Isotopies of 3-manifolds

Abstract

An isotopy of a manifold M that starts and ends at the identity diffeomorphism determines an element of π 1(Diff( M)). For compact orientable 3-manifolds with at least three nonsimply connected prime summands, or with one S 2 × S 1 summand and one other prime summand with infinite fundamental group, infinitely many integrally linearly independent isotopies are constructed, showing that π 1(Diff( M)) is not finitely generated. The proof requires the assumption that the fundamental group of each prime summand with finite fundamental group imbeds as a subgroup of SO(4) that acts freely on S 3 (conjecturally, all 3-manifolds with finite fundamental group satisfy this assumption). On the other hand, if M is the connected sum of two irreducible summands, and for each irreducible summand P of M, π 1(Diff( P)) is finitely generated, then results of Jahren and Hatcher imply that π 1(Diff( M)) is finitely generated. The isotopies are constructed on submanifolds of M which are homotopy equivalent to a 1-point union of two 2-spheres and some finite number of circles. The integral linear independence is proven by obstruction-theoretic methods.