Small covers over a product of simplices

Abstract

In this paper, we determine the number of equivariant homeomorphism classes of small covers over a product of m simplices for m ≤ 3 or for the dimension of each simplex being greater than 1 and m > 3. Moreover, we calculate the number of equivariant homeomorphism classes of all orientable small covers over a product of at most three simplices.

On small covers over a product of simplices

“…Therefore, the number of equivariant small covers over a product of simplices of dimension greater than 1 is the quotient of the number of the small covers and the order of the automorphism group of the face poset. In [2], Chen and Wang directly counted the number of equivariant homeomorphism classes of small covers over ∆ 1 × ∆ n1 × ∆ n2 and ∆ 1 × ∆ n3 , where ∆ ni is an n i -simplex with n i ≥ 1 for 1 ≤ i ≤ 3 . In this paper, we use Choi's argument to generalize these formulas to an arbitrary product of simplices.…”

“…Theorem 4.1,[2]) If P = I × ∆ n × ∆ m then the number of Z n 2 -equivariant homeomorphism classes of small covers over P is 1.…”