Some Non-Linear Equivariant Sphere Bundles

509

where T' is still a Л„ manifold corresponding to хещ{0^^.и SO^j,i), Because of the above diagram, x is representable by 7^2°^' | (-Z'^) x 5"*. On the other hand, ing to our remark at the beginning of this proof, for the knot -л; given by F'((-r*) x xO)ciS^xD^-'^czS^^^^\ œ^{"-x) is also represented by П20р'\{-I^)xS'^, Thus û>3 {^)— —^- This completes the proof of Theorem 5.2.

By Theorem 5.2, the problem of deciding whether the total spaces of the bundles constructed in Theorem 4.5 are equivariantly diffeomorphic to I^ x 52^«-!^ j^ largely reduced to homotopy theory. As сок(я^505 -^я^О5) = 0 for A: = 3,4, there are no such non-Hnear symplectic bundles in these dimensions. We do not know whether our linear unitary bundles have famihar total spaces for к — Ъ,А. Now assume k^5. We have Levine's exact sequence [13]

According to Theorem 5.2, the total spaces of all bundles constructed in Theorem 4.5 are equivariantly diffeomorphic to some I'' x S^^"~^ if and only ifkQT{d:n,,{Ga+u SOa+i)-^nk-i^SOa+^)^cok{nkSOd+i-^nkGd+i) is contained in 1тшз. As щ{Оа+и SOd+i) is finite for all k'^5, d-2, 4, CO3 is certainly surjective unless A: = 2 mod4. In the latter case, Р,^ = ^2» ^^^ ^ъ is an epimorphism if and only if a codimension 2 knot in 5^"^^ with Arf invariant 1 remains non-trivial after {d- l)-fold suspension. (Exactly then Pj^-^0^'^^'^~'^ is injective.) As the Kervaire sphere is not diffeomorphic to the standard sphere in dimensions different from 2'*-3 [4, Corollary 2], 0)3 is surjective for all кФТ-1. For A: = 6, 14, using [20], Ш3 can be computed to be surjective in the unitary case (öf=2). For A: = 6, c/=4 (symplectic action), Q^''^^^"^ is zero for sional reasons [13]. As ker5= n^{G^, S0s) = Z2, 0)3 is not surjective in this case, and we have spotted a non-linear symplectic S^"~^ bundle over S^ whose total space is not equivariantly diffeomorphic to 5^ x 5®""Ч We summarize:

PROPOSITION 5.4. If k^5, кфТ-1, then the total spaces of the nonlinear equivariant S^dw-i huddles over S^, constructed in Theorem 4.5, are equivariantly diffeomorphic to a product of a homotopy k-sphere with trivial action andS^^'^'K This is also true for к = 6, \4 in the unitary case. For к = 6, there is a non-linear symplectic S^"~^ bundle over S^ whose total space is not equivariantly diffeomorphic to a product of a homotopy sphere with trivial action and 5 ® " " ^

REFERENCES

[ 1 ] Antonelli, p., Burghelea, D., and Kahn, P. J., GromoU groups, DiffS^, and bilinear tions of exotic spheres. Bull. Amer. Math. Soc. 76 (1970), 772-777. [2] BoREL, A., Groupes d'homotopie des groupes de Lie /, Séminaire H. Cartan 1949/50 N0.12. [3] Browder, W., Torsion in H-spaces, Ann. Math. 74 (1961), 24-51.