Isotopies of 3-manifolds

AbstractAn isotopy of a manifold M that starts and ends at the identity diffeomorphism determines an element of π1(Diff(M)). For compact orientable 3-manifolds with at least three nonsimply connected prime summands, or with one S2 × S1 summand and one other prime summand with infinite fundamental group, infinitely many integrally linearly independent isotopies are constructed, showing that π1(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 S3 (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, π1(Diff(P)) is finitely generated, then results of Jahren and Hatcher imply that π1(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