Cobordism invariants, the Kervaire invariant and fixed point free involutions

Browder, William

doi:10.1090/S0002-9947-1973-0324717-0

Cobordism invariants, the Kervaire invariant and fixed point free involutions
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Trans. Amer. Math. Soc. 178 (1973), 193-225 Request permission

Abstract:

Conditions are found which allow one to define an absolute version of the Kervaire invariant in ${Z_2}$ of a ${\text {Wu - }}(q + 1)$ oriented 2q-manifold. The condition is given in terms of a new invariant called the spectral cobordism invariant. Calculations are then made for the Kervaire invariant of the n-fold disjoint union of a manifold M with itself, which are then applied with $M = {P^{2q}}$, the real protective space. These give examples where the Kervaire invariant is not defined, and other examples where it has value $1 \in {{\mathbf {Z}}_2}$. These results are then applied to construct examples of smooth fixed point free involutions of homotopy spheres of dimension $4k + 1$ with nonzero desuspension obstruction, of which some Brieskorn spheres are examples (results obtained also by Berstein and Giffen). The spectral cobordism invariant is also applied directly to these examples to give another proof of a result of Atiyah-Bott. The question of which values can be realized as the sequence of Kervaire invariants of characteristic submanifolds of a smooth homotopy real projective space is discussed with some examples. Finally a condition is given which yields smooth embeddings of homotopy ${P^m}$’s in ${R^{m + k}}$ (which has been applied by E. Rees).

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