Jason (jcreed) wrote,

I think I made a sort of breakthrough understanding chain homotopies today. The definition is, if f and g are chain maps from a chain complex F to a chain complex G, then a homotopy from f to g is a family of maps φn : Fn → Gn satisfying (leaving off all the subscripts) the equation
∂φ + φ∂ = g - f

So f and g are the endpoints of the homotopy; if I think of F as standing in for the unit interval, then f and g would be paths. To exhibit a continuous deformation from f to g, I'd have to fill in a 2-chain between them. Since F is not really just the unit interval, what I do is take every n-chain u that lives in F, and explain how the n-chain f(u) evolves into g(u) by means of an (n+1)-chain in G. The fact that Massey presents all this with singular cubes instead of singular simplices makes this totally straightforward. (I wonder, incidentally, how it would go with simplices...) Anyway, the term ∂φ gives the boundary of this deformation region. It includes g-f, (well, it ought to if it's actually a homotopy from f to g!) but it also includes all the junk around the sides, where the endpoints of f are becoming the endpoints of g (this is not a homotopy-with-basepoints, notice) That's exactly what the φ∂ takes into account. If you take the boundary points of the common domain of f,g, and then lift, you get minus of the "breadcrust" of the boundary of the deformation region. Add minus breadcrust to breadcrust, and it vanishes: a homotopy that even a choosy three-year-old would gladly accept. If served with enough peanut butter and jelly.

Went to the TG, chatted with a few people. I met a physicist named Diana, a girlfriend of a friend of James Hendricks's. I swear she reminded me so much of a younger version of Jodie Foster in Contact, the way she talked and laughed. It was almost creepy.

Talked with william lovas a lot after that, about cake-cutting algorithms and pure type systems. I really wonder if there's no structural proof already in the literature of SN for PTS. If not, I ought to chug through it.

At lunch the Veracruz waitress was totally flirting with me. I found it rather flattering.
Tags: homology, math, normalization, social
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