rybu's blog

homology of knot spaces, known unknowns -- part 1: torsion

There's a fair amount of papers in the literature on the homotopy and homology of spaces of knots. An oddity that probably isn't apparent to the casual reader is that very little is known about torsion in the homology of knot spaces. To be precise, let's `normalize' this discussion and consider knot spaces to be the space smooth of embeddings of Rj in Rn which agree with a fixed linear embedding outside of a fixed ball.

Favourite classifications of the 3-dimensional lens spaces

With the proof of the Poincare conjecture, it's a great time to look at the whole of 3-manifold theory and poke at it. Some things are "more proven" than others. For example: We only have one proof of the Poincare conjecture. Similarly, there's only one proof of the Smale conjecture. Some things are "super proven", like Dehn's Lemma, the Loop Theorem and the Sphere Theorem -- there's even more than one proof of their equivariant versions.

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