1
vote
0answers
48 views

A Homeomorphism that is not unique even upto Isotopy

I'm currently reading the following paper by Richard Skora, entitled Cantor sets in $S^3$ with simply connected complements found here, and on page 2, just before Theorem 1, it says "the homeomorphism ...
1
vote
2answers
24 views

Questions about the Nature of Chirality (with some focus on dimensionality)

Are all chiralities the same? (Not in the sense of "is the right hand the same as the left hand?" but in the sense of "is the way in which X is chiral the same (or negative of the same) as the way in ...
8
votes
1answer
209 views

Surgery results in a cylinder

While reading a proof of a theorem about Reshetikhin Turaev topological quantum field theory, I encountered the following problem. Suppose we have several unlinked unknots $K_i$, $i=1, \dots, g$ in ...
1
vote
1answer
65 views

Number of connected components of this complement

Let $X$ be a locally finite simplicial complex and let $K$ be a finite subcomplex of $X$. Why is the number of connected components of the complement $X-K$ finite?
5
votes
2answers
197 views

Does the Euler characteristic of a manifold depend upon the field of coefficients?

Define the Euler characteristic of a space $X$ to be $$\chi(X)= \sum_i \dim H_i(X, \mathbb Q)$$ This is obviously not necessarily well-defined for an arbitrary space $X$, so let $X$ be a manifold ...
1
vote
0answers
63 views

Homeomorphic or Homotopic

Q1: Are the Fig (a) and (b), the equivalence "=" is Homeomorphic or Homotopic? ps. for details of the figures see Ref here. I learned that "The characterization of a homeomorphism often ...
1
vote
1answer
47 views

Projection onto a convex closed set

H, If $K$ is a non-empty convex and closed subset of a uniformly convex Banach space $X$ (Hilbert for example) and $v \notin K$, we know that there exists a unique $k_0\in K$ such that ...
0
votes
1answer
50 views

Extending homeomorphism of unit circle to unit disk

What is the best best way to prove that any homeomorphism of the unit circle onto itself can be extended to a homeomophism of the closure of the unit disk onto itself?
0
votes
0answers
24 views

History of vectorial bundles in articles or papers?

I'm looking for an article or book that gives a thorough and interesting history of bundles and vectorial bundles in algebraic topology. I'm looking for it for my own learning, please help its ...
1
vote
0answers
28 views

Open Book Decompositions of 3-manifold and Associated Heegard Splittings

In page 13 of the paper: http://arxiv.org/pdf/math/0510639v1.pdf It is stated that "An open book decomposition (S,h, K) , gives rise to a special Heegard decomposition of M ". Here, S is a surface , ...
0
votes
1answer
57 views

Manifolds Resulting from Gluing Tori

I'm trying to show that if solid tori $T_1, T_2; T_i=S^1 \times D^2$ ,are glued by homeomorphisms between their respective boundaries, then the homeomorphism type of the identification space depends ...
3
votes
2answers
44 views

Identification Space and Isotopy

Original Question: Let $X$ and $Y$ be topological spaces and let $f:X \to Y$ and $g:X \to Y$ be isotopic embeddings. Is it true that $X \cup_f Y$ is homeomorphic to $X \cup_g Y$? Edit: I meant to ...
0
votes
1answer
50 views

How to enclose a ball more than once with a surface homeomorphic to $S^2$? In 3D.

In 3 dimensional space, how to enclose a monopole (either point-like or ball-like, both types exist.) more than once with a surface homeomorphic to $S^2$?
2
votes
1answer
104 views

The image of homomorphism of fundamental group of closed surface

$\phi: \pi_1(S)\to \pi_1(S)$ is a homomorphism of fundamental group of closed orientable surface $S$ of genus $\geq 2$. If $\phi$ is not an epimorphism, can we find a non-surjective self map $f: S\to ...
3
votes
2answers
355 views

Simply connected does not imply contractible. Is there a nice counter example in $R^2$?

The standard counter example to the claim that a simply connected space might be contractible is a sphere $S^n$, with $n > 1$, which is simply connected but not contractible. Suppose that I were ...
3
votes
2answers
140 views

Generalization of the hairy ball theorem.

The hairy ball theorem of states that there is no nonvanishing continuous tangent vector field on even dimensional n-spheres. Can the hairy ball theorem be strengthened to say that there is no ...
14
votes
0answers
269 views

In $n>5$, topology = algebra

During the study of the surgery theory I faced following sentence: Surgery theory works best for $n > 5$, when "topology = algebra". I don't know what is the meaning of topology=algebra. ...
1
vote
1answer
62 views

Lifting homeomorphisms covering

Hello I had a question regarding a lemma from the paper: http://www.math.columbia.edu/~jb/bir-hilden-annals.pdf I don't understand the proof of Lemma 5.1. Notation: $T_{0,0}$ is the 2-sphere, ...
3
votes
0answers
70 views

What is the “Standard” Open Book Decomposition for $\mathbb R^n$, and why does this matter?

I am trying to understand better Open book decompositions. To that effect, I tried to work out a couple of (relatively-simple) examples, specifically, for $\mathbb R^2 $ and higher. But I have not ...
0
votes
1answer
45 views

Prove $f\colon X/{\sim} \to Y \text{ is continuous} \iff \pi\circ f\colon X \to Y \text{ is continuous}$

I need to show that $$f\colon X/{\sim} \to Y \text{ is continuous} \iff \pi\circ f\colon X \to Y \text{ is continuous}$$ where $X/{\sim}$ is a quotient topology and $\pi$ is the quotient map. I ...
1
vote
1answer
70 views

How can you prove that the winding number around two zeros of a vector field is the sum of the two indices?

If v is a continuous vector field with two isolated zeros, then the winding number around one zero is its index. The winding number on a circle with both zeros in its interior is the sum of the two ...
5
votes
3answers
132 views

when is the region bounded by a Jordan curve “skinny”?

How can I formalize and prove the following intuition?: Picture a very skinny rectangle, one with base length 1 and sides length $\epsilon$. Or imagine a very flattened ellipse. The interiors of ...
2
votes
1answer
80 views

Density of continuous knots in the plane transversal to some circles

This is an exercise from the book "Knots and Links" by Rolfsen (exercise 6 in section 2C) Let $\kappa : S^1 \rightarrow \mathbb{R}^2-(0,0)$ be a continuous imbedding. Let $M := \{ x \in ...
2
votes
1answer
71 views

Uniqueness of “Punctured” Tubular Neighborhoods (?)

Here is a question that has been haunting me for a while: Let $\mathbb{R}^{n-1} \times [0, \infty)$ be the upper half space of $\mathbb{R}^n$ and suppose we have a smooth homeomorphism (not a diffeo) ...
3
votes
1answer
145 views

Annulus Theorem

I'm trying to read Rolfsen's "Knots and Links" and I'm a little discouraged that I can't do one of the first and seemingly more important exercises. The question is Use the Schoenflies theorem ...
11
votes
1answer
148 views

Equivalence of definitions of $S^\infty$

Consider the following two definitions of the infinity-sphere $S^\infty$. Why do they define homeomorphic spaces? $1)$ The set of points in $\mathbb R^\infty$ with distance $1$ from the origin. $2)$ ...
1
vote
1answer
129 views

Simple Sphere Suspension Question

I have heard it said that the suspension of an $n$-sphere is an $n+1$-sphere. This is stated without proof in chapter 0 of Hatcher's book on algebraic topology. More generally, it seems that the ...
11
votes
1answer
257 views

Space of homeomorphisms Homeo$(S^1)$ of $S^1$ deformation retracts onto $O(2)$

How can we prove that the space of homeomorphisms Homeo$(S^1)$ of $S^1$ (strong) deformation retracts onto the orthogonal group $O(2)$? I know that this result is proved by Hellmuth Kneser in his ...
1
vote
1answer
127 views

How is PL knot theory related to smooth knot theory?

I really want to like knot theory but the PL condition seems sort of ugly. I was hoping someone could give me a justification for secretly thinking about smooth knots as I read through a book like ...
2
votes
2answers
113 views

Compact subvarieties in $\mathbb{C}^n$

I ran across a statement, the maximum principle, which states $X\subset \mathbb{C}^n$ is compact in the Euclidean topology iff $X$ is a finite set of point. Unfortunately, a proof didn't come along ...
2
votes
0answers
98 views

Why does a circle cut a torus into an annulus?

Let $\phi : S^1 \rightarrow T^2$ be an (topological. Not necessarily smooth) imbedding of the circle in the 2-torus and let $\iota : S^1 \rightarrow T^2, \theta \mapsto (\theta,0)$ be the imbedding ...
1
vote
1answer
103 views

How to proof M is a compact connected n-manifold,if for any point $p\in M,M\backslash\{p\}\cong R^n$ then M is homeomorphic to $S^n$.

M is a compact connected n-manifold,if for any point $p\in M,M\backslash\{p\}\cong R^n$ then M is homeomorphic to $S^n$. I have the guess from ...
2
votes
0answers
104 views

Homeomorphism between simply connected, closed 3 - manifold and 3-sphere.

The Poincare conjecture states that a simply-connected, closed 3-manifold is homeomorphic to the 3-sphere. Now that the conjecture has been settled, could someone tell me what this homeomorphism is ...
1
vote
3answers
281 views

Infinitely sheeted covering spaces!

I was wondering what the fundamental group of an infinitely sheeted covering space of say some surface might be? I'm thinking it should be an infinite cyclic group, but this is more intuitively, i ...
6
votes
1answer
189 views

Orientability of Manifolds

Given that $f \colon \mathbb R^n \rightarrow \mathbb R$ is a smooth function and if $c \in \mathbb R$ is a regular value how would I go about showing that $f^{-1} (c)$ is an orientable manifold? ...
3
votes
0answers
132 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 ...
3
votes
0answers
71 views

Curvature on topological spaces

On what subsets of the category of topological spaces are different notions of curvature defined?
2
votes
3answers
132 views

Finding the rank of subgroups of free groups?

How do you find the rank of a subgroup(of finite index) of a free group? I was thinking of looking at the fundamental group of a graph.
0
votes
1answer
179 views

Higher homotopy groups!

How would you show that $\pi_n, n>1$ of the Klein bottle is the trivial group? I was thinking Seifert-Van Kampen could be applicable?
1
vote
2answers
138 views

intersection of decreasing path-connected spaces

If we have path-connected spaces $A_0 \supseteq A_1 \supseteq A_2 \supseteq \ldots$, is $\bigcap^\infty A_i$ path-connected? I was thinking that if we take $A_i$ to be a $1/i$-neighborhood of the ...
3
votes
0answers
166 views

Fundamental Group!

Say you have two surfaces of genus 2, say $X$ and $Y$ and you want to attach them via homotopy attaching maps $f$ along their waist curves. Then what will the fundamental group of the adjunction space ...
3
votes
0answers
68 views

Do we really need to use the Jordan-Schönflies Theorem to prove that every surface can be triangulated?

I have read that most proofs of the triangulability of surfaces require the use of the Jordan-Schönflies Theorem. However, is such high-tech machinery really needed? The problem is that 3-manifolds ...
4
votes
1answer
137 views

Prime decomposition of 3-manifolds

Let $H_g$ be a three dimensional handlebody bounded by a genus $g$ surface. Let $M_g$ be a manifold obtained by gluing two copies of $H_g$ via an orientation reversing homeomorphism of the surface of ...
4
votes
1answer
76 views

Gluing axiom of a TQFT

In the book, Lectures on tensor categories and modular functors by Bakalov and Kirillov they construct a TQFT. When they come to prove the gluing axiom, they just mention that "...This statement is ...
3
votes
2answers
377 views

Homotopy Question Help?

Let $X$ be a topological space and suppose $X_1$ and $X_2$ are spaces obtained by attaching an n-cell to $X$ via homotopic attaching maps. Show that $X_1$ and $X_2$ are homotopy equivalent. Proof: ...
0
votes
2answers
1k views

Retraction of the Moebius strip to its boundary

Prove that there is no retraction (i.e. continuous function constant on the codomain) $r: M \rightarrow S^1 = \partial M$ where $M$ is the Moebius strip. I've tried to find a contradiction using ...
2
votes
1answer
51 views

Nonhomeomorphic CW-complexes that are stably homeomorphic

Can one find two finite CW-complexes $X$ and $Y$ such that $X \times I$ is homeomorphic to $Y \times I$, where $I = [0, 1]$, but $X$ is not homeomorphic to $Y$? I know how to find such topological ...
3
votes
1answer
347 views

The Join of Two Copies of $S^1$

So I know the fact that the join of $S^1$ and $S^1$ is homeomorphic to the 3-sphere, but I'm having trouble "seeing" this. I'd prefer something that appeals to geometric intuition, but more formal ...
10
votes
2answers
390 views

$\{(x,y)\!\in\!\mathbb{B}^n; -\varepsilon\leq-\|x\|^2\!+\!\|y\|^2\leq\varepsilon\}\approx\mathbb{B}^k\!\times\!\mathbb{B}^{n-k}$

The question is motivated by the notion of handle attachment, Morse theory, critical points of index $k$, Morse lemma, sublevel sets, etc. For $0\!\leq\!k\!\leq\!n$ and ...
5
votes
1answer
198 views

Question on “up to isotopy” when attaching two spaces

Let $M$, $N$, $A$, $B$ be topological spaces (or manifold) such that $A$ and $B$ are subspaces in $M$, $N$ respectively. Let $f: A \to B$ and $g:A \to B$ be homeomorphism and assume that $f$ and $g$ ...