The cohomology of holomorphic self maps of the Riemann sphere (Q1346342)

Let \(\text{Hol}_ k\) denote the space of degree \(k\) holomorphic self maps of the Riemann sphere, \(\mathbb{P}^ 1\), and let \(\text{Rat}_ k \subset \text{Hol}_ k\) denote the subspace of based maps. The cohomology groups \(H^*(\text{Rat}_ k;\mathbb{Z}_ p)\) (\(p\) prime) have been computed by \textit{F. R. Cohen, R. L. Cohen, B. M. Mann} and \textit{R. J. Milgram} [Acta Math. 166, No. 3/4, 163-221 (1991; Zbl 0741.55005)] and the algebra structure has been given by \textit{B. Totaro} [The cohomology ring of the space of rational functions (preprint MSRI 1990)] for \(p\) odd. In this note we compute the cohomology algebra \(H^*(\text{Hol}_ k;\mathbb{Z}_ p)\) when \(p\) does not divide \(k\). We also determine the cohomology groups and a graded version of the cohomology algebra when \(k = pm\). Direct analysis of the Leray-Serre spectral sequence for the standard bundle \(\text{Rat}_ k \to \text{Hol}_ k \to \mathbb{P}^ 1\) leads to difficulties, and so we make use of the principal bundle \(\text{SO}(3) \to \text{Hol}_ k \to \text{Rat}_ k/S^ 1\). Our computations rely heavily on Milgram's calculation of the groups \(H^*(\text{Rat}_ k/S^ 1;\mathbb{Z}_ p)\).