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.