Homotopy of spaces of diffeomorphisms of some three-dimensional manifolds

Abstract

The goal of the paper is to calculate the homotopy type of the space of diffeomorphisms for most orientable three-dimensional manifolds with finite fundamental group containing the Klein bottle. The fundamental group of such a manifold Q has the form . As m and n one can have any relatively prime natural numbers; these numbers m, n determine the manifold Q up to diffeomorphism. Let K be a Klein bottle lying in Q and let P be a closed tubular neighborhood in Q of this Klein bottle K. We denote by Diffo(Q) the connected component of the space of diffeomorphisms Q→Q containing id Q, and by E0(K, Q) the connected component of the space of imbeddings K→Q containing the inclusion K→Q; analogously we define E0(K, P). The main results of the paper are the following two theorems. THEOREM 1. If m, n≠1, then the space Diffo(Q) is homotopy equivalent with a circle. THEOREM 2. If m, n≠1, then the inclusion E0(K, P) ↺ E0(K, Q) is a homotopy equivalence. With the help of familiar results on spaces of diffeomorphisms of irreducible manifolds which are sufficiently large, Theorem 1 reduces without difficulty to Theorem 2. The main difficulty is the proof of Theorem 2. This proof develops a technique of Hatcher and the author which deals with spaces of PL-homeomorphisms and diffeomorphisms of irreducible manifolds which are sufficiently large. In the paper we use a different structure definition of the class of manifolds considered. It is easy to verify that these definitions are equivalent.