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The abstract

It has been 35 years since Stanley proved that f-vectors of boundaries of simplicial polytopes satisfy McMullen’s conjectured g-conditions. Since then one of the outstanding questions in the realm of face enumeration is whether or not Stanley’s proof could be extended to larger classes of spheres. Here we hope to give an overview of various attempts to accomplish this and why we feel this is so important. In particular, we will see a strong connection to f-vectors of manifolds and pseudomanifolds. Along the way we have included several previously unpublished results involving how the g-conjecture relates to bistellar moves and small g_2, the topology and combinatorics of stacked manifolds introduced independently by Bagchi and Datta, and Murai and Nevo, and counterexamples to over optimistic generalizations of the g-theorem. ]]>

Here’s a followup which maybe makes sense. The set of random sorting networks carries a natural “Hurwitz action” of the braid group on (n choose 2) strands — namely, twining strands i and i+1 sends

(t_1, ..t_i,t_{i+1}, …) to

(t_1, …. t_i t_{i+1} t_i^{-1}, t_i, ….)

If whp a sorting network tracks a great circle on the sphere, is there a meaningful way to talk about how a braid “moves” that circle?

]]>I see, but then I guess the letters Q and T got swapped again at some point. Poya’s congruence result holds for T rather than Q?

*GK:oops corrected*