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 “Stallingsʹ fibration theorem” for semibundles.