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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3
votes
1answer
228 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. ...
13
votes
4answers
594 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
99 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
201 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
213 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
371 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
121 views

Planar kelvin problem

What is the minimal possible value of the maximal total sidelength shared by any two tiles in a tiling of the plane if all tiles have the same area A? total sidelength = Length-integral of the curve ...
19
votes
1answer
429 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
119 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
163 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
269 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 ...
3
votes
1answer
505 views

the restriction of a homeomorphism on a subset

Let $X$ be a topological space and $f:X\to X$ be a homeomorphism. Then the induced map $f_1:\pi_1(X,x)\to\pi_1(X,fx)(\cong \pi_1(X,x))$ is an isomorphism (automorphism up to conjugate). In the ...
2
votes
3answers
142 views

Structure theorem for foliations on $\mathbb R$

In my research, it so happens that it might be useful to look at foliations on a surface. I am however having a hard time absorbing a new subject. So, could somebody point me to some suitable ...
1
vote
1answer
135 views

Number of components of complement to a reducible real algebraic hypersurface

Let $X_1,\ldots X_k$ be irreducible(may be singular) affine real algebraic hypersurfaces in $R^n$ with $x_1,\ldots, x_k$ connected components, respectively. Let $G_1,\ldots, G_l$ be their ...
7
votes
3answers
418 views

Putting Geometries on Knot Complements

I have two different, but related, questions about the type of geometry one can get on a knot complement. Quickly some notation: $K$ will be a non-trivial smooth knot - living in $S^3$ - and $M$ will ...
11
votes
1answer
284 views

Fractional versions of euclidean space?

This is going to be a somewhat vague question, but I'll be happy if you indulge me. Euclidean space $\mathbb{R}^n$ is equipped with a lot of nice (algebraic, metric, topological,...) structure and ...
3
votes
1answer
82 views

solid ball with “boundary modulo antipody”

Consider the standard solid ball $\{(x,y,z)\in \mathbb{R}^3\mid x^2+y^2+z^2\le 1\}$, and the equivalence relation doing nothing in the interior and identifying antipodal points of the boundary. What ...
2
votes
1answer
289 views

Proof: The complement of an annulus embedded in a sphere has two connected components

By the Jordan Curve Theorem we know that the complement of an $S^{n-1}$ embedded into the $S^n$ has exactly two connected components. What if -- instead of a sphere -- we embed an annulus, i.e. ...
27
votes
3answers
2k views

Why is the Jordan Curve Theorem not “obvious”?

I am horribly confused about Jordan's Curve Theorem (henceforth JCT). Could you give me some reason why should the validity of this theorem be in doubt? I mean for anyone who trusts the eye theorem is ...
5
votes
1answer
235 views

Tangent bundle of a noncompact surface

Let $\Sigma$ be a connected noncompact orientable surface. I'm not assuming that $\Sigma$ is of finite type or anything -- for instance, I'm allowing $\Sigma$ to be the $2$-sphere minus a Cantor set. ...
5
votes
2answers
549 views

Which mapping tori are Seifert manifolds?

According to Orlik's lecture notes on Seifert manifolds (and the Wikipedia page on Seifert fiber spaces), a mapping torus over a 2-torus is a Seifert manifold if and only if it is the mapping torus of ...
52
votes
1answer
3k views

Trigonometric sums related to the Verlinde formula

Original question (see also the revised, possibly simpler, version below): Let $g > 1, r > 1$ be integers. Playing around with the Verlinde formula (see below), I came across the expression ...
2
votes
1answer
194 views

Are satellite knots prime?

Which satellite knots are prime? I do know that connected sum of knots is a satellite operation, but I found this statement: "the satellite knots all have structures which are well known and ...
11
votes
1answer
283 views

A quadratic reciprocity formula

Inspired by a problem of calculating explicitly the invariants by Reshetikhin and Turaev for certain 3-manifolds, I have come across the following problem involving Gauss sums: I would like to prove ...