Counting faces of cubical spheres modulo two

AbstractSeveral 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 (Discrete Geometry and Convexity, Annals of the New York Academy of Sciences, Vol. 440, 1985, pp. 212–223) and Billera and Lee (Bull. Amer. Math. Soc. 2 (1980) 181–185) in 1980. Along these lines Blind and Blind (Discrete Comput. Geom. 11(3) (1994) 351–356) 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, Lecture Notes in Mathematics, Vol. 788, Springer, Berlin, 1980, pp. 23–38; Herbert, Mem. Amer. Math. Soc. 34 (250) (1981); Lannes, Lecture Notes in Mathematics, Vol. 1051, Springer, Berlin, 1984, pp. 263–270) 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

Cubulations, immersions, mappability and a problem of Habegger

The aim of this paper (inspired from a problem of Habegger) is to describe the set of cubical decompositions of compact manifolds mod out by a set of combinatorial moves analogous to the bistellar moves considered by Pachner, which we call bubble moves. One constructs a surjection from this set onto the the bordism group of codimension one immersions in the manifold. The connected sums of manifolds and immersions induce multiplicative structures which are respected by this surjection. We prove that those cubulations which map combinatorially into the standard decomposition of ${\bf R}^n$ for large enough $n$ (called mappable), are equivalent. Finally we classify the cubulations of the 2-sphere.Comment: Revised version, Ann.Sci.Ecole Norm. Sup. (to appear

Embedding calculus knot invariants are of finite type

We show that the map on components from the space of classical long knots to the n-th stage of its Goodwillie-Weiss embedding calculus tower is a map of monoids whose target is an abelian group and which is invariant under clasper surgery. We deduce that this map on components is a finite type-(n-1) knot invariant. We also compute the second page in total degree zero for the spectral sequence converging to the components of this tower as Z-modules of primitive chord diagrams, providing evidence for the conjecture that the tower is a universal finite-type invariant over the integers. Key to these results is the development of a group structure on the tower compatible with connect-sum of knots, which in contrast with the corresponding results for the (weaker) homology tower requires novel techniques involving operad actions, evaluation maps, and cosimplicial and subcubical diagrams.Comment: Revised maps to the infinitesimal mapping space model in Sections 3 and 4 and analysis of cubical diagrams in Section 5. Minor expository and organizational changes throughout. Now 28 pages, 4 figure

Surface cubications mod flips

Let $\Sigma$ be a compact surface. We prove that the set of surface cubications modulo flips, up to isotopy, is in one-to-one correspondence with $\Z/2\Z\oplus H_1(\Sigma,\Z/2\Z)$.Comment: revised version, 18