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 ...

learn more… | top users | synonyms (1)

1
vote
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 ...
3
votes
3answers
196 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 ...
0
votes
2answers
41 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 ...
1
vote
0answers
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. ...
8
votes
1answer
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 ...
2
votes
1answer
31 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 ...
0
votes
0answers
27 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?
3
votes
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 ...
2
votes
0answers
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, ...
1
vote
0answers
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 ...
10
votes
1answer
104 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 ...
13
votes
1answer
298 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 ...
1
vote
0answers
34 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 ...
0
votes
0answers
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 ...
1
vote
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?
4
votes
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 $P^3$. ...
1
vote
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 ...
2
votes
0answers
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 ...
1
vote
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 ...
3
votes
0answers
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 ...
0
votes
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 ...
3
votes
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 ...
0
votes
0answers
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 ...
1
vote
0answers
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 ...
7
votes
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
votes
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 ...
2
votes
1answer
55 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 ...
0
votes
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 ...
3
votes
0answers
39 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 ...
1
vote
1answer
28 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 ...
4
votes
2answers
83 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 ...
2
votes
1answer
150 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 ...
0
votes
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
votes
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 ...
1
vote
0answers
51 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 $S^1$ ...
0
votes
0answers
29 views

Homology of Subspace vs. Homology of Ambient Space.

Let $M$ be a manifold embedded in $\mathbb R^n$ , so that the manifold has non-trivial $k-th$ homology for some $n \geq k\geq 0$ . How do we identify the fact that while there is a non-trivial cycle ...
0
votes
2answers
86 views

Showing that stereographic projection is a homeomorphism

For any $n\geq 0$,the unit $n$-sphere is the space $S^{n}\subset \mathbb{R^{n+1}}$ defined by $$S^{n}=S^{n}(1) :=\left\{ (x_{1},...,x_{n+1}) \left\vert\,\sum_{i=1}^{n+1} x_{i}^{2}=1\right.\right\}$$ ...
2
votes
1answer
32 views

Homeomorphisms of different spheres

It is pretty straightforward to show that if $X$ is a Banach space under two equivalent norms, then the respective open unit balls $B_1$ and $B_2$ are homeomorphic only by showing that $B_i$ is ...
0
votes
1answer
18 views

geometric triangle parallel

$DC$ is parallel to $AB$. Find the value of $BE$ and $DC$. I've tried too many times but still can't figure it out.
0
votes
0answers
13 views

Geometric Parallel Triangle Find Value

DC parallel to AB Find Value of BE and DC. I've tried too many times but still can't figure it out.
1
vote
1answer
56 views

Completely self-contained (and as elementary as possible) introduction to Teichmuller Theory

Can you recommend a completely self-contained and elementary (as much as it can be) introduction to Teichmuller Theory?
1
vote
0answers
16 views

use homology groups to obtain minimal cell structure

On Hatcher's book Algebraic Topology, Section 4.C Prop. 4C.1, for a simply-connected CW-complex $X$, if $H_*(X;\mathbb{Z})$ is known as a graded module over $\mathbb{Z}$, then the minimal cell ...
0
votes
1answer
25 views

a generalization of punctured cylinder

Let $S^1\times \mathbb{R}$ be the infinite cylinder. Pucture it, we have $S^1\times \mathbb{R}-*$. Then $(S^1\times \mathbb{R}-*)\simeq Skeleton^1(S^1\times \mathbb{R})\simeq S^1\vee S^1$. How ...
3
votes
1answer
36 views

Example expectation of an exponential function

Given a geometric random variable $Y$ with $p = 1/3$, I know that $E[Y] = 1/p = 3$. However, what is $E[e^{aY}]$ ? for a small value $a$. Thanks
0
votes
1answer
28 views

Shapes of a simplicial complex

In Bridson and Haefliger's book, page 98, there is a definition of shape. Here is a link to the book. The definiton is not very clear to me. It says set of isometry classes: Is it the isometry ...
0
votes
0answers
23 views

Simplicial complex and link

Theorem 7.16 in Bridson and Haefliger's book states that: Theorem: Let K be an $M_k$-simplicial complex, and let $x\in K $. If the number $\epsilon(x)$ defined in (7.8) is strictly positive, ...
0
votes
1answer
46 views

What is the definition of a properly-immersed arc?

I am going over a paper in which a theorem includes a condition that certain arcs are "properly-immersed geodesic arcs" in a surface $\Sigma$ with nonempty boundary. I have done a search, but I have ...
4
votes
2answers
102 views

Homomorphism between homotopy groups of spheres induced by the fibration and the multiplication map of $SO(n)$

Let $n$ be a nonnegative integer and $x\in S^n$ a point in the n-sphere. Combining the map $\alpha\colon SO_{n+1}\longrightarrow S^n$ induced by matrix multiplication with $x$ and the connecting ...
0
votes
0answers
27 views

Right Veering Property of elements in MCG(S)

Let h be an element of MCG(S), the mapping class group of a surface S. I was going over : I was going over :Geometric Intuition for "Right-Veering" Property of $f$ in MCG(S)? Where a p.e ...
2
votes
1answer
52 views

Homeomorphism type of the cone on a cylinder

Let $X$ be a topological space. The cone $CX$ on $X$ is the cylinder $X \times I$ with the top $X \times 1$ identified to a point. Clearly for every $X$, $CX$ is contractible. Looking at the ...