Hidden Stasheff polytopes in algebraic K-theory and the space of Morse functions, by Mikhail Kapranov and Masahico Saito

We show how the Stasheff polytopes (associahedra) appear naturally as higher syzygies among elementary matrices. The first instance of the associahedron is the pentagon and the corresponding syzygy is the Steinberg relation

                [x  , x  ] = x
                  ij   jk     ik
We also show that the associahedron can be realized as a bifurcation diagram for "Smalefications" for a Morse but not Morse-Smale function f which has a chain of gradient trajectories joining critical points of the same index:
                x  ---> x  ---> ... ---> x
                 1       2                n
We state a conjecture about the existence of a CW model for the Volodin space whose cells are labelled by some "hieroglyphs", i.e., pictures formally describing various patterns of gradient trajectories of a Morse function, joining critical points of the same index.

Comments: latex, 40 pages, with 17 pictures in .eps format. The file K3.dvi calls for the 17 picture files, but the postscript file K3.ps incorporates them.

Mikhail Kapranov <kapranov@math.nwu.edu>
Masahico Saito <saito@math.usf.edu>