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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4
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2answers
189 views

What is smoothness needed for?

We can either define a knot to be (1) a smooth embedding $S^1 \hookrightarrow \mathbb R^3$ or (2) a piecewise linear, simple closed curve in $\mathbb R^3$ Then these two definitions are ...
4
votes
0answers
78 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
155 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 ...
8
votes
0answers
261 views

When are maps between topological manifolds automatically surjective?

Take an injective (continuous) map $T:\mathbb{S}_1\to\mathbb{S}_1$. It's an obvious fact (though I can only prove it with nontrivial facts about homotopy) that $T$ is automatically surjective. I have ...
4
votes
1answer
99 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 ...
8
votes
4answers
660 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 ...
3
votes
2answers
258 views

line equidistant from two sets in the plane

Suppose $A$ and $B$ are disjoint subsets of the plane, both closed, nonempty, and connected. Define $E(A, B)$ as the set of points in the plane equidistant from $A$ and $B$. For example, if $A$ is a ...
1
vote
0answers
57 views

ENR spaces which group of homeomorphisms is not locally contractable

Siebenmann on page 1 of his manuscript gives an example of an ENR space (euclidean neighborhood retract) whose group of homeomorphisms is not locally contactable. Take the sphere ...
4
votes
2answers
571 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: ...
2
votes
1answer
55 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
640 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 ...
1
vote
1answer
103 views

a question about connected sum

I know this probably has a really straightforward answer, specially if it is as standard as it is entuitive to visualize. Still, because i'm not experienced at all on working with this objects and im ...
2
votes
1answer
131 views

Could a surface bundle over a circle have free fundamental group?

Specifically, I was wondering if the surface was non-compact with infinitely generated free fundamental group, could the surface bundle itself have infinitely generated free fundamental group. In this ...
10
votes
2answers
397 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 ...
4
votes
0answers
245 views

Partitioning a triangulated 2-sphere into two triangulated discs

Take a triangulation of the 2-sphere, $S^2$. Let the triangulation be denoted $T$. The Euler characteristic tells you that the number of triangles in $T$ is even. Since triangulations of the ...
1
vote
1answer
68 views

Why can we consider only torus switch rather than arbitrary Dehn filling in Lickorish theorem?

I understand that any orientable $3$-manifold can be obtained by doing Dehn surgery on $S^3$ along a set of circles sitting in it; but why can we further assume the slop to be $0$, i.e. we can obtain ...
1
vote
1answer
189 views

Given lattice G; parameters of torus R^2/G?

This should be a simple, known result, but I can't seem to find it. Given a lattice $\Gamma = m\mathbb{Z} \times n\mathbb{Z}$, $\mathbb{R}^2/\Gamma$ is topologically a torus. For suitable $m$ and ...
5
votes
2answers
135 views

Are bounded open regions in $\mathbb{R}^n$ determined by their boundary?

Let $U$ and $V$ be two bounded open regions in $\mathbb{R}^n$, and let us further assume that their topological boundaries are nice enough that they are homeomorphic to finite simplicial complexes. ...
0
votes
0answers
49 views

Spinning construction

The p-spinning construction works as follows: You start with $A$ an $n$-ball embedded in an $m$-ball $B$ such that $\partial A \subset \partial B$ and $A^\circ \subset B^\circ$. The $p$-spin of ...
2
votes
1answer
124 views

Morton Brown's theorem

Suppose that a topological space $X$ is the union of an increasing sequence of open subsets $U_i$c each of which is homeomorphic to the Euclidean space $\mathbb{R}^n$. How does one show that $X$ is ...
4
votes
1answer
110 views

Isotopy to the identity on disk

Let $D^2 \subset \mathbb{R}^2$ the unit disk and $f: D^2 \rightarrow D^2$ a homeomorphism with the property that $f$ restricted to the boundary $\partial D^2$ is the identity. Then $f$ is ambient ...
1
vote
1answer
67 views

A knot which intersects $S^2$ transversely once in 3-connected manifold

I am reading the paper,ON ATTACHING 3-HANDLES TO A 1-CONNECTED 4-MANIFOLD by BRUCE TRACE here.He says in this paper that we need only construct a knot $K\subset \partial W^4$ which meets $Σ ^2$ ...
1
vote
0answers
101 views

About manifolds after attaching handles.

I'm reading R.E. Gompf and A.I. Stipsicz, 4-Manifolds and Kirby Calculus. I don't understand Remarks 4.4.1 on page 116-117 Google books here. At first I can't understand why we take immersed disk $D ...
4
votes
1answer
160 views

About Kirby Diagrams

I'm reading R.E. Gompf and A.I. Stipsicz, 4-Manifolds and Kirby Calculus. There is something I don't understand on page 116 (Google Books link to page 116; alternatively, here are images of page 115 ...
15
votes
1answer
428 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 ...
3
votes
0answers
197 views

Hairy ball theorem: references to applications

I'm looking for references to applications of the Hairy ball theorem. I already visited wikipedia and cited references, but I need a little more explanation in both meteorology and applications in ...
6
votes
1answer
251 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 $g$ ...
11
votes
1answer
450 views

Covering points on a sphere with a disk

Suppose $m$ points ("sites") are selected on the unit sphere $S^2$. For a given radius $r < \pi$, we can define a disk around any point on the sphere as the set of points at geodesic distance at ...
3
votes
0answers
119 views

Multiple Dehn twists and minimal position

I have a question about a proof that I am reading in "A primer on Mapping Class Groups" by Farb and Margalit. Let $a$ be a simple closed curve in a compact surface $S$ (possibly with marked points ...
2
votes
1answer
158 views

Bigon question related to Dehn twists

Perhaps someone can help me with this: For simple closed curves on an orientable compact surface, if they form a bigon, then is it true that at the intersection points the orientations must be ...
8
votes
1answer
616 views

For an $n$-dimensional object, how many types of holes are possible?

Update 2012-06-06: At some point I'll attempt to answer my own question by using a dual-fluid model that places the dimensionality and connectivity of "solids" and "holes" on an equal footing. With ...
2
votes
1answer
130 views

Poincaré conjectures for other shapes

If you replace spheres in the Poincaré conjecture with objects-with-some-kinds-of-holes can you say that every manifold with the same number and type of holes is homeomorphic to every other such ...
3
votes
1answer
182 views

Topological shape of sphere-like equations with complex radius

For fixed complex number $s≠0$, what 4-dim shape is given by complex solutions $z,w$ of $z^2+w^2=s^2$ and higher dimensional version 2N-dim shapes of $z_1^2+z_2^2+...+z_N^2=s^2$ ?
11
votes
1answer
711 views

Why is the knot group of the trefoil isomorphic to the group of 3-braids?

I apologise in advance for the vagueness of this question but I have not been able to find very much info on the topic and have made very little progress on my own. I am trying to understand why the ...
3
votes
0answers
132 views

Something like the weak Whitney embedding theorem for continuous maps and homotopy.

This is sort of a reference request. Consider a continuous map of orientable topological manifolds $f:N\longrightarrow M$ of dimension $n$ and $m$ respectively. I have been told that there is a ...
3
votes
1answer
97 views

Remove links by Kirby moves

I am trying to prove the following proposition. proposition; If in a framed link $L$ a component $K$ is an unknot with framing zero which links only one other component $H$ geometrically once, ...
5
votes
1answer
140 views

Status of PL topology

I'm starting to learn about geometric topology and manifold theory. I know that there are three big important categories of manifolds: topological, smooth and PL. But I'm seeing that while topological ...
12
votes
1answer
616 views

Gluing a solid torus to a solid torus with annulus inside.

I was thinking the fact that if two genus $1$ handlebodies (solid tori) are glued via an orientation preserving homeomorphism of boundaries, the resulting manifold depends only on (up to isotopy) ...
4
votes
1answer
384 views

How does handle attachment work in Morse Theory

I am reading R.E.Gompf and A.I. Stipsicz, 4-Manifolds and Kirby Calculus. I can't understand the 2nd-paragraph of p.101, where they explain framings on the attaching sphere. In particular I cannot ...
2
votes
1answer
128 views

Various types of TQFTs

I am interested in topological quantum field theory (TQFT). It seems that there are many types of TQFTs. The first book I pick up is "Quantum invariants of knots and 3-manifolds" by Turaev. But it ...
3
votes
1answer
583 views

Showing a bijective, continuous function between connected, locally euclidean spaces is a homeomorphism.

This question comes from Conlon's Differentiable Manifolds (it's Exercise 1.1.13). Let $X$ and $Y$ be connected, locally Euclidean spaces of the same dimension. If $f:X \rightarrow Y$ is bijective ...
12
votes
4answers
1k views

Reference on Geometric Topology

Geometric topology is more motivated by objects it wants to prove theorems about. Geometric topology is very much motivated by low-dimensional phenomena -- and the very notion of low-dimensional ...
5
votes
1answer
873 views

Homeomorphism vs. Homotopy (Equivalence)

Trying to brush up on some geometric and algebraic topology, I got a little confused about the following: Suppose we have the standard unit sphere $S^2$, but we remove its north and south poles. Is ...
4
votes
0answers
152 views

Can the disk bundle associated to a vector bundle over a finite CW-complex be obtained by attaching cells?

Let $V$ be a vector bundle (with a chosen metric) over a finite CW-complex $X$ and $B(V), S(V)$ the associated (unit) disk resp. sphere bundles. In the paper Clifford-Modules the authors remark that ...
5
votes
0answers
150 views

Heegaard Splitting of Brieskorn homology 3-spheres

For pairwise coprime integers $(p,q,r)$, we define a Brieskorn homology 3-spheres by $\Sigma(p,q,r)=\{(z_1,z_2,z_3)\in\mathbb{C}^3 | z_1^p+z_2^q+z_3^r=0, |z_1|^2+|z_2|^2+|z_3|^2=1\}$. I want to know ...
1
vote
1answer
277 views

Pinched torus generalization

The pinched torus is homeomorphic to a sphere with two (different) points identified.           What is the name and topological structure of the ...
5
votes
0answers
242 views

Points in the plane at integer distances

Does there exist a set of $n$ points $p_1,p_2,...,p_n$ in the plane, all at mutual integer distances to each other, and an $e>0$, such that the following statement holds: For all $a,b$ with ...
3
votes
3answers
584 views

Orientation reversing diffeomorphism

Why each sphere admits an orientation reversing diffeomorphism onto itself? (For even dimensional ones can we take conjugation map?) And why complex projective spaces do not admit? Is there a ...
9
votes
1answer
689 views

Understanding the Hopf fibration

I'm taking a class in manifolds, and the Hopf fibration recently came up. I'm trying to get a handle on it, so I'm going to try and explain what I think is going on, and hopefully math.stackexchange ...
1
vote
1answer
281 views

Finding a one-to-one correspondence from the middle third Cantor set $C$ to the unit sphere $S^2$

I'm familiar with the fact that, if I'm not mistaken, there is a one-to-one correspondence between the unit interval $[0, 1]$ and the unit sphere $S^2$ though I'm not sure explicitly how to find it. ...