A NEW PROOF OF THE BROUWER PLANE TRANSLATION THEOREM

Let f be an orientation-preserving homeomorphism of R2 which is fixed point free. The Brouwer `plane translation theorem' asserts that every x0 is-an-element-of R2 is contained in a domain of translation for f, i.e. an open connected subset of R2 whose boundary is L or f(L) where L is the image of a proper embedding of R in R2, such that L separates f(L) and f-1(L). In addition to a short new proof of this result we show that there exists a smooth Morse function g:R2 --> R such that g(f(x)) < g(x) for all x and the level set of g containing x0 is connected and non-compact (and hence the image of a properly embedded line).