Homeomorphisms of a Surface Which Act Trivially on Homology

Let 911 be the mapping class group of a surface of genus g > 3, and 9 the subgroup of those classes acting trivially on homology. An infinite set of generators for 9, involving three conjugacy classes, was obtained by Powell. In this paper we improve Powell's result to show that 9 is generated by a single conjugacy class and that [911, 9] = 5. I. Let M = M , be an orientable surface of genus g > 3 with one boundary component. (We shall frequently refer to the boundary curve as "the hole".) Let 91L = 9ILgil be its mapping class group (that is, homeomorphisms of M which are 1 on the boundary modulo homeomorphisms which are isotopic to 1 by an isotopy which is fixed on the boundary), and let í = íg , be the mapping classes of 9ILwhich induce the identity map on the homology group HX(M, Z). The group í is of specific interest to topologists for a number of reasons. For example, every hommology 3-sphere is obtained as a Heegaard decomposition with glueing map in 5 ; more precise knowledge about í could thus conceivably give some information about homology spheres. For the group-theoretically inclined, í supports a number of interesting problems. For example, it is an open question as to whether it is finitely generated. At present, information concerning í is scarce; the main references are given at the end. Let a be any bounding simple closed curve (BSCC) in M. It has then a well defined genus g(a), namely, the genus of the surface it bounds. (In contrast to the case of a closed surface, a bounds only on one side; the other side contains the hole.) Consider the group ^ c í generated by all twists Ta on BSCC's a of genus k; <ök is clearly a normal subgroup of 911, since the genus of a is invariant under any homeomorphism h of M, and Th^a) = hTah~x. If ax, a2 are two disjoint, homologous SCC's with a, not homologous to zero (we shall write "~" for "homologous") then (ctx, ot^ also has a genus g(otx, ot2), since otx, cx2 bound a piece of M. If we let 6¡¡Sk be the group generated by all maps of the form Ta Ta~x with g(ax, aj = k then el£k c Í is also normal in 9IL. We shall speak of such a map as "a generator of sl£k"; likewise, if a is a BSCC and g(a) = k, Ta will be called "a generator of 9*". Note that all generators of a given type are conjugate in 91L; this is just the same as saying, for example, that if a, ß are BSCC's of the same genus, then Received by the editors April 13, 1978 and, in revised form, July 25, 1978. AMS (MOS) subject classifications (1970). Primary 57A05, 55A99. © 1979 American Mathematical Society 0002-9939/79/0000-0277/S02.75 119 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use