John Kalliongis

Isotopies of 3-manifolds

Abstract An isotopy of a manifold M that starts and ends at the identity diffeomorphism determines an element of π 1 (Diff( M )). For compact orientable 3-manifolds with at least three nonsimply connected prime summands, or with one S 2 × S 1 summand and one other prime summand with infinite fundamental group, infinitely many integrally linearly independent isotopies are constructed, showing that π 1 (Diff( M )) is not finitely generated. The proof requires the assumption that the fundamental group of each prime summand with finite fundamental group imbeds as a subgroup of SO(4) that acts freely on S 3 (conjecturally, all 3-manifolds with finite fundamental group satisfy this assumption). On the other hand, if M is the connected sum of two irreducible summands, and for each irreducible summand P of M , π 1 (Diff( P )) is finitely generated, then results of Jahren and Hatcher imply that π 1 (Diff( M )) is finitely generated. The isotopies are constructed on submanifolds of M which are homotopy equivalent to a 1-point union of two 2-spheres and some finite number of circles. The integral linear independence is proven by obstruction-theoretic methods.

Equivalence and strong equivalence of actions on handlebodies

An algebraic characterization is given for the equivalence and strong equivalence classes of finite group actions on 3-dimensional handlebodies. As one application it is shown that each handlebody whose genus is bigger than one admits only finitely many finite group actions up to equivalence. In another direction, the algebraic characterization is used as a basis for deriving an explicit combinatorial description of the equivalence and strong equivalence classes of the cyclic group actions of prime order on handlebodies with genus larger than one. This combinatorial description is used to give a complete closed-formula enumeration of the prime order cyclic group actions on such handlebodies. The study of finite group actions on compact 2-manifolds has an intricate history which may be traced back well into the 19th century. (For background on this see the survey article and bibliography [El].) Many of the results which have been established in this study may be obtained through the following algebraic setup: if G is a finite group acting effectively on the compact surface S, then there is an associated group extension 1 ill(S) -+E -+G -+1 obtained by lifting the action to the universal covering of S. Given such a group extension whose abstract kernel G -Out(HI (S)) preserves the peripheral structure of HII(S) (if S is bounded), the affirmative solution to the Nielsen realization problem [K] implies that there is a G-action which corresponds to it. Furthermore it is known that two G-actions on S associated with the same group extension (in the sense of equivalence of exact sequences) are equivalent in fact even strongly equivalent. (In the closed case a proof may be found in [ZZ] for the bounded case a similar approach works using [M].) Within this setting an enumeration of the equivalence classes of G-actions on a given surface is theoretically possible through the techniques employed in [S] and again in [E3]. The key idea is that the set of free G-actions with a fixed quotient surface are algebraically categorized by epimorphisms from the fundamental group of the quotient surface to G. Actions which are not free may be categorized in a similar way using the orbifold fundamental group of the quotient orbifold. In this paper we will utilize a similar approach to study the equivalence and strong equivalence classes of finite group actions on 3-dimensional handlebodies. In Received by the editors April 17, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 57M99; Secondary 57S25, 57M12, 57M15.

Diffeomorphisms of elliptic 3-manifolds

1 Elliptic 3-manifolds and the Smale Conjecture.- 2 Diffeomorphisms and Embeddings of Manifolds.- 3 The Method of Cerf and Palais.- 4 Elliptic 3-manifolds Containing One-sided Klein Bottles.- 5 Lens Spaces

Actions on lens spaces which respect a Heegaard decomposition

Abstract Let L = L ( p , q ) be a 3-dimensional lens space. We consider smooth finite group actions on L which leave a Heegaard torus and each of its complementary components invariant. Such an action is said to have rotational type if each element is isotopic to the identity on L , and to have dirotational type otherwise. In this paper we enumerate and classify these two types of actions up to equivalence, where two actions are equivalent if their images are conjugate in the group of self-diffeomorphisms of L . When q 2 ≠±1 ( mod p) this results in a conjecturally complete classification of all finite group actions on L .

Involutions of nonirreducible 3-manifolds

Abstract The paper is concerned with the problem of distinguishing the equivalence classes of involutions of a compact orientable 3-manifold M , each of whose summands in a prime decomposition is irreducible and has infinite first homology, by considering their representations in Out( π 1 ( M )). We determine how close these involutions are to being slide equivalent and give conditions under which they are equivalent.

Finite actions on the 2-sphere, the projective plane and I-bundles over the projective plane

In this paper, we consider the finite groups which act on the 2 -sphere S 2 and the projective plane P 2 , and show how to visualize these actions which are explicitly defined. We obtain their quotient types by distinguishing a fundamental domain for each action and identifying its boundary. If G is an action on P 2 , then G is isomorphic to one of the following groups: S 4 , A 5 , A 4 , Z m or Dih( Z m ) . For each group, there is only one equivalence class (conjugation), and G leaves an orientation reversing loop invariant if and only if G is isomorphic to either Z m or Dih( Z m ) . Using these preliminary results, we classify and enumerate the finite groups, up to equivalence, which act on P 2  ×  I and the twisted I-bundle over P 2 . As an example, if m  > 2 is an even integer and m /2 is odd, there are three equivalence classes of orientation reversing Dih( Z m ) -actions on the twisted I-bundle over P 2 . However if m /2 is even, then there are two equivalence classes.

Diffeomorphisms and Embeddings of Manifolds

This chapter contains foundational material on spaces of diffeomorphisms and embeddings. Such spaces are known to be Frechet manifolds, separable when the manifolds involved are compact. Versions of these and related facts are developed for manifolds with boundary, as well as in the context of fiber-preserving diffeomorphisms and maps. The latter utilizes a modification of the exponential map, called the aligned exponential, adapted to the fibered structure.

Elliptic Three-Manifolds Containing One-Sided Klein Bottles

Extending work of N. Ivanov, the Smale Conjecture is proven for all elliptic three-manifolds containing one-sided Klein bottles, other than the lens space L(4, 1). The technique takes a parameterized family of diffeomorphisms and uses its restriction to embeddings of the Klein bottles to deform the diffeomorphisms to preserve a Seifert fibration of the manifolds. The Conjecture is deduced from this. Another key element of the proof is a collection of techniques for working with parameterized families developed by A. Hatcher.In this chapter, we will prove Theorem 1.3. Section 4.1 gives a construction of the elliptic three-manifolds that contain a one-sided geometrically incompressible Klein bottle; they are described as a family of manifolds M(m, n) that depend on two integer parameters \(m,n \geq 1\). Section 4.2 is a section-by-section outline of the entire proof, which constitutes the remaining sections of the chapter.

Elliptic Three-Manifolds and the Smale Conjecture

After a discussion of the (Generalized) Smale Conjecture, the main results of the monograph are summarized. The extent to which the Smale Conjecture extends to larger classes of three-manifolds—usually in a limited form called the Weak Smale Conjecture, if at all—is detailed. The chapter closes with a brief discussion of why Perelman’s methods appear not to give progress on the Smale Conjecture. As noted in the Preface, theSmale Conjecture is the assertion that the inclusion \(\mathrm{Isom}(M) \rightarrow \mathrm{Diff}(M)\) is a homotopy equivalence whenever M is an elliptic three-manifold, that is, a three-manifold with a Riemannian metric of constant positive curvature (which may be assumed to be 1). TheGeometrization Conjecture, now proven byPerelman, shows that all closed three-manifolds with finite fundamental group are elliptic.In this chapter, we will first review elliptic three-manifolds and their isometry groups. In the second section, we will state our main results on the Smale Conjecture, and provide some historical context. In the final two sections, we discuss isometries of nonelliptic three-manifolds, and address the possibility of applying Perelman’s methods to the Smale Conjecture.

The Method of Cerf and Palais

A fundamental theorem of R. Palais and J. Cerf shows that the map sending a diffeomorphism to its restriction to an imbedding of a submanifold is a locally trivial fibration of the spaces of mappings involved. In this chapter, this theorem is extended to other maps between spaces of mappings, in particular to the map sending each fiber-preserving diffeomorphism of a bundle to the diffeomorphism it induces on the base manifold. Versions with boundary control are obtained, as well as versions for singular fiberings, a class that includes Seifert-fibered three-manifolds. A final section gives a proof that sending the space of diffeomorphisms of a singularly fibered manifold to its space of cosets by the subgroup of fiber-preserving diffeomorphisms is a fibration. This coset space may be regarded as the space of fibered structures diffeomorphic to the given one.R. Palais (Comment. Math. Helv. 34:305–312, 1960) proved a very useful result relating diffeomorphisms and embeddings. For closed M, it says that if \(W \subseteq V\) are submanifolds of M, then the mappings \(\mathrm{Diff}(M) \rightarrow \mathrm{Emb}(V,M)\) and \(\mathrm{Emb}(V,M) \rightarrow \mathrm{Emb}(W,M)\) obtained by restricting diffeomorphisms and embeddings are locally trivial, and hence are Serre fibrations. The same results, with variants for manifolds with boundary and more complicated additional boundary structure, were proven by J. Cerf (Bull. Soc. Math. Fr. 89:227–380, 1961). Among various applications of these results, the Isotopy Extension Theorem follows by lifting a path in \(\mathrm{Emb}(V,M)\) starting at the inclusion map of V to a path in \(\mathrm{Diff}(M)\) starting at \({1}_{M}\). Moreover, parameterized versions of isotopy extension follow just as easily from the homotopy lifting property for \(\mathrm{Diff}(M) \rightarrow \mathrm{Emb}(V,M)\) (see Corollary 3.3).In this chapter, we will extend the theorem of Palais in various ways. Many of our results concern fiber-preserving maps. For example, in Sect. 3.3 we will prove the