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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10
votes
2answers
396 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
242 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
67 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
183 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
109 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
159 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
426 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 ...
2
votes
0answers
195 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
248 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
449 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
155 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
600 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
181 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
702 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
130 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
95 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
611 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
379 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
571 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 ...
11
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
864 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
150 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
148 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
271 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
239 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
579 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
683 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. ...
1
vote
0answers
128 views

Topology of “almost convex” sets

Let $C$ be a compact set in $\mathbb{R}^n$. If $C$ is convex, it must be homeomorphic to a closed ball. Now suppose that instead of convexity we require the intersection of $C$ with any line to have ...
3
votes
1answer
235 views

Attaching Two Discs Along the Boundary

Let $A := \mathbb{S}^1 \subset \overline{\mathbb{B}^2}$, and let $f : A \hookrightarrow \overline{\mathbb{B}^2}$ be the inclusion map. Consider the adjunction space $\overline{\mathbb{B}^2} \cup_f ...
3
votes
0answers
70 views

Are PL maps determined by PL-paths?

Let $X,Y$ be polyhedra. If $f\colon X\rightarrow Y$ is a PL map and $g\colon I\rightarrow X$ is a PL map (where $I$ denotes the interval), then $f\circ g$ is a PL map. Is the converse true? i.e. ...
14
votes
4answers
663 views

Useful fibrations

What are the most useful fibrations that one be familiar with in order to use spectral sequences effectively in algebraic topology? There's at least the four different Hopf fibrations and $S^1\to ...
4
votes
1answer
104 views

Upper bound on the number of charts needed to cover a topological manifold

If $M^n$ is a compact topological manifold (not necessarily with additional structure), is there an upper bound on the number of charts needed to cover $M$ ? Does this bound depend on the dimension of ...
2
votes
3answers
204 views

Real analytic diffeomorphisms of the disk

Is there any real analytic diffeomorphism from two dimensional disk to itself, except to the identity, such that whose restriction to the boundary is identity?
1
vote
2answers
229 views

When does an embedded $2$-torus bound a solid torus in $3$-manifolds?

This is a simple version of the question asked here. Let $M$ be a compact 3-manifold without boundary, $\mathbb{T}^2$ be the standard 2-torus, and $i:\mathbb{T}^2\to M$ be an embedding. (*) Assume ...
6
votes
2answers
379 views

Euler characteristic 1: Half a hole?

The Euler characteristic of a two-dimensional disk is $\chi=1$. If one blindly interprets the disk as a closed, orientable surface, then $\chi = 2 - 2g$, and the genus is $g=\frac{1}{2}$. Is there ...
3
votes
1answer
128 views

Planar kelvin problem

What is the minimal possible value of the maximal total side length shared by any two tiles in a tiling of the plane if all tiles have the same area $A$? $\text{Total side length} = ...
19
votes
1answer
433 views

Decomposition of a manifold

As a kind of aside to this question, where one of the answers assumed that if $S^n=X \times Y$ then we can assume that $X$ and $Y$ are manifolds. If we have a manifold $M$, such that $M$ is ...
8
votes
2answers
120 views

$n$ points on every line

For which integers $n$ is it possible to find a subset $S$ of $\mathbb R^2$ such that every infinite line contains exactly $n$ points of $S$?
2
votes
2answers
172 views

Relationship between the $2$-plane bundles over $S^2$ and $\mathbb{Z}$

I want to follow up on this answer by asking a few more questions (posting directly on the question didn't seem to "bump" the thread). I was trying to read the referenced text (Husemoller's Fiber ...
7
votes
1answer
285 views

How to Classify $2$-Plane Bundles over $S^2$?

I'm curious how one can classify the bundles over a given manifold. I recently read this paper on classifying $2$-sphere bundles over compact surfaces. A lot of the concepts went over my head since ...