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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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##### References:
 [1] [CCMM] Cohen, F.R., Cohen, R.L., Mann, B.M., Milgram, R.J.: The topology of rational functions and divisors of surfaces. Acta Math.166, 163–221 (1991) · Zbl 0741.55005 · doi:10.1007/BF02398886 [2] [CLM] Cohen, F.R., Lada, T.J., May, J.P.: The homology of iterated loop spaces. (Lect. Notes Math., vol. 533) New York Berlin Heidelberg: Springer 1976 · Zbl 0334.55009 [3] [CS] Cohen, R.L., Shimamoto, D.H.: Rational functions, labelled configurations, and Hilbert schemes. J. London Math. Soc.43, 509–528 (1991) · Zbl 0756.55005 · doi:10.1112/jlms/s2-43.3.509 [4] [DL] Dyer, E., Lashof, R.K.: Homology of iterated loop spaces. Am. J. Math.84, 35–88 (1962) · Zbl 0119.18206 · doi:10.2307/2372804 [5] [G] Guest, M.A.: Topology of the space of absolute minima of the energy functional. Am. J. of Math.106, 21–42 (1984) · Zbl 0564.58014 · doi:10.2307/2374428 [6] [K] Kirwan, F.C.: On spaces of maps from Riemann surfaces to Grassmannians and applications to the cohomology of moduli of vector bundles. Ark. Mat.24, 221–275 (1986) · Zbl 0625.14026 · doi:10.1007/BF02384399 [7] [MaM1] Mann, B.M., Milgram, R.J.: Some spaces of holomorphic maps to complex Grassmann manifolds. J. Differ. Geom.33, 301–324 (1991) · Zbl 0736.54008 [8] [MaM2] Mann, B.M., Milgram, R.J.: On the moduli of SU(n) monopoles and holomorphic maps to flag manifolds. Preprint, University of New Mexico and Stanford University 1991 [9] [May] May, J.P.: The geometry of iterated loop spaces. (Lect. Notes Math., vol. 271) New York Berlin Heidelberg: Springer 1972 · Zbl 0244.55009 [10] [M1] Milgram, R.J.: Interated loop spaces. Ann. of Math.84, 386–403 (1966) · Zbl 0145.19901 · doi:10.2307/1970453 [11] [M2] Milgram, R.J.: The structure of spaces of Toeplitz matrices. Preprint, Stanford University and the University of New Mexico 1992 [12] [MiS] Milnor, J.W., Stasheff, J.D.: Characteristic classes. (Ann. of Math. Studies, no. 76) Princeton University Press 1974 · Zbl 0298.57008 [13] [S] segal, G.: The topology of spaces of rational functions. Acta Math.143, 39–72 (1979) · Zbl 0427.55006 · doi:10.1007/BF02392088 [14] [T] Totaro, B.: The coholomogy ring of the space of rational functions. Preprint, MSRI 1990
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