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If you look in Brayton Gray's book Homotopy Theory, I think that you will find that this theorem gives any generalized homology theory is representable by homotopy classes into a spectrum. I may have …

I was re-reading sections of Whitehead's book the other day, and I found it very helpful to think in the way he was writing. For a historical perspective, I would ask Clarence Wilkerson, Peter May, Bi …

I first learned Poincare duality from Milnor's "Lectures on the h-cobordism theorem," published in the Princeton yellow series. It will seem a little old now adays, but it develops the Morse theory fr …

I have been thinking about this as well. My approach would be to construct a broken surface diagram or chart of an immersed disk that a classical knot bounds. For simplicity take the untwisted double …

About $\pi_5(S^4)$, this is already in the stable range, so that it is isomorphic to the cobordism group of immersed curves in the plane. That is clearly ${\mathbb Z}/2$ since you can cancel double po …

Expanding upon Andre Henriques's answer, $\pi_3^s(P^\infty) \cong {\mathbb Z}/8$. The homotopy group is isomorphic to the cobordism group of non-orientable surfaces in 3-space. Boy's surface is a gen …

I think the reference that you are looking for is this article by Cannon, Floyd, and Parry.

Originally, the idea of a 4D TQFT was to be found in a Hopf Category as defined by Crane and Igor Frenkel. Crane and Yetter gave an example via certain cocycles over a finite group. Kauffman, Saito, a …

Let me expand upon jc's comment above. For convenience, I'll pass to the stable homotopy group for a little while. The first stable stem is $\pi_1^s = {\mathbb Z}/(2)$. A representative class is the …

I always thought that Haefliger's result applied in dimensions higher than 5, but in skimming the manuscript, it looks like 5 might be a critical case. Please read at page 404 and 405 carefully. To …

Which categorifications give explicit braided monoidal 2-categories with duals? This question is in response to Ben Webster's questions in recent history. The point is that given a braided monoidal 2 …