Open book decompositions for odd dimensional manifolds

Abstract

AN OPEN book decomposition of a PL manifold M is a decomposition of M as V,, U(aV x D’), where V, = V x I/(X, 1) (h(x), O), h a PL homeomorphism of V which restricts to the identity on aV, and M is formed by joining V,, and aV x D* by a PL homeomorphism of their boundaries. The terminology was introduced by H. Winkelnkemper, who proved that simply connected closed PL manifolds of dimension r7 possess open book decompositions if their index is zero (cf. [16, 171). In particular, odd dimensional simply connected closed PL manifolds of dimension 27 always have open book decompositions. I. Tamura gave an independent proof [ 141 of the existence of open book decompositions (which he calls spinnable structures) in the odd dimensional simply connected case. Both Tamura[ 141 and Winkelnkemper [ 161 conjectured that the hypothesis of simple connectivity could be removed, but no proof of this conjecture has appeared. We wish to furnish a proof here. The reader can consult[5,6, IO, 12-16, 181 for various applications of open book decompositions. Our approach will be to start with a decomposition of M*‘+’ as W, UEWZ, where W, denotes the handles of index sk and W, denotes the handles of index zk + 1 in a handle decomposition, and show that after stabilization M= n W,rN-QS~ x Dk+‘)j) u D+‘+s ,j) k wJ-I(q,Dk+l X S’)j) = W U EW$, we can imbed V I in E’ so that W: = V x I. It is easy to get from this condition to the open book decomposition of M (cf. [16, 171). Note that we get as a corollary that M is a double. The representation of a manifold as a double under various hypotheses was first given by Smale[l I] (M simply connected, tors J&(M) = 0, dim M = 2k + I L 7), and has since been proved by Barden[3] (M orientable, dim M = 2k + 1 L 7), Levitt[S] (dim M = 4m + 2 z 6, M simply connected, tors H2,,,+, (M) = 0), Winkelnkemper [ 151 (M simply connected, dim M 2 7), and Alexander [ 1,2] (dim M L 7, rr,M finite if dim M even). Unfortunately, none of the proofs besides Smale’s is readily available in the literature. We are indebted to John Alexander for providing us with a copy of [2], which has influenced greatly our presentation here.