Allen Hatcher

A presentation for the mapping class group of a closed orientable surface

THE CENTRAL objects of study in this paper are collections {C,, . . . , C,} of g disjoint circles on a closed orientable surface M of genus g, whose complement M-(C, u . . . U C,) is a 2g-punctured sphere. We call an isotopy class of such collections a cut system. Of course, any two cut systems are related by a diffeomorphism of M, by the classification of surfaces. We show that any two cut systems are also joined by a finite sequence of simple moues, in which just one Cj changes at a time, to a circle intersecting it transversely in one point and disjoint from the other Ci’s. Furthermore, we find a short list of relations between sequences of simple moves, sufficient to pass between any two sequences of simple moves joining the same pair of cut systems. From these properties of cut systems it is a routine matter to read off a finite presentation for the mapping class group of M, the group of isotopy classes of orientation preserving self-diffeomorphisms of M. Unfortunately, the presentation so obtained is rather complicated, and stands in need of considerable simplification before much light will be shed on the structure of the mapping class group. Qualitatively, one can at least deduce from the presentation that all relations follow from relations supported in certain subsurfaces of M, finite in number, of genus at most two. This may be compared with the result of Dehn [3] and Lickorish [4] that the mapping class group is generated by diffeomorphisms supported in finitely many annuli. A finite presentation in genus two was obtained by Birman-Hilden[2], completing a program begun by Bergau-Mennicke [l]. For higher genus the existence of finite presentations was shown by McCool[ lo], using more algebraic techniques. For another approach to finite presentations, see [12], and for general background on mapping class groups, see [ 111. Our methods apply also to maximal systems of disjoint, non-contractible, nonisotopic circles on M. This is discussed briefly in an appendix.

On triangulations of surfaces

Abstract Our purpose here is to give a simple topological proof of a theorem of Harer, that the simplicial complex having as its top-dimensional simplices the isotopy classes of triangulations of a compact surface with a fixed set of vertices is contractible, except in a few special cases. The proof yields mild generalizations of Harers theorem, allowing more general vertex sets, as well as extending to a larger complex whose simplices correspond to curve systems consisting of circles as well as arcs. As a corollary we deduce the well-known and useful classical fact that any two isotopy classes of triangulations of a compact surface with a fixed set of vertices are related by a finite sequence of elementary moves in which only one edge changes at a time.

Homological stability for automorphism groups of free groups

Let F, be a free group on n generators, Aut(F,) its group of automorphisms, and Out(F,) its outer automorphism group, the quotient of Aut(F,) by inner automorphisms. There has been much progress of late in the study of these groups via the one-dimensional model which arises from regarding F, as the fundamental group of a graph; see e.g., [BH] and [CV]. In this paper we return to a three-dimensional model first used by J. H. C. Whitehead in the 1930s, which involves looking at embedded 2-spheres in a connected sum of S 1 x S2s. Refining Whiteheads techniques and applying subsequent results of Laudenbach, we use this three-dimensional model to prove:

A proof of the Smale Conjecture, Diff(S3) 0(4)

The Smale Conjecture [9] is the assertion that the inclusion of the orthogonal group 0(4) into Diff(S3), the diffeomorphism group of the 3-sphere with the Cw topology, is a homotopy equivalence. There are many equivalent forms of this conjecture, some of which are listed in the appendix to this paper. We shall prove

Incompressible surfaces in punctured-torus bundles

Abstract We derive in this paper the classification up to isotopy of the incompressible surfaces in hyperbolic 3-manifolds which fiber over the circle with fiber a once-punctured torus. From this classification it follows that most of the 3-manifolds obtained by compactifying these bundles via a circle at infinity are closed hyperbolic 3-manifolds which contain 1.0 incompressible surfaces, i.e., are not Haken manifolds.

Stabilization for mapping class groups of 3-manifolds

We prove that the homology of the mapping class group of any 3-manifold stabilizes under connected sum and boundary connected sum with an arbitrary 3-manifold when both manifolds are compact and orientable. The stabilization also holds for the quotient group by twists along spheres and disks, and includes as particular cases homological stability for symmetric automorphisms of free groups, automorphisms of certain free products, and handlebody mapping class groups. Our methods also apply to manifolds of other dimensions in the case of stabilization by punctures.

Boundary slopes for Montesinos knots

FOR A KNOT K c S3, let S(K) c Q u {CQ} be the set of slopes of boundary curves of incompressible, %incompressible orientable surfaces in the knot exterior, slopes being normalized in the standard way so that a longitude has slope 0, a meridian slope co. These sets S(K) of %slopes are of special interest because of their relation with Dehn surgery and character varieties; see e.g., [2]. The only general results known so far are that S(K) is always finite [S] and contains at least two elements [3], including of course 0 (coming from a minimal genus Seifert surface). Only for special classes of knots has S(K) been determined exactly. For the (p, q) torus knot, S(K) = (0, pq}. For 2-bridge knots, S(K) is an arbitrarily large set of even integers computable via continued fractions [7], A non-integer s for the motivating example (- 2,3,7) we find S(K) = (0, 16, 18; .20). For somewhat more complicated cases, a small computer can do the work rather quickly. Some examples of these computer calculations are given in the last section of the paper. These include the Montesinos knots of I 10 crossings, plus a few other random examples of greater complexity.

CERF THEORY FOR GRAPHS

We develop a deformation theory for k-parameter families of pointed marked graphs with fixed fundamental group Fn. Applications include a simple geometric proof of stability of the rational homology of Aut(Fn), computations of the rational homology in small dimensions, proofs that various natural complexes of free factorizations of Fn are highly connected, and an improvement on the stability range for the integral homology of Aut(Fn).

Homology stability for outer automorphism groups of free groups

We prove that the quotient map from Aut(Fn) to Out(Fn) induces an isomorphism on homology in dimension i for n at least 2i + 4. This corrects an earlier proof by the first author and significantly improves the stability range. In the course of the proof, we also prove homology stability for a sequence of groups which are natural analogs of mapping class groups of surfaces with punctures. In particular, this leads to a slight improvement on the known stability range for Aut(Fn), showing that its ith homology is independent of n for n at least 2i + 2. AMS Classification 20F65; 20F28, 57M07

Homeomorphisms of sufficiently large P2-irreducible 3-manifolds

LET V BE a compact connected PL 3-manifold which is irreducible, sufficiently large, and contains no embedded projective plane having a trivial normal bundle. Denote by PL( V rel a), G( V rel a) the simplicial spaces of PL homeomorphisms, respectively, homotopy equivalences of V which restrict to the identity on dV. Waldhausen[3] showed that the inclusion PL( V rel ~3) + G( V rel a) induces an isomorphism on 7~~. Laudenbach [ l] extended this to r,. Pushing their techniques further, we prove in this paper: