86i:57002 57-06 57N10 57S17 57S25 The Smith conjecture. (English) Papers presented at the symposium held at Columbia University, New York, 1979. Edited by John W. Morgan and Hyman Bass. Pure and Applied Mathematics, 112. Academic Press, Inc., Orlando, Fla., 1984. xv+243 pp. $49.50. ISBN 0-12-506980-4 _________________________________________________________________ Contents: John W. Morgan, The Smith conjecture (pp. 3--6); John W. Morgan, History of the Smith conjecture and early progress (pp. 7--9); John W. Morgan, An outline of the proof (pp. 11--16); Peter Shalen, The proof in the case of no incompressible surface (pp. 21--36); John W. Morgan, On Thurston's uniformization theorem for three-dimensional manifolds (pp. 37--125); Hyman Bass, Finitely generated subgroups of ${\rm GL}\sb{2}$ (pp. 127--136); C. McA. Gordon and R. A. Litherland, Incompressible surfaces in branched coverings (pp. 139--152); Shing Tung Yau and William H. Meeks, III, The equivariant loop theorem for three-dimensional manifolds and a review of the existence theorems for minimal surfaces (pp. 153--163); William H. Meeks, III and Shing Tung Yau, Group actions on ${\bf R}\sp{3}$ (pp. 167--179); Michael W. Davis and John W. Morgan, Finite group actions on homotopy $3$-spheres (pp. 181--225); Michael W. Davis, A survey of results in higher dimensions (pp. 227--240). Let the finite cyclic group $Z\sb p$ act by orientation-preserving homeomorphisms on the $3$-sphere $S\sp 3$. P. A. Smith [Ann. of Math. 40 (1939), 670--711; MR 1, 30] proved that the fixed point set of the action, if nonempty, is a simple closed curve $K$. He asked whether $K$ could be knotted. (One may show that $K$ is unknotted if and only if the action is conjugate to a linear action. This is easy in the smooth or piecewise linear cases, but requires more work in the topological category [see E. E. Moise , Trans. Amer. Math. Soc. 252 (1979), 1--47; ibid. 259 (1980), no. 1, 255--280; MR 83g:57030a b; Smith, Illinois J. Math. 9 (1965), 343--348; MR 30 #5311].) The Smith conjecture stands in the first rank of mathematical problems when measured by the amount and depth of new mathematics required to solve it. The solution of the Smith conjecture is remarkable in several more ways: its solution required input into $3$-manifold topology from hyperbolic geometry, algebra, and minimal surfaces; the solution required a remarkable synthesis of the work of several people (and a catalyst to make it all work together); and the deepest mathematics (existence of hyperbolic structures on certain $3$-manifolds, due to W. Thurston), now over eight years old, still has not found a complete treatment in the literature (in fact there may be at most one or two people in the world besides Thurston willing to say that they understand the complete proof); further, higher-dimensional analogues of the Smith conjecture have generated a substantial amount of work in transformation groups. Analogous questions in dimensions less than three were earlier answered affirmatively by L. E. J. Brouwer [Math. Ann. 80 (1919/20), 39--41; Jbuch 47, 527] and B. von Kerekjarto [ibid. 80 (1919/20), 36--38; Jbuch 47, 526] (and completed by S. Eilenberg [Fund. Math. 22 (1934), 28--41; Jbuch 60, 1228]): Every finite cyclic group action on the $2$-sphere is equivalent to a rotation. This certainly gave some credence to the possibility that the Smith conjecture might be true. There intervened, however, some sobering evidence that things might be more complicated than previously thought. First R. H. Bing [Ann. of Math. (2) 56 (1952), 354--362; MR 14, 192] showed that an orientation-reversing topological involution on the $3$-sphere may have a wild $2$-sphere (like the Alexander horned sphere) as its fixed point set. D. Montgomery and L. Zippin [Proc. Amer. Math. Soc. 5 (1954), 460--465; MR 15, 978] modified Bing's example to produce an orientation-preserving involution on the $3$-sphere with fixed point set a wildly embedded simple closed curve. Later Bing [Ann. of Math. (2) 80 (1964), 78--93; MR 29 #611] gave voluminous families of such periodic maps. Henceforth we will therefore restrict attention to the case of smooth (or, equivalently, piecewise linear) periodic maps. Soon the analogous question in higher dimensions began to receive some attention, and various examples of nonlinear actions on higher-dimensional spheres were discovered by various people. In particular, C. H. Giffen [Amer. J. Math. 88 (1966), 187--198; MR 33 #6620] showed that the generalized Smith conjecture is false in dimensions greater than three: If $n>3$ then the $n$-sphere admits a smooth periodic map whose fixed point set is a knotted $(n-2)$-sphere. Subsequently, F. Waldhausen [Topology 8 (1969), 81--91; MR 38 #5209] succeeded in showing that the classical Smith conjecture is indeed true for piecewise linear involutions on the $3$-sphere. His techniques were relatively elementary, but complicated, and seemed to have no application to the case of maps of odd periods greater than two. During this period of time various restrictions on the possible knot types which could be fixed point sets were discovered by a number of people. The first such result was due to Montgomery and H. Samelson [Canad. J. Math. 7 (1955), 208--220; MR 16, 946] who proved that the fixed point set of a piecewise linear involution could not be a "two-stranded cable"---that is, it could not bound a Mobius band. S. Kinoshita and R. H. Fox gave restrictions on the Alexander polynomial of a knot which is the fixed point set of a $Z\sb p$-action. Because their arguments are essentially algebraic their results still have content (even in the light of later developments) in the case of actions on homology spheres. The survey by Fox [Topology of $3$-manifolds and related topics, 177--182, Prentice-Hall, Englewood Cliffs, N.J., 1962; MR 25 #3524] gives an overview of these things as of 1962. Various other knot types were systematically ruled out as the fixed point set of a periodic map: torus knots [Giffen, "Fibered knots and periodic transformations", Ph.D. Thesis, Princeton Univ., Princeton, N.J., 1964]; $2$-bridge knots [S. Cappell and J. Shaneson, Topology 17 (1978), no. 1, 105--107; MR 58 #2819]; all cabled knots, doubled knots, nonfibered knots with unique incompressible spanning surface, etc. [R. Myers, "Companionship of knots and the Smith conjecture", Ph.D. Thesis, Rice Univ., Houston, Tex., 1977; Trans. Amer. Math. Soc. 259 (1980), no. 1, 1--32; MR 81e:57008]; cabled knots [G. A. Swarup, Quart. J. Math. Oxford Ser. (2) 31 (1980), no. 121, 105--108; MR 81c:57006]. Finally, in the fall of 1978, the Smith conjecture was completely verified in the affirmative. The present volume is the outgrowth of a symposium on the Smith conjecture held at Columbia University (where Paul Smith spent most of his professional life) in the spring of 1979. First there is a historical introduction to the problem, followed by an overview of the solution, by Morgan. After some preliminary reductions the solution splits naturally into two parts: (1) no nontrivial incompressible surfaces in the complement of the fixed point set, and (2) a nontrivial incompressible surface in the complement. The first part involves Thurston's theorem on the existence of hyperbolic structures and Bass's analysis of finitely generated subgroups of ${\rm GL}\sb 2({\bf C})$. Shalen shows how these ingredients fit together, Morgan provides an excellent long description of Thurston's theorem (incomplete, but the best that exists), and Bass provides the proof of his result. For the second part Gordon and Litherland show how to invoke the equivariant loop theorem of Meeks and Yau to reach a contradiction, and Meeks and Yau describe their minimal surfaces approach to the loop theorem. The volume concludes with three sections that describe various generalizations of the Smith conjecture. Meeks and Yau analyze smooth group actions on Euclidean $3$-space, showing that any such compact group is isomorphic to a subgroup of ${\rm SO}(3)$, and, except in the case of the alternating group $A\sb 5$, the action is equivalent to an orthogonal action. Davis and Morgan analyze actions of noncyclic groups on the $3$-sphere, showing that in most cases a nonfree action is equivalent to an orthogonal action. Finally, Davis gives a sampler of related results in higher dimensions. The most significant subsequent results are due to Thurston and to Meeks and G. P. Scott . Thurston has announced a far-reaching extension of his hyperbolization theorem, showing in particular that a nonfree, orientation-preserving finite group action on a compact $3$-manifold that contains no incompressible sphere or torus, preserves a geometric structure on the $3$-manifold. This can be used to show, for example, that any $A\sb 5$ action on $3$-space is standard. Meeks and Scott have shown, using the techniques of minimal surfaces, that a group action on a Seifert fiber space (which preserves the homotopy class of the fiber) is equivalent to a fiber-preserving action. As a corollary they observe that any orientation- and end-preserving action on $M\sp 2\times [0,1]$ is equivalent to a product action. (This is a direct generalization of the Smith problem which corresponds to the case $M\sp 2=S\sp 2$.) The solution of the Smith conjecture seems to require an analytic attack. Recent works of the reviewer, M. J. Dunwoody , and W. Jaco and J. H. Rubinstein have provided "elementary" proofs of the equivariant Dehn lemma, sphere theorem, and loop theorem using classical three-dimensional techniques. This removes the reliance upon the theory of minimal surfaces in the second case of the proof. But no one has even approached being able to find an alternative to the existence of hyperbolic structures in the first case described above. Reviewed by Allan Edmonds Cited in: 98c:52013 97m:57008 94e:57015 94c:57024 93h:57024 93b:57040 89g:57012 � Copyright American Mathematical Society 1986, 1998