Free groups and handlebodies
Proceedings of the American Mathematical Society
In this note we state a conjecture about free groups and give some topological consequences which would follow if the conjecture is true. The conjecture seems to be a natural extension of a theorem of Nielsen, and the main topological consequence is the theorem that regular neighborhoods of contractible 2-dimensional subcomplexes of combinatorial 5-manifolds are S-cells. This has some interesting consequences relative to the 3-and 4-dimensional Poincaré conjectures. The referee has remarked on
... ee has remarked on a weakening of the conjecture and on the relation of the topological problem to a more general problem as follows: "The results of the paper follow from a somewhat weaker conjecture: Suppose P is (xi, ■ ■ • , xn: ri, • • • , rn), and is a presentation of the trivial group ; define an operation of type (i), (ii), (iii), (iv) on this presentation to be the result of applying such an operation as defined in the paper on the set of relators. An operation of type (v) will consist of adding an additional generator, say y, and the additional relator y; an operation of type (vi) will be the inverse of an operation of type (v). The weaker conjecture would be that P can be reduced to the empty presentation by a finite sequence of operations of types (i)-(vi). "The problem is related to the question: If the 2-dimensional polyhedra P and Q have the same simple homotopy type, can they both be embedded in a 3-dimensional polyhedron Z such that Z geometrically contracts to P and Z geometrically contracts to Q? If you replace "2" and "3" by "re" and "re + 1," this question has the affirmative answer for re greater than 2, according to J. H. C. Whitehead in Simplicia! spaces, nuclei, and m-groups. It is equivalent to the above conjecture when P is a contractible 2-dimensional polyhedron and Q is a point." 1. A conjecture about free groups. Let F be the free group on Xi, • • ■ , xn. If Oi, • • • , flp is any (finite) set of elements of F, then a Nielsen operation on Oi, • • • , ap changes ait ■ • • , av to äi, • • • , äp in one of the following three ways: (i) öi = ar\ äi = a2, • • • , äp = ap. (ii) Interchange ai and a¿, leaving others the same. (iii) âi = axa2, all others fixed.