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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7
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3answers
441 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
299 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
85 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
296 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
243 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
565 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 ...
53
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
199 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
284 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 ...