Let f be an orientation-preserving homeomorphism of ℝ2 which is fixed point free. The Brouwer ‘plane translation theorem’ asserts that every x0 ∈ ℝ2 is contained in a domain of translation for f i.e. an open connected subset of ℝ2 whose boundary is L ∪ 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).
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