Some topological proofs and extensions of Grusko's theorem

Homomorphisms h:F ! * G are studied, where F is a free group and * G is the free product. In Part I, the conditions imposed on the homomorphism h relate to certain commutators in F . The method of proof is to obtain a topological realization of h, interpreting F as the fundamental group of a surface with boundary and * G as the fundamental group of a wedge of complexes. Standard general position arguments and reductions are made; the theorems are obtained from the well-known classi cation of surfaces. In Part II, it is assumed that h is an epimorphism. Gru sko's Theorem is obtained. Again the method of proof is topological, based on general position arguments, a direct limit construction, and a combinatorial argument. Besides Gru sko's Theorem, several related facts are proved. One of these is Kneser's Conjecture; if M is a closed 3-manifold, then any free factorization of 1(M) is mirrored by a factorization of M into a sum of manifolds. Another is a result on free products with amalgamation; if (G H)A = Q has rank n, then there are n elements of G [H, whose consequence is Q. A new algebraic proof of Gru sko's Theorem is given. This is more or less a translation of the topological proof. A DISSERTATION Presented to the Faculty of Princeton University in Candidacy for the Degree of Doctor of Philosophy. Recommended for Acceptance by the Department of Mathematics. May, 1959. Some topological proofs and extensions of Gru sko's theorem John R. Stallings