Atlas: Another extension of Grushko's Theorem by John R. Stallings

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G^3 = Geometric Groups on the Gulf coast
May 28-29, 1999
University of South Alabama
Mobile, AL, USA

Organizers
Phil Bowers, Stephen Brick, Jon Corson

Another extension of Grushko's Theorem
by
John R. Stallings
University of California, Berkeley

Let K be a finite simplicial complex, such that (a) K is connected; (b) the link of every vertex is connected; (c) every 1-simplex is the face of at least two 2-simplexes. Let A and B be connected, basepointed spaces, and let A \/ B denote the union of disjoint copies of A and B joined along an arc between their basepoints; let p denote the midpoint of this arc. Suppose that f:K --> A \/ B is a continuous function which is surjective on fundamental group. Then f is homotopic to g for which g^-1(p) is 2-sided and connected. (The delicate point is that K remains unchanged.)

One interesting case of this is when K is a closed 2-manifold; in this case, one can write down explicit corollaries about groups; the hope is that some generalizations really do exist. The case of Grushko's original theorem about a free group mapping onto a free product can almost be derived from this result by letting K be the 2-skeleton of a connected sum of S¹ x S²'s.

Date received: May 17, 1999