A NOTE ON THE W-COMPONENT OF (4« - l)-DIMENSIONAL HOMOTOPY SPHERES

The /»/'-component of a (An l)-dimensional homotopy sphere 2 e 64„ | = hP4n © (Coker7)4„_ , bounding a spin manifold M is shown to be computable in terms of the signature and the decomposable Pontrjagin numbers of M. Let Bm_i be the group of /»-cobordism classes of (m l)-dimensional homotopy spheres and let bPm C 8m_1 be the subgroup of those homotopy spheres bounding parallelizable w-manifolds. Using results of Kervaire and Milnor [5], G. Brumfiel showed that 64ll_x has a direct sum decomposition 04,,-i = ^4,, ©<,-i/im(/), where J: ir4n_l(SO) -> ■n-¡n_1 is the stable ./-homomorphism [1]. The group bP4l} is cyclic and its order IW4J can be expressed in terms of the «th Bernoulli number (see below). To define the projection map s:84n_x^bP4n^T/\bP4n\L Brumfiel shows that every homotopy sphere 2 E 04„_, bounds a spin manifold M with vanishing decomposable Pontrjagin numbers and that the signature of such an M is divisible by eight. Then he defines s by s(2) := | sign(M) G Z/\bP4l,\Z [1]. The above definition is not suitable to compute s(2) for a homotopy sphere 2 given explicitly by some geometric construction. The reason is that it is usually not possible to find an explicit spin manifold bounding 2 whose decomposable Pontrjagin numbers vanish. For example, if 2 is constructed by plumbing it bounds a manifold M by construction, but in general the decomposable Pontrjagin numbers of M do not vanish. In this note we show how to compute s(2) from the signature and the decomposable Pontrjagin numbers of a spin manifold M bounding 2. To describe explicitly which linear combination of decomposable Pontrjagin numbers is involved, let L(M) (resp. Â(Mj) be the L-class (resp. the /i-class) of M, which are power series in the Pontrjagin classes of M [4]. For any power series K(M) in the Pontrjagin classes, let Kn{M) be its 4«-dimensional component. Let ph(M) be the Pontrjagin character of M, i.e. the Chern character of the complexified tangent bundle of M. Here we think of the tangent bundle as an element of KO(M), in particular. Received by the editors December 20. 1984 and. in revised form, January 7, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 57R60, 55R50. ©1987 American Mathematical Society 0002-9939/87 $1.00 + $.25 per page 581 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use