Some Calculations with Milnor Hypersurfaces and an Application to Ginzburg’s Symplectic Bordism Ring

whose domain is the symplectic bordism ring of V. L. Ginzburg [2]. As Ginzburg proved σ to be injective, this establishes it to be an isomorphism, a result first proved by J. Morava [3] by a more topological argument. We also make some observations on the associated homology theory. For the benefit of topologists, we remark that the notion of manifold with symplectic structure is distinct from that of manifold with normal reduction to some compact symplectic group Sp(N) ⊂ U(N). The resulting bordism ring Ω ∗ may not be the homotopy ring of a Thom complex since there is apparently no transversality theory for such symplectic manifolds; the more familiar symplectic bordism ring is of course the homotopy of the Thom spectrum MSp.