Homotopy of spaces of diffeomorphisms of some three-dimensional manifolds

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 <a, b ¦abab^–1=1,a^mb²ⁿ=1>. 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 Diff_o(Q) the connected component of the space of diffeomorphisms QQ containing id Q, and by E₀(K, Q) the connected component of the space of imbeddings KQ containing the inclusion KQ; analogously we define E₀(K, P). The main results of the paper are the following two theorems. THEOREM 1. If m, n1, then the space Diff_o(Q) is homotopy equivalent with a circle. THEOREM 2. If m, n1, 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.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 122, pp. 72–103, 1982.