## ON THE HOMOTOPY TYPE OF SPACES OF PROJECTABLE HOMEOMORPHISMS

### Abstract (summary)

Let X be a polyhedron and let H(X) denote its group of (PL-) homeomorphisms with the compact-open topology. Observe that neither the homeomorphism group correspondence X(' )(--->)(, )H(X) nor the homeotopy group correspondence X(' )(--->)(, )(pi)(,k)(H(X)) are functors between the expected categories. Nevertheless, it is reasonable to expect that if f : X (--->) Y is a map which strongly reflects the structure of X then there should be a relationship between (pi)(,k)(H(X)) and (pi)(,k)(H(Y)). With this dissertation I propose a study of such relationships. In particular the case where f : X (--->) Y is a covering map is studied. The context of the work is geared towards applications where X and Y are 'sufficiently large' low dimensional manifolds.(,)

Let f : X (--->) Y be an arbitray surjective map. Of central importance in relating H(X) with H(Y) is H(,f), the space of projectable homeomorphisms. A homeomorphism G (epsilon) H(X) is projectable with respect to f if f G f('-1) defines a homeomorphism on Y. In this case f G f('-1) is a liftable homeomorphism of Y. The space of liftable homeomorphisms is denoted by H('f). We study the projectable homeotopy group (pi)(,k)(H(,f)).

Corresponding to k = 0 we have M(X) = (pi)(,0)(H(X)), M(,f) =(pi)(,0)(H(,f)) and M('f) = (pi)(,0)(H('f)) and these groups are referred to as mapping class groups. When f : X (--->) Y is a covering map we develop three diagrams which allow us to effectively relate the groups M(X), M(,f), M('f) and M(Y). In this framework an important problem is to compute the kernel of i(,*) : M(,f )(--->) M(X) where i : H(,f) (--->) H(X) is the inclusion. f is said to have the projecting isotopy property (PIP) provided that ker i(,*) = 1. In Chapter III we utilize the diagrams mentioned above to determine for a large class of covering maps between low dimensional manifolds precisely which maps fail to have the PIP. Furthermore examples illustrating the computation of M(,f) and its relationships with M(X) and M(Y) are discussed.

For the determination of (pi)(,k)(H(,f)) when k > 0, we show that if f : X (--->) Y is a 'reasonable' covering map then there is a covering map (rho) : H(,f) (--->) H('f). Here (rho) is the epimorphism defined by G (--->) f G f('-1). Since, for covering maps, (pi)(,k)(H('f)) (TURNEQ) (pi)(,k)(H(Y)), we deduce that (pi)(,k)(H(,f)) (TURNEQ) (pi)(,k)(H(Y)) for k > 1 and (pi)(,k)(H(,f)) (pi)(,k)(H(Y)) for k = 1. For the class of coverings mentioned above this leads to a complete description of the groups (pi)(,k)(H(,f)).

Finally, to apply these results to a wider class of covering maps f : X (--->) Y, we develop a parallel theory for E(,f), the space of projectable self homotopy equivalences of f. When X is aspherical we obtain a complete description of the groups (pi)(,k)(E(,f)).