2-torus manifolds, cobordism and small covers

Zhi Lü

Vol. 241 (2009), No. 2, 285–308

DOI: 10.2140/pjm.2009.241.285

Abstract

Define M_n to be the set of equivariant, unoriented cobordism classes of n-dimensional 2-torus manifolds, where any such manifold is smooth, closed and n-dimensional, and has an effective smooth action of a rank n 2-torus group (ℤ₂)ⁿ. Then M_n forms an abelian group with respect to disjoint union. For n = 3, we determine the group structure of M_n and show that each class of M_n contains a small cover as its representative.

Keywords

2-torus manifolds, cobordism, small cover

Mathematical Subject Classification 2000

Primary: 55N22, 57R85, 57S17

Secondary: 05C10, 57M60

Milestones

Received: 7 September 2008

Revised: 27 March 2009

Accepted: 11 April 2009

Published: 1 June 2009