Celebratio Mathematica

V. W. Adkisson and S. MacLane: “Planar graphs whose homeomorphisms can all be extended for any mapping on the sphere,” Am. J. Math. 59 : 4 (October 1937), pp. 823–832. MR 1507285 JFM 63.1158.05 Zbl 0017.42703 article

S. MacLane and V. W. Adkisson: “Fixed points and the extension of the homeomorphisms of a planar graph,” Am. J. Math. 60 : 3 (July 1938), pp. 611–639. MR 1507342 JFM 64.0596.03 Zbl 0019.14201 article

Gehman [1926; 1929; 1936], Adkisson [1930; 1934], and Moore [1926] have found conditions under which a given homeomorphism of a continuous curve on the sphere can be extended to the sphere. In this paper we consider the problem of characterizing those graphs of the sphere for which every homeomorphism of the graph can be extended to the sphere. We restrict ourselves to cyclicly connected graphs and obtain a necessary and sufficient condition that such a graph has some one map on the sphere in which all of its homeomorphisms are extendable. This necessary and sufficient condition depends essentially on the behavior of certain fixed points, for it asserts merely that the fixed points of all homeomorphisms of order 2 must be distributed on the graph as are the fixed points of periodic homeomorphisms of the sphere.

V. W. Adkisson and S. Mac Lane: “Extending maps of plane Peano continua,” Duke Math. J. 6 : 1 (1940), pp. 216–228. MR 0001345 JFM 66.0970.03 Zbl 0026.36302 article

When can a homeomorphism \( T \) of a Peano continuum \( M \) on a sphere \( S \) to a set \( M^{\prime} \) on a sphere \( S^{\prime} \) be extended to a homeomorphism \( T^{\prime} \) which carries the whole sphere \( S \) into \( S^{\prime} \)? Gehman has solved this problem for the extension of \( T \) to a map of a plane containing such a space \( M \) [1926]. His condition can easily be modified to include the case of \( M \) and \( M^{\prime} \) on spheres [Adkisson 1934]. But this condition, which requires that certain “sides” of \( M \) be preserved by \( T \), is extrinsic to the given \( M \), and requires in fact that one establish the existence of certain arcs of \( S \) and \( S^{\prime} \) which may cut through \( M \) and its complement \( S - M \) in a complicated fashion. We establish here another necessary and sufficient condition for the extendibility of \( T \). This condition is intrinsic, and applies only to the triods of \( M \), where by a triod in \( M \) we mean a set \( \tau = [\alpha,\beta,\gamma] \) of three arcs of \( M \) with a common end point, the vertex of the triod, such that any two of these arcs intersect only in this vertex.

S. Mac Lane and V. W. Adkisson: “Extensions of homeomorphisms on the sphere,” pp. 223–236 in Lectures in topology (University of Michigan, Ann Arbor, MI, 24 June–6 July 1940). Edited by R. L. Wilder and W. L. Ayres. University of Michigan Press (Ann Arbor, MI), 1941. MR 0005304 Zbl 0063.03694 incollection

Virgil William Adkisson	Related
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