Chapter III An Outline of the Proof

Abstract

Publisher Summary The chapter presents an outline of the proof of the Smith conjecture with illustrations of theorems. Making some reductions, the chapter proceeds to sketch an argument to complete the proof with the help of a few case studies. The techniques are those of classical piecewise-linear (PL) topology, plus an essential new fact—namely, the equivariant version of Dehn's lemma and the loop theorem. The hypotheses of one of these cases are exactly those of Thurston's uniformization theorem, which argues that Σ–K admits a hyperbolic structure. This representation is automatically irreducible. At this point, the argument becomes purely algebraic. A flow chart shown in the chapter gives a pictorial representation of the logical structure of the argument in support of the proof.