Гомотопии пространств диффеоморфизмов некоторых трёхмерных многообразий

N. V. Ivanov

Zapiski Nauchnykh Seminarov POMI, Investigations in topology. Part IV, Tome 122 (1982), pp. 72-103

Résumé

The purpose of the paper is to compute the homotopy type of the space of diffeomorphisms for most of orientable three-dimensional manifolds which have finite fundamental group and contain Klein bottles. The fundamental group of a manifold $Q$ of that kind is of the form $\langle a, b\mid abab^{-1}, a^mb^{2n}=1\rangle$. Any pair of relatively prime natural numbers can arise as $m, n$; these numbers $m, n$ determine the manifold $Q$ up to diffeomorphism. Let $K$ be some Klein bottle embedded in $Q$, and let $P$ be a tubular neighbourhood in $Q$ of the Klein bottle $K$. We denote by $\operatorname{Diff}_0(Q)$ the connected component of $\operatorname{id} Q$ in the space of diffeomorphisms $Q\to Q$ and denote by $E_0(K, Q)$ the connected component of the inclusion map $K\hookrightarrow Q$ in the space of embeddings $K\to Q$; we define $E_0(K, P)$ analogously. The main results of the paper are the following two theorems. Theorem 1. If $m, n\ne1$ then the space $\operatorname{Diff}_0(Q)$ homotopically equivalent to the circle. Theorem 2. If $m, n\ne1$ then the inclusion $E_0(K, P)\hookrightarrow E_0(K, Q)$ is a homotopy equivalence. Basing of the known theorems on the spaces of diffeomorphisms of irreducible sufficiently large manifolds, theorem 1 can be reduced to theorem 2. The main difficulties present the proot of theorem 2. The technique of this proof extends the techique of Hatcher's and autor's papers on spaces of $PL$-homeomorphisms and diffeomorphisms of irreducible sufficiently large manifolds. (The text of the paper make use of a different, constructive definition of the studied class of manifolds. One can check without difficulties that these definitions are equivalent.)