Homeotopy groups of irreducible 3-manifolds which may contain two-sided projective planes (Q1207870)

The homeotopy groups (isotopy classes of homeomorphisms) of Haken 3- manifolds, i.e. of compact, orientable, irreducible and sufficiently large 3-manifolds, are reasonably well understood. In the absence of projective planes \(\mathbb{P}^ 2\), most of the results can be generalized to the nonorientable case. On the other hand, if the manifold contains 2- sided projective planes, things become more difficult. In the present paper an algebraic description is given of the homeotopy group \({\mathcal H}(M)\) of a compact irreducible sufficiently large 3-manifold \(M\) which may contain 2-sided projective planes, in terms of homeotopy groups in dimension 2. This depends on a suitable definition of ``sufficiently large'': following Swarup, the authors call \(M\) sufficiently large if there exists a hierarchy of 2-sided incompressible surfaces cutting \(M\) into a collection of 3-balls and \(\mathbb{P}^ 2\times I\)'s. The computations and the description of \({\mathcal H}(M)\) are given modulo finite groups, i.e. neglecting finite kernels and finite cokernels (extensions). As a main application it follows that \({\mathcal H}(M)\) is finitely presented. The main steps of the proof are as follows. The manifold \(M\) contains a finite maximal system of disjoint 2-sided projective planes, unique up to ambient isotopy, and the homeotopy group of \(M\) splits along this system. Now \(M\) is sufficiently large if and only if this is the case for each component \(N\) of the decomposition of \(M\) along this system. In particular, after capping off the 2-sphere boundary components, the 2- fold orientable cover \(N_ 0\) of each component \(N\) is a Haken 3- manifold. It is shown that \({\mathcal H}(N)\) is isomorphic to the normalizer of the isotopy class of the covering involution in \({\mathcal H}(N_ 0)\). The proof reduces then to the computation of the normalizer of an involution in the homeotopy group of a Haken 3-manifold.