The Klee–Novik Complexes B(i,d) Triangulate S^i × B^(d−i−1)

For 0 ≤ i ≤ d − 2, Klee and Novik defined B(i,d) as the subcomplex of the boundary of the d-dimensional cross-polytope generated by the facets whose xy-words have at most i switches, and asked whether B(i,d) is a combinatorial triangulation of S^i × B^(d−i−1). We show that it is, for all 0 ≤ i ≤ d − 2. Klee and Novik observed that B(i,d) collapses onto the boundary of the (i+1)-dimensional cross-polytope and noted that it is therefore a disc bundle over S^i. We show that |B(i,d)| is in fact a product. That sphere is a join factor of the boundary of the d-dimensional cross-polytope, so it has a regular neighbourhood that is a product, and |B(i,d)| together with an outer collar is another regular neighbourhood of it; uniqueness of regular neighbourhoods gives the result. Consequently ∂B(i,d) is PL homeomorphic to S^i × S^(d−i−2), and a conjecture of Cohen, Klee and Pannell holds. This is an unrefereed note.