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Given a free group Γ of finite rank n and a prime number p, denote by Γ • k the k th layer of the Stallings (• = S) or Zassenhaus (In this paper we prove that there is a non-centralwhich splits if and only if k = 1 and p is odd if • = Z or, k = 1 and (p, n) = (3, 2), (2, 2) if • = S. Moreover, if we denote by IA p (N • k ) the subgroup of Aut N • k formed by the automorphisms that acts trivially on N • 1 , then the restriction of this extension to IA p (N • k+1 ) give us a non-split central extension of groups 0 → Hom(N • 1

In this paper we prove that the Rohlin invariant is the unique invariant inducing a homomorphism on the Torelli group. Using this result we generalize the construction of invariants of homology [Formula: see text]-spheres from families of trivial 2-cocycles on the Torelli group given by Pitsch to include invariants with values on an abelian group with [Formula: see text]-torsion.

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