Allan L. Edmonds

Aspects of group actions on four-manifolds☆

Abstract A collection of various consequences of the existence of finite cyclic group actions on simply connected topological 4-manifolds, with an emphasis on involutions, is presented as a complement to various known constructions of group actions on these manifolds. The relationship between the representation on integral cohomology and the fixed-point set is described. The significance of a spin structure on the 4-manifold for the fixed-point set of a locally linear involution is described, leading to examples of 4-manifolds admitting no locally linear involution. Further restrictions on the two-dimensional parts of the fixed-point set of an involution are derived.

Group actions on fibered three-manifolds.

In this paper we présent several results about finite group actions on threedimensional manifolds. The results are primarily directed toward a géométrie understanding of periodic knots in the 3-sphere, that is, knots left invariant by periodic homeomorphism of S3 which fix a simple closed curve in the complément of the knot. One noteworthy application of the présent techniques is that nontrivial periodic knots hâve "Property i?," that is, surgery on such a knot cannot produce S^S2. Let G be a finite group acting efïectively and smoothly on an orientable 3-manifold M, preserving orientation. The first resuit is that if the orbit manifold M* contains an incompressible surface F*, then (after suitably adjusting the embedding) the preimage of F* in M is an incompressible surface. The proof of this makes use of the Equivariant Loop Theorem of Meeks and Yau [12,13] and isr given in Section 2. An immédiate corollary of this resuit is that a periodic knot has an invariant incompressible Seifert surface. If there is a bound on the possible gênera of the incompressible Seifert surfaces of a given knot K (as is the case for fibered knots), then the Riemann-Hurwitz formula places nontrivial bounds on the possible periods of K. See Section 2. There is an extensive literature devoted to the problem of determining the possible periods of a given knot [1,5,6,7,11,15,16,18,22]. Most previous work on this problem has been heavily algebraic in nature, in contrast to the présent more géométrie approach. In Section 3 the preceding work is applied to prove that periodic knots hâve Property jR. Now suppose that F is a compact, orientable surface and that G acts on Fx[0,1] preserving orientation and leaving both Fx{0} and Fx{1} invariant. We show that the action is équivalent to the level-preserving action which is the product of the action of Fx{0} with the trivial action on the interval [0,1], except

Least area Seifert surfaces and periodic knots

Abstract It is shown that if a nontrivial knot K in S 3 has genus g and is invariant under a smooth action of a cyclic group C m of order m which fixes a simple closed curve disjoint from K , then K is the boundary of an invariant Seifert surface of genus g . As a corollary one has that m ⩽2 g +1. The proof utilizes the theory of least area surfaces in 3-manifolds.

Remarks on the Cobordism Group of Surface Diffeomorphisms

In this note we sketch our computation of the group A 2 of cobordism classes of orientation-preserving diffeomorphisms of closed, oriented surfaces. See Sect. 2 for precise definitions. This computation was made independently and a little earlier by Bonahon [2]. Our development follows in part the same lines, extending approaches to related questions by Scharleman [13] and Johannson and Johnson [10], who were the first to apply the characteristic manifold theory of Johannson [8] and Jaco and Shalen [7] to questions about surface diffeomorphisms. Bonahons approach depends crucially on the Mostow Rigidity Theorem and the unpublished and as yet inaccessible Hyperbolization Theorem of Thurston. In contrast, we get by with the G-Signature Theorem (in dimension two, a classical formula) and some number theory. For this reason we have belatedly decided to publish our own approach. Our main contribution is a proof, independent of deep three-dimensional geometry, that an orientation-preserving periodic surface diffeomorphism which bounds as a diffeomorphism also bounds periodically. Cf. I-2 ; Proposition B]. The proof is given in Sect. 3. In Sect. 2 we develop the proof of the main calculation, modulo the analysis of periodic diffeomorphisms deferred to Sect. 3.

Symmetric representations of knot groups

Abstract In this paper we give an explicit and constructive description of the pairs (μ, λ) of elements in the symmetric group S n which can be realized as the image of the meridian-longitude pair of some knot K in the 3-sphere S 3 under a representation π 1 ( S 3 − K ) → S n . The result is applied to give an otherwise nonobvious restriction on the numbers of branch curves of a branched covering of S 3 , answering a question of R. Fox and K. Perko.

Sommerville's Missing Tetrahedra

D.M.Y. Sommervilles 1923 classification of tetrahedra that can tile 3-space in a proper, face-to-face manner is completed, by showing that the case he failed to consider cannot occur, although it does occur in non-proper and in non-face-to-face tilings.

Pseudofree group actions on spheres

R. S. Kulkarni showed that a finite group acting pseudofreely, but not freely, preserving orientation, on an even-dimensional sphere (or suitable sphere-like space) is either a periodic group acting semifreely with two fixed points, a dihedral group acting with three singular orbits, or one of the polyhedral groups, occurring only in dimension 2. It is shown here that the dihedral group does not act pseudofreely and locally linearly on an actual n-sphere when n is congruent to 0 mod 4. The possibility of such an action when n is congruent to 2 mod 4 and n>2 remains open. Orientation-reversing actions are also considered.

Involutions on odd four-manifolds

Abstract It is shown that every closed, simply connected topological 4-manifold having an odd intersection pairing, with the possible exception of the fake C P2, admits an involution. We show that in many cases the involutions described here can be constructed to be locally linear, provided the Kirby-Siebenmann triangulation obstruction vanishes. It remains an open question, reduced to a question about characteristic elements of quadratic forms, whether this is true in general.

Torsion free subgroups of fuchsian groups and tessellations of surfaces

It has been known for many years that a finitely generated fuchsian group G i.e. a finitely generated discrete subgroup of orientation-preserving isometries of the hyperbolic plane contains a torsion free subgroup of finite index. The known proofs are by representations in the symmetric groups cf. Fox [3] or by the method of congruence subgroups cf. Mennicke [4]. The latter method extends to all finitely generated matrix groups cf. Selberg [5]. There is no information about the possible indices of subgroups in these proofs. Here we announce the precise determination of the possible indices of torsion free subgroups of finite index in terms of the torsion in G. Using the connection between fuchsian groups and uniformization of Riemann surfaces the results may be interpreted as a step in determining a class of intermediate uniformizations, or looked in a different way, a step towards a topological classification of holomorphic maps between Riemann surfaces of finite type. Contained herein are some results of a naive geometric interest. Namely they imply the existence of certain interesting tessellations of surfaces which are natural generalizations of the tessellations of the sphere determined by Platonic solids. We remark that we completely leave aside the questions of normality of subgroups. Determining the indices of normal torsion-free subgroups of finite index in fuchsian groups appears to involve deeper number theoretic considerations which are probably yet to be understood. To formulate our main result let G have a standard presentation with generators ax, bx, . . . , ag, bgy xt, . . . , xr, yx, . . . , ys and relations xTM 1 = • • • = xTM = 1 and axbxa\ b\ • • • agbga~ b~xl • • • xry1 • • • ys = 1. Let / = LCM{mt, . . . , mr}9 and let L2\ denote the 2-primary part of /. We say that G has odd type if s = 0, l,2^ > 1, and the number of m^s such that L2)\ i * °&&* Otherwise G has even type.

Periodic maps of composite order on positive definite 4-manifolds

The possibilities for new or unusual kinds of topological, locally linear periodic maps of non-prime order on closed, simply connected 4-manifolds with positive definite intersection pairings are explored. On the one hand, certain permuta- tion representations on homology are ruled out under appropriate hypotheses. On the other hand, an interesting homologically nontrivial, pseudofree, action of the cyclic group of order 25 on a connected sum of ten copies of the complex projective plane is constructed.