Counting faces of cubical spheres modulo two
  Several recent papers have addressed the problem of characterizing the $f$-vectors of cubical polytopes. This is largely motivated by the complete characterization of the $f$-vectors of simplicial polytopes given by Stanley, Billera, and Lee in 1980. Along these lines Blind and Blind have shown that unlike in the simplicial case, there are parity restrictions on the $f$-vectors of cubical polytopes. In particular, except for polygons, all even dimensional cubical polytopes must have an even number of vertices. Here this result is extended to a class of zonotopal complexes which includes simply connected odd dimensional manifolds. This paper then shows that the only modular equations which hold for the $f$-vectors of all d-dimensional cubical polytopes (and hence spheres) are modulo two. Finally, the question of which mod two equations hold for the $f$-vectors of PL cubical spheres is reduced to a question about the Euler characteristics of multiple point loci from codimension one PL immersions into the $d$-sphere. Some results about this topological question are known (Eccles,Herbert,Lannes) and Herbert's result we translate into the cubical setting, thereby removing the PL requirement. A central definition in this paper is that of the derivative complex, which captures the correspondence between cubical spheres and codimension one immersions. 
E. Babson (U. of Washington), C. Chan