Relations among smooth Kervaire classes and smooth involutions on homotopy spheres (Q1824914)

The main result of this paper is: Theorem. If \(k_{4i+2}\in H^{4i+2} (G/0,\mathbb{Z}_ 2)\) denotes the smooth Kervaire class, then for integers r, s with \(r>s\geq 2\) the following relation holds: \[ Sq^{2r-1} Sq^{2r-2} \ldots Sq^{2s-1} Sq^{2s-2} k_{2s-2} = Sq^{2s-1} Sq^{2s-2} k_{2r-2}. \] As an application of this result it is shown that for integers m, n with \(0\leq m\leq 2n\), if \(n>0\) and \(n+1\) is not a power of 2, then the Kervaire obstruction \[ c: [S^{2m+1} \times P^{4n-2m+1}, G/0] \to \mathbb{Z}_2 \] is trivial. As a consequence of this it is proved that there does not exist an orientation reversing almost linear smooth involution of the Kervaire sphere of dimension \(4n+1\) which is t-isovariantly homotopy equivalent to a linear involution on the standard sphere, and that the connected sum of the real projective space \(P^{4n+1}\) with the Kervaire sphere is not diffeomorphic to \(P^{4n+1}\) when \(4n+4\) is not a power of 2. The proof of the main theorem requires considerable manipulation of mod 2 binomial coefficients.