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