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I don't believe it's written up anywhere. edit: in the comments Charlie Frohman corrects me: MR2128054 (2005m:57041) Roseman, Dennis(1-IA) Elementary moves for higher dimensional knots. (Eng …

This is a longish comment on the $B_3$ case. $B_3$ is isomorphic to the fundamental group of the complement of a trefoil knot. A trefoil complement fibres over $S^1$ with fiber a once-punctured sur …

This is only a partial answer, and it only addresses the compact case. If you take developable to have the original sense of the word (provided by jc's Wikipedia link above), a developable 3-manifol …

I imagine bridge index is readily computable, at least for "small enough" knots. Bridge index is a lot like Heegaard genus of a 3-manifold. Heegaard splitting surfaces can be found via (almost) norm …

There's a lot of basic facts about Grassmannians and Stiefel manifolds like yours, for which I imagine there are plenty of references out there, yet they also don't appear to show-up in commonly-used …

B. Everitt. 3-manifolds from platonic solids. Topology and its applications, 2004. Covers everything you're asking for and more.

A standard way to get at $\pi_2$ of any space is that it is $H_2$ of the universal cover of the space. This is because higher homotopy groups are invariant under covering maps plus the Hurewicz theor …

In the smooth case the idea is to take a linear height function on the plane, which is generically Morse on the curve. Apply the Jordan curve theorem + basic Morse theory, this tells you the compact r …

If you're okay going the extra step and assuming a smooth structure, the standard argument of Whitehead goes like this: take a smooth embedding of your manifold (of any dimension) into euclidean space …

I'll convert my comments to an answer. Characteristic classes were originally defined as obstruction classes, going back to the work of Whitney and Stiefel. The top Chern class of a complex bundl …

The theorem you're looking for is proven in Cerf's dissertation. J. Cerf, Topologie de certains espaces de plongements, Bull S.M.F., tome 89 (1961) 227-380. This is for the case you mention, when …

Although this does not answer your question, there are partial answers in dimension 3. For example, if you construct an orientable 3-manifold via a random Heegaard splitting (constructing the gluin …

I think Kevin's suggestion is likely to be a productive one. GAP has good routines for counting conjugacy classes of representations to a large number of finite groups. A similar strategy you coul …

Knot theory was fundamentally connected to 3-manifold theory (making planar diagrams relatively peripheral) by Schubert, in his pair of papers, where he proved the prime factorization of knots and the …

To me it looks like you're asking for the equivalence relation on the set of atlases on $M$, where one atlas is considered equivalent to the other if the identity map $Id : M \to M$ (where $Id(x)=x$ a …