Semibundle decompositions of 3-manifolds and the twisted cofundamental group

In this paper we study ''semibundle decompositions'' of 3-manifolds, that is, decompositions of 3-manifolds as unions of twisted I-bundles over nonorientable surfaces. The primary algebraic tool in this study is a certain easily defined, homotopy-theoretic, abelian group, which we call the ''twisted cofundamental group''. The main technical work in this paper is the construction of a natural isomorphism between the twisted cofundamental group and a certain one-dimensional cohomology group with ''twisted integer'' coefficients. We use this isomorphism to determine the structure of the twisted cofundamental group, and we also formulate a ''Stallings' fibration theorem'' for semibundles. (C) 1997 Elsevier Science B.V.