James K. Whittemore· Lectures in Mathematics Given at Yale University GROUP THEORY and THREE-DIMENSIONAL MANIFOLDS
James K. Whittemore· Lectures in Mathematics given at Yale University GROUP THEORY AND THREE-DIMENSIONAL MANIFOLDS by John Stallings New Haven and London, Yale University Press, 1971 1 BACKGROUND AND SIGNIFICANCE OF THREE-DIMENSIONAL MANIFOLDS I.A Introduction The study of three-dimensional manifolds has often interacted with a certain stream of group theory, which is concerned with free groups, free products, finite presentations of groups, and simUar combinatorial mat ters. Thus, Kneser's fundamental paper [4] had latent implications toward Grushko's Theorem [8]; one of the sections of that paper dealt with the theorem that, if a manifold's fundamental group is a fre.e product, then the manifold exhibits this geometrically, being divided into two regions by a sphere with appropriate properties. Kneser's proof is fraught with geo metric hazards, but one of the steps, consisting of modifying the I-skele ton of the manifold and dividing it up, contains obscurely something like Grushko's ,Theorem: that a set of ~enerators of a free product can be modified in a certain st"mple way so as to be the union of sets of genera tors of the factors. Similarly, in the sequence of theorems by PapakyriakopoUlos, the Loop ~eorem [14], Def1:n's Lemma, and the ~here ~eorem [15], there are implicit facts about group theory, which form the \~tn. subject of these If.'' chapt~rs. " ) Philosophically speaking, the depth and beauty of 3~manifold theory is, it seems to me, mainly due to the fact that its theorems have offshoots that eventually blossom in a different subject, namely group theory.
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