Groups of free involutions of homotopy $S^{[n/2]}\times S^{[(n+1)/2]}$’s

Abstract:

Let $M$ be an oriented $n$-dimensional manifold which is homotopy equivalent to ${S^l} \times {S^{n - l}}$, where $l$ is the greatest integer in $n/2$. Let $Q$ be the quotient manifold of $M$ by a fixed point free involution. Associated to each such $Q$ are a unique integer $k\bmod {2^{\varphi (l)}}$, called the type of $Q$, and a cohomology class $\omega$ in ${H^1}(Q;{Z_2})$ which is the image of the generator of the first cohomology group of the classifying space for the double cover of $Q$ by $M$. Let ${I_n}(k)$ be the set of equivalence classes of such manifolds $Q$ of type $k$ for which ${\omega ^{l + 1}} = 0$, where two such manifolds are equivalent if there is a diffeomorphism, orientation preserving if $k$ is even, between them. It is shown in this paper that if $n \geq 6$, then ${I_n}(k)$ can be given the structure of an abelian group. The groups ${I_8}(k)$ are partially calculated for $k$ even.