$e$-invariants and finite covers. II

Abstract:

Let $\widetilde {M} \downarrow M$ be a finite covering of closed framed manifolds. By the Pontrijagin-Thom construction both $\widetilde {M}$ and $M$ define elements in the stable homotopy ring of spheres $\pi _*^s$. Associated to $\widetilde {M}$ and $M$ are their $e$invariants ${e_L}(\widetilde {M})$, ${e_L}(M) \in \mathbb {Q}/\mathbb {Z}$. If $\widetilde {N} \downarrow N$ is a finite covering of closed oriented manifolds, then there is a related invariant ${I_\Delta }(\widetilde {N} \downarrow N) \in \mathbb {Q}$ of the diffeomorphism class of the covering. In a previous paper we examined the relation between these invariants. We reduced the determination of ${e_L}(\widetilde {M}) - p{e_L}(M)$, as well as ${I_\Delta }(\widetilde {N} \downarrow N)$, for a $p$-fold cover, to the evaluation of certain sums of roots of unity. In this sequel we show how the invariant theory of the cyclic group $\mathbb {Z}/p$ may be used to evaluate these rums. For example we obtain \[ \sum _{\substack {\zeta ^p = 1\\\zeta \ne 1}} \frac {(1 + \zeta )(1 + \zeta ^{-1})}{(1 - \zeta )(1 - \zeta ^{-1})} = \frac {(p - 1)(p - 2)}{3} \] which may be used to determine the value of ${I_\Delta }$ in degrees congruent to $3$ $\mod 2(p - 1)$ for odd primes $p$.