Involutions of the 3-sphere which fix 2-spheres
Robert Craggs · Pacific Journal of Mathematics · 1970
We show in that the space of involutions of the 3-sphere whose fixed point sets are 2-spheres is pathwise and locally pathwise connected.From Smith theory it is known that these involutions are orientation reversing.The fixed point sets need not be tame 2-spheres; Bing and others have many examples of involutions of the 3-sphere whose fixed point sets are wild 2-spheres.In order to prove the connectivity theorem ( § 6) just mentioned we develop an approximation theory for involutions of the 3-sphere in §'s 3, 4. Some of the results there are interesting in their own right.Corollary 3.1 states that involutions which fix wild 2-spheres can be approximated by involutions which fix tame 2-spheres.Theorem 4.6 states that if an involution g fixing a 2-sphere R approximates an involution / fixing a 2-sphere S very closely then R approximates S.We also make use of Theorem 5.2, a modified version of the Alexander deformation theorem, which states that if the boundary of a 3-cell C in the 3-sphere approximates a given 2-sphere very closely then very small homeomorphisms of C onto itself which fix Bd (C) can be deformed back to the identity by small isotopies of C which fix Bd (C).NOTATION.