Homotopy commutativity in quasitoric manifolds

Abstract

We prove that the loop space of a quasitoric manifold is homotopy commutative if and only if the underlying polytope is a product of $3$-simplices $(\Delta^3)^n$ and the characteristic matrix is equivalent to a matrix of certain type. Quasitoric manifolds over $(\Delta^3)^n$ include generalized Bott manifolds, and we also construct an infinite family of homotopy nonequivalent generalized Bott manifolds over $(\Delta^3)^n$, only half of them have homotopy commutative loop spaces. In particular, for each $n\ge 2$, there are infinitely many homotopy types in $6n$-dimensional quasitoric manifolds having homotopy (non)commutative loop spaces.