Combinatorial constructions of three-dimensional small covers

Details Combinatorial constructions of three-dimensional small covers Combinatorial constructions of three-dimensional small covers

2011-04-10

arxiv.org/abs/1104.1744v1

Mathematics

Algebraic Topology

18 pages, 20 figures

Scientific paper

A small cover was introduced by Davis and Januszkiewicz as an $n$-dimensional closed manifold with a locally standard $Z_2)^n$-action such that its orbit space is a simple convex polytope. There exist a one-to-one correspondence between small covers and $(Z_2)^n$-colored polytopes. In this paper we study a construction of 3-dimensional small covers by using two operations called a connected sum and a surgery. These operations correspondent to combinatorial operations on $(Z_2)^3$-colored simple convex polytopes. We shall show that each 3-dimensional small cover can be constructed from $T^3$, $RP^3$ and $S^1 \times RP^2$ with two different $(Z_2)^3$-actions by using these operations. This result is a generalization and an improvement of L\"{u}-Yu's result.

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