there any “Brunnian-like” braid which is not a pure braid in $B_n$ with $n\geq 3$?

A braid $\beta$ is called Brunnian if it satisfies $\beta$ is a pure braid; $\beta$ becomes a trivial braid after removing any of its strands. Obviously in $B_2$, every braid satisfies condition 2....
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1answer
146 views

Fundamental group of open subsets of $\mathbb{R}^n$

Suppose that $U$ is an open subset of $\mathbb{R}^n$. What can be said about its fundamental group? I'm sure that the answer should be well known, since this is rather natural question.
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43 views

Linear vs smooth actions of finite groups on spheres, Euclidean spaces and closed disks

I would like to know examples (with references, if possible) of the following: (1) a finite group $G$ acting effectively and smoothly on a sphere $S^n$ but admitting no effective linear action on $S^...
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2answers
146 views

actions of $\mathbb{Z}_2$ on spheres

Let $S^m$ be the $m$-sphere and $$F(S^m,2)/\mathbb{Z}_2=\{(a,b)\mid a,b\in S^m, a\neq b\}/(a,b)\sim (b,a)$$ be the $2$-nd unordered configuration space on $S^m$. Why $F(S^m,2)/\mathbb{Z}_2$ is ...
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1answer
70 views

Mapping the intersection of hyperplanes/simplex to lower-dimensional unit-simplex

Suppose I have an object in $\mathbb{R}^5$ described by: $$x_1+x_2+x_3+x_4+x_5=1$$ $$x_1+2x_2+3x_3+4x_4+5x_5=6$$ $$x_1+7x_2+8x_3+9x_4+10x_5=11$$ $$x_1,x_2,x_3,x_4,x_5 \geq 0$$ Is there a way that I ...
3
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1answer
48 views

Thom space of unit circle

Say we embed $S^1$ into $\mathbb{R}^2$ as the unit circle. What is the Thom space $Th(i)$ associated to this embedding $i:S^1 \to \mathbb{R}^2$? By definition, the Thom space is the one point ...
9
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1answer
164 views

General relationship between braid groups and mapping class groups

I just finished correcting my answer on visualizing braid groups as fundamental groups of configuration spaces, and in the process became interested in the other pictorial definition of the braid ...
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1answer
58 views

Open condition given by inequality on functions

Let's say we have two functions $f,g\in C^\infty(D)$, $D$ an open domain in $\mathbb{R}^2$. The condition $f(x,y)<g(x,y)$ is an open condition on $D$? With this I mean: do the points $(x,y)\in D$ ...
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125 views

finite graphs with homeomorphic covering space that do not cover the same graph.

I came across this exercise from section 1.3 in Hatcher's "Algebric topology". Construct finite graphs $X_1$ and $X_2$ having a common finite-sheeted covering space $X_1 \cong X_2$ , but such ...