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The cohomology of holomorphic self maps of the Riemann sphere. (English) Zbl 0814.55003
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 F. R. Cohen, R. L. Cohen, B. M. Mann and R. J. Milgram [Acta Math. 166, No. 3/4, 163-221 (1991; Zbl 0741.55005)] and the algebra structure has been given by 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 \(H^*(\text{Rat}_ k/S^ 1;\mathbb{Z}_ p)\).

MSC:
55N99 Homology and cohomology theories in algebraic topology
55R20 Spectral sequences and homology of fiber spaces in algebraic topology
58D15 Manifolds of mappings
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