A Brouwer Translation Theorem for Free Homeomorphisms.
Edward E. Slaminka · Deep Blue (University of Michigan) · 1984
From 1909 through 1920 L. E. J. Brouwer wrote a series of eight papers, each having the same title, "Continuous one-one transformations of surfaces in themselves". In these papers he proved the fixed point property for orientation preserving homeomorphisms of the two-sphere, the translation arc lemma, and the plane translation theorem. Subsequent authors, doubting the validity of Brouwer's proof of the plane translation theorem, endeavored to present a corrected proof of that theorem. The purpose of this dissertation is both to present a new proof of the above theorem, and to extend the result from fixed point free orientation preserving homeomorphisms to free homeomorphisms with finite fixed point set. A homeomorphism h:S('2) (--->) S('2) is called free it it satisfies the conclusion of Brouwer's translation arc lemma; that is, if h has the property that for any disc D (L-HOOK) S('2) such that D (INTERSECT) h(D) = 0, D (INTERSECT) h('n)(D) = 0 for all n (NOT=) 0. These homeomorphisms may have fixed points. The version of the Brouwer translation theorem that is proved is as follows: THEOREM: Let h:S('2) (--->) S('2) be a free homeomorphism of the two-sphere with finite fixed point set F. Then for any p (epsilon) S('2) - F there exists an embedding (phi)(,p):(R('2),0) (--->) (S('2) - F,p) such that: (i) h(phi)(,p) = (phi)(,p)(tau), where (tau)(z) = z + 1 is the canonical translation of the plane; and , (ii) the image of each vertical line under (phi)(,p) is closed. As a consequence of this theorem, the x-axis is mapped under (phi)(,p) to an invariant translation line through p.