A Bilinear Form for Spin Manifolds

Peter S. Landweber, Robert E. Stong
1987 Transactions of the American Mathematical Society  
This paper studies the bilinear form on Hi( M; Z2) defined by [x, y] = xSq2 y[M] when M is a closed Spin manifold of dimension 2j + 2. In analogy with the work of Lusztig, Milnor, and Peterson for oriented manifolds, the rank of this form on integral classes gives rise to a cobordism invariant. Of course, these results are completely analogous to the work of Lusztig, Milnor, and Peterson [LMP], or originally Browder [Bt], on the form (x, y) = xSql y [M] for oriented manifolds of dimension 4k +
more » ... . The proofs are, unfortunately, rather more complicated, and involve the calculation of the Spin bordism of Eilenberg-Mac Lane spaces just outside the stable range. As a sidelight, this work helps to explain the work of Wilson [W] on the vanishing of Stiefel-Whitney classes in Spin manifolds. Knowledge of the form gives COROLLARY 1.3. For a closed Spin manifold M Sk + 2 of dimension 8k + 2, the Stiefel-Whitney class Sq3 V 4k is zero. In §2, the proof is begun by showing that there is a class 8 E H *( B Spin; Z2) for which pzSq2 pz[M] = pz· T*(8) [M]. In §3, the elementary properties of 8 are described, and in the following section, 8 is shown to be unique by a nasty
doi:10.2307/2000360 fatcat:2cpzblvcurdtbjs7kw245njqhi