On 2-Dimensional Homotopy Invariants of Complements of Knotted Surfaces

We prove that ifM is a compact 4-manifold provided with a handle decomposition with 1-skeleton X , and if G is a finite crossed module, then the number of crossed module morphisms from the fundamental crossed module Π2(M,X , ∗) = (π1(X , ∗), π2(M,X , ∗), ∂, ⊲) into G can be re-scaled to a manifold invariant IG (i. e. not dependent on the choice of 1-skeleton), a construction similar to David Yetter’s in [Y], or Tim Porter’s in [P1, P2]. Therefore, we elucidate an algorithm to calculate π2(M,X , ∗) as a (crossed) module over π1(X , ∗), in the case when M is the complement of a knotted surface in S4 and X is the 1-skeleton of a handle decomposition of M . We prove that in this case the invariant IG yields a non-trivial invariant of knotted surfaces, which we conjecture to coincide with the invariant defined in [FM]. Introduction Throughout most of this article, manifold means piecewise linear manifold, and diffeomorphism means piecewise linear diffeomorphism. The exceptions are 2.3 and 2.4, where we need to move to the smooth category, so that we can use transversality and Cerf Theory. In [FM] we defined an invariant of knotted surfaces from any finite categorical group (or what is the same a finite crossed module). The