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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Examples of connections between bounded cohomology and geometric properties of groups

I think that the question is self explanatory. The only example that comes to my mind is the characterization of hyperbolic groups given by Mineyev ("Straightening and bounded cohomology"). There is ...
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1answer
27 views

Embeddability of connected sum of non-embeddable surfaces

Let $X$ be a surface which can not be embedded into $\Bbb R^n$. Let $X \# X $ denotes the connected sum of two copies of $X$. Then is it true that the connected sum $ X \# X $ is also not embeddable ...
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60 views

John Hempel's proof of residual finiteness of surface groups

John Hempel proved that fundamental groups of $2$-manifolds are residually finite. I want to understand this proof so I have some questions: Why if we have $S(f)=\emptyset$ then $f$ represents ...
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1answer
34 views

A differentiable manifold of class $\mathcal{C}^{r}$ tangent to $E^{\pm}$ and representable as graph

The stable manifold theorem tell us: A local stable manifold $W^{s}_{loc}(x^{*})$, which is a differentiable manifold of class $\mathcal{C}^{r}$ and dimension $n_{-}, $ tangent to the ...
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26 views

Linking of $S^p$ and $S^q$ in the $\mathbb{R}^d$ space

Can we have a nontrivial linking of a $S^p$ sphere and a $S^q$ sphere in the $\mathbb{R}^d$ space (or in the ${S}^d$ space)? I suppose that it can happen only if $p+q<d$. For example, we can have: ...
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Homology group and homotopy group of the standard twin

Given a 4-sphere, if we cut out a solid 3-torus $B^2 \times S^1 \times S^1$ from a 4-sphere $S^4$ (with an unknotted torus), the remained exterior is called "the standard twin," say $M$. What are ...
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0answers
45 views

Embed $S^{p} \times S^q$ in $S^d$?

Can we embed $S^{p} \times S^q$ in $S^d$ with all the nice properties, what are the allowed values of $p$ and $q$ for $d=2,3,4$ where $p+q \leq d$? =For $d=2$= I suppose that we cannot embed $S^{1} ...
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300 views

Maps from $D^n$ to $D^n$ with a single inverse set are open.

Let $D^n$ denote the closed unit ball in $\Bbb R^n$. In multiple sources proving Brown's generalized Schoenflies theorem (including a version in the original paper), the following consequence of ...
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1answer
27 views

1-surgery on the figure-eight knot: reference request

As far as I know, 1-surgery on the figure-eight knot gives ($\pm$) the Brieskorn sphere $\Sigma(2,3,7)$. However, is there a citeable source for this? Sometimes Thurston's notes are mentioned, but I ...
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2answers
42 views

Orbits of mapping class group on four-punctured sphere

Let $\mathcal{M}_{0,4}$ be the mapping class group of the four-punctured sphere $S_{0,4}$. Denote the simple closed curve around boundary components $b_1$ and $b_2$ by $x$, the one around $b_2$ and ...
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199 views

Question about closed sets

Let $A$ and $B$ be subsets of $\mathbb{R}^n$ (where $\mathbb{R}^n$ is Euclidean n-space). Define $A + B = \{ x + y : x \in A , y \in B \}.$ Now If $A$ and $B$ are closed sets, is $A+B$ also a closed ...
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1answer
105 views

How equivalent are the homeomorphism and diffeomorphism groups of 3-manifolds?

Every topological 3-manifold carries a smooth structure, unique up to diffeomorphism. In addition, "up to homotopy", the maps between them are the same: every continuous map is homotopic to a smooth ...
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33 views

Decomposition of ball in Banach Tarski paradox and covering a soccer ball

Banach Tarski paradox says that it's possible to decompose a ball in $R^3$ into a finite number of disjoint subsets, which can be then reassembled into 2 identical copies of the original ball. ...
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61 views

Parallelizing lines

Let $n \geq 1$ be an integer, and $L_1,\ldots,L_n$ be $n$ lines in $\mathbb{R}^3$ which are pairwise disjoint. Is it possible to move all $n$ lines continuously so that they never cross, and so as to ...
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1answer
163 views

Covering spaces!

If one wanted to find all connected covering spaces of a product of two spaces, say $S^1\times RP(3)$, how would you go about it? I'm thinking finding the fundamental group of $S^1\times RP(3)$, and ...
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1answer
33 views

Are knot complements prime 3-manifolds?

Well, that is basically the question. Is the complement of a knot, i.e. an smoothly embedded copy of $S^1$ in $S^3$ a prime $3$-manifold? Here I mean by prime: A connected $3$-manifold $M$ is prime ...
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401 views

Can one cancel $\mathbb R$ in a bi-Lipschitz embedding?

Let $X$ be a metric space. Suppose that the product $X\times\mathbb R$ admits a bi-Lipschitz embedding into $\mathbb R^{n}$. Does it follow that $X$ admits a bi-Lipschitz embedding into $\mathbb ...
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2answers
88 views

A complex line bundle is trivial if and only if the first Chern class is zero

Let $\xi$ be a complex line bundle over a CW-complex $B$. I want to prove that $\xi$ is trivial if and only if $c_1(\xi)=0$. My attempt: Suppose $c_1(\xi)=0$. Then the Euler class $e(\xi)=0$. Since ...
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0answers
31 views

Determining slice knots

Lately I have been thinking about slice knots. Is there any known effective procedure for determining whether a knot is a slice knot?
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62 views

A homotopy equivalence between two sets

I was trying to prove that the set consisting of the union of the circles $\{\langle x,y\rangle\mid(x-10)^2 +y^2 = 1\}$, $\{\langle x,y\rangle\mid(x+10)^2 +y^2 = 1\}$ and line segment $\{\langle x, ...
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1answer
79 views

Stiefel-Whitney class of complex projective spaces

Let $T\mathbb{C}P^m$ be the tangent bundle of complex projective space. What is the total Stiefel-Whitney class $w(T\mathbb{C}P^m)$? Let $a_m$ be the maximal integer such that the $a_m$-th dual ...
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27 views

Jordan curve interior and curvilinear coordinates

One of the popular ways of evaluating areas of some plane subsets is change of variables. One classic example is the lemniscate of Bernoulli $\gamma: (x^2+y^2)^2=x^2-y^2, x>0$. Using polar ...
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37 views

Side identification on a hexagon

Apparently giving a hexagon side identification aabbcc results in a sphere. I'm struggling to see this, can someone explain? perhaps with a diagram? It seems to be all the vertices are identified, but ...
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24 views

Euler Characteristic of Side identification

Let $S$ be a surface obtained by identifying the sides of a regular hexagon in pairs. I want to show $\chi(S) > -1 $. I can see how we can obtain surfaces with $\chi(S) = 0,1,2$ but I think I'm ...
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1answer
35 views

Is the closure of $[\sigma_1^2,\sigma_2^2]$ in $B_3$ equal to the Borromean rings?

Is the closure of $[\sigma_1^2,\sigma_2^2]\in B_3$ (the braid group with $3$ strings) equal to the Borromean rings? If yes, is there any simple proof?
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1answer
33 views

Description of real projective space $P^3$

I know that the real projective plane $P^2$ can be thought of as a union of a mobius band and a disk, where the union occurs among the common boundary of the two (circle). My question is about $P^3$. ...
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1answer
100 views

Torsion Subgroup of Mapping class group.

What is the cardinality of finite order elements in Mapping class group of a surface $S_{g,n}$ of genus g and n boundary components. 1) If it is infinite then how can I generate a collection of ...
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4answers
596 views

Embedding compact (boundaryless?) n-manifolds in n-dimensional real space

I know the embedding theorems that allow you to embed $n$-manifolds into $\mathbb{R}^k$, provided $k$ is chosen large enough. Here I'm interested in the possibility of taking $k=n$ in the case of ...
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2answers
63 views

A reference for a result by A. Casson

I was reading this article about the disproof of Triangulation conjecture: it says that A. Casson disproved this conjecture in dimension 4 in the '80s In 1982, Michael Freedman, then at the ...
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1answer
47 views

Several questions concerning Alexander's Theorem

I'm reading Hatcher's proof of Alexander's theorem in his 3 manifolds notes. The statement is the following: Let $\Sigma \subset \mathbb{R}^3$ be an embedded $2$-sphere; then $\Sigma$ bounds a ...
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29 views

Zeros of vector field extended from field lines

Suppose I have a finite number of 1-dimensional curves embedded in a 3-dimensional space. What can I say topologically about vector fields chosen so that these closed curves are field lines? For ...
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1answer
27 views

Framing of embedding induces an isotopy of embeddings

Let $M$ be a smooth manifold of dimension $m$ and $\phi : S \to M$ a smooth embedding (dim S = k < m) such that the normal bundle $T_S M$ is trivializable. Let $f: T_S M \to S \times ...
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1answer
108 views

About generalized Schoenflies problem in the smooth category

Reading some books and comparing with Wikipedia I found some different statements about how the smooth Schoenflies problem is solved in high dimension, and I wanted to know which one is the correct ...
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65 views

Homotopically equivalent manifolds and product with $\mathbb{R}$

I know that some manifolds which are homotopically equivalent become homeomorphic after taking the product with $\mathbb{R}$, e.g. $\mathbb{T}^{2}$ minus a point and $\mathbb{S}^{2}$ minus three ...
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1answer
28 views

punctured Mobius band in high dimension

Let $M^{n+1}$ be the non-trivial line bundle over sphere $S^n$. When $n=1$, $M^2$ is Mobius band and the punctured space $M^2\setminus *\simeq S^1$. How about $M^{n+1}\setminus *$ in general? Does ...
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1answer
102 views

Mapping Class Group

$\newcommand{\MCG}{\mbox{MCG}}$Let $\alpha$, $\beta$ be non-isotopic, non-separating curves on a surface $S$ (meaning that "cutting along " them will not disconnect the surface). How do we show that ...
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1answer
1k views

Understanding the Equation of a Möbius Strip

I am in HL Math and trying to finish my IA. My topic is the Möbius band. The only problem is, I do not understand the formula that defines it and everywhere I have looked has just given me a ...
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16 views

given a specific vector bundle how to see whether the first Pontryagin class is zero

Let $Z_2$ be the group with $2$ elements. Let $a\in Z_2$ be the nontrivial element. Let $S^n$ be the $n$-sphere. Let $C(S^n,2)=\{(x,y)\in S^n\times S^n\mid x\neq y\}$. Let $a$ act on $C(S^n,2)$ by ...
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45 views

Maximum principle of harmonic function on compact manifold

Thm . (Maximum Principle) Let h be a harmonic function on a domain D in C . (a) If h attains a local maximum in D then h is constant. (b) Suppose that D is bounded and h extends continuously to the ...
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2answers
86 views

Are $\{(x,y)\in \mathbb{R}^2 : (x,y)\neq(0,0)\}$ and $\{(x,y)\in \mathbb{R}^2 : (x,y)\notin [0,1]\times\{0\}\}$ homeomorphic?

Let $X_1$ and $X_2$ be the spaces \begin{align*} X_1&=\{(x,y)\in \mathbb{R}^2 : (x,y)\neq(0,0)\}, \\ X_2&=\{(x,y)\in \mathbb{R}^2 : (x,y)\notin [0,1]\times\{0\}\}. \end{align*} Are these ...
3
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1answer
23 views

how to see whether a bundle is trivial or not?

Let $Z_2$ be the group with $2$ elements. Let $a\in Z_2$ be the nontrivial element. Let $S^n$ be the $n$-sphere. Let $C(S^n,2)=\{(x,y)\in S^n\times S^n\mid x\neq y\}$. Let $a$ act on $C(S^n,2)$ by ...
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1answer
56 views

triviality of tensor product of vector bundles

Let $\xi$ be a $O(n)$-bundle with fibre $\mathbb{R}^n$. Let $\xi\otimes \mathbb{C}$, $\xi\otimes \mathbb{H}$ be complex vector bundles and quaternionic vector bundles. If $\xi$ is not a trivial ...
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0answers
25 views

principal bundle and the associated bundle

Let $G\leq O(n)$ be a subgroup of orthogonal group. Let $\xi$ be a principal $G$-bundle. Let $\xi[\mathbb{R}^n]$ be the associated vector bundle. If $\xi$ is not a trivial bundle, can we obtain that ...
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40 views

What is the relation between the construction of Reshetikhin-Turaev and Witten's paper?

I have read that the papers of Reshetikhin and Turaev provide a mathematically rigorous framework for the ideas contained in Witten's paper "Quantum field theory and the Jones polynomial". Can ...
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1answer
29 views

a question on oriented bundles and Euler class

In Characteristic classes, J. Milnor, J. Stasheff, Prop. 9.7, it is proved that: if the oriented vector bundle $\xi$ possesses a nowhere zero cross section, then the Euler class $e(\xi)=0$. I want ...
2
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1answer
151 views

Software for drawing braid-related graphs

I am looking for a (preferably free) software that can draw braid-related graphs, such as and (Quoted from A Study of Braids By Kunio Murasugi, B. Kurpita) I have seen this question, but the ...
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1answer
21 views

How to prove that the minimum parameters required is equal to the dimension of an object?

Based on my (very limited) understanding, if an object needs to be expressed in terms of at least $n$-parameters, then it is an $n$-dimensional object. You can parameterise a circle as: ...
2
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1answer
34 views

ways to see whether the Pontryagin class of a quaternionic line bundle over a CW-complex is zero

the first pontryagin class of a quaternionic line bundle over a CW-complex is zero if and only if the quaternionic line bundle is trivial or not? Let $\xi^\mathbb{H}$ be a given quaternionic line ...
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1answer
294 views

how to prove that quotient group $\mathbb{R}/\mathbb{Z}$ and the circle are diffeomorphism?

Are quotient group $\mathbb{R}/\mathbb{Z}$ and the circle are diffeomorphism? How to prove? Hope someone give me some advise or some reference documents. Thank you.
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53 views

Is every $S^1$ knot orientable?

let $K$ be an $S^1$ knot, i.e., $K$ is a homeomorphic copy of the "standard" $S^1$ living in $\mathbb R^3$ ). I am trying to see if $K$ is orientable. This is what I have: 1) If the map sending $S^1$ ...