Free Piecewise Linear Involutions on Spheres

If T is a piecewise linear fixed-point free involution on S, the orbit space Q = S/T is a PL-manifold homotopy equivalent to P»(JR) ~ P n [2J; the affirmative solution to the Poincaré conjecture implies that conversely for n&Z, 4 the double covering manifold of any such Q can be identified with S. Write In for the set of (oriented if n is even) PL-homeomorphism classes of manifolds Q homotopy equivalent to P \ We will compute In for ti^Z, 4. Let Q be as above. We define a normal invariant rj(Q). Take a homotopy equivalence h: P-*Q (orientation-preserving if n is odd): this is unique up to homotopy. Approximate AX0 by a PL-embedding PXQ-*R{N>n); let v be the normal bundle of the embedding, which exists if N is large enough [5], and F: p—>e the fibre homotopy trivialisation induced by the homotopy equivalence [7], [10, 3.5]. Then (*>, F) induces a homotopy class TJ(Q) of maps P—»G/PL, which depends only on the PL-homeomorphism class of Q. We have thus defined 77: ƒ»—*[P, G/PL]: our description follows Sullivan [8], the main idea goes back to Novlkov [6]. We next compute [P, G/PL]. The homotopy groups of G/PL are known to be Z (in dimensions 4i), Z2 (in dimensions 4i+2), and 0 (in odd dimensions). Further, Sullivan [8] has shown that if finite groups of odd order are ignored, the only nonzero &-invariant is the first (which is 8Sq). We choose fundamental classes x2i G# (G/PLi Z2) (*V2), aE.H {P\ Z2). Because of the fe-invariant, [P, G/PL]=~Z4: let y be an isomorphism. Further, denote by r the restriction [P»+\ G/PL]-»[P, G/PL]. Then we have