zbMATH — the first resource for mathematics

The homotopy type of rational functions. (English) Zbl 0790.55005
Let \(\text{Rat}_ k(\mathbb{C}\mathbb{P}^ n)\) denote the space of degree \(k\), basepoint preserving, holomorphic (equivalently, rational) maps from the Riemann sphere \(S^ 2\) to complex projective space \(\mathbb{C}\mathbb{P}^ n\). In this note we give a relatively short, combinatorial proof that \(\text{Rat}_ k (\mathbb{C}\mathbb{P}^ n)\) is stably homotopy equivalent to a wedge of subquotients of the May-Milgram filtered model for the double loop space of the 3-sphere, \(\Omega^ 2 S^ 3\). This implies that, when \(n=1\), \(\text{Rat}_ k(\mathbb{C}\mathbb{P}^ 1)\) is stably homotopy equivalent to \(K(\beta_{2k},1)\), the Eilenberg-MacLane space associated to Artin’s braid group on \(2k\) strings. A more complicated homological proof of this theorem that extends to other mapping spaces and various applications are discussed in our companion paper [Acta Math. 166, No. 3/4, 163-221 (1991; Zbl 0741.55005)].

55P42 Stable homotopy theory, spectra
55P45 \(H\)-spaces and duals
Full Text: DOI EuDML
[1] Boyer, C.P., Mann, B.M.: Monopoles, non-linear \(\sigma\)-models, and two-fold loop spaces. Commun. Math. Phys.115, 571–594 (1988) · Zbl 0656.58049 · doi:10.1007/BF01224128
[2] Brown, E.H., Peterson, F.P.: On the stable decomposition of 2 S r+2. Trans. Am. Math. Soc.243, 287–298 (1978) · Zbl 0404.55003 · doi:10.1090/S0002-9947-1978-0500933-4
[3] Cohen, F.R., Cohen, R.L., Mann, B.M., Milgram, R.J.: The topology of rational functions and divisors of surfaces. Acta Math.166(3), 163–221 (1991) · Zbl 0741.55005 · doi:10.1007/BF02398886
[4] Cohen, F.R., Mahowald, M., Milgram, R.J.: The stable decomposition of the double loop space of a sphere. In: Milgram, R.J. (ed.) Algebraic and Geometric Topology. (Proc. Symp. Pure Math., vol. 32, part 2, pp. 225–228) Providence, RI: Am. Math. Soc. 1978 · Zbl 0406.55007
[5] Cohen, F.R., Taylor, L.R., May, J.P.: Splitting of certain spacesC(X). Math. Proc. Camb. Philos. Soc.84, 465–496 (1978) · Zbl 0408.55006 · doi:10.1017/S0305004100055298
[6] Cohen, R.L.: Stable proofs of stable splittings. Math. Proc. Camb. Philos. Soc.88, 149–151 (1980) · Zbl 0453.55009 · doi:10.1017/S030500410005742X
[7] Dold, A., Thom, R.: Quasifaserungen und unendliche symmetrische Produkte. Ann. Math.67, 239–281 (1958) · Zbl 0091.37102 · doi:10.2307/1970005
[8] Kahn, D.S.: On the stable decomposition of S A. In: Barratt, M.G., Mahowald, M.E. (eds) Geometric Applications of Homotopy Theory II. (Lect. Notes Math., vol. 658, pp. 206–214) Berlin Heidelberg New York: Springer 1978
[9] Kahn, D.S., Priddy, S.: Applications of the transfer to stable homotopy theory. Bull. Am. Math. Soc.78, 981–987 (1972) · Zbl 0265.55009 · doi:10.1090/S0002-9904-1972-13076-3
[10] May, J.P.: The Geometry of Iterated Loop Spaces. (Lect. Notes Math., vol. 271). Berlin Heidelberg New York: Springer 1972 · Zbl 0244.55009
[11] Milgram, R.J.: Iterated loop spaces. Ann. Math.84, 386–403 (1966) · Zbl 0145.19901 · doi:10.2307/1970453
[12] Segal, G.B.: The topology of rational functions. Acta Math.143, 39–72 (1979) · Zbl 0427.55006 · doi:10.1007/BF02392088
[13] Smith, L.: Transfer and ramified coverings. Math. Proc. Camb. Philos. Soc.93, 485–493 (1983) · Zbl 0525.57031 · doi:10.1017/S0305004100060795
[14] Snaith, V.P.: A stable decomposition of n X. J. Lond. Math. Soc., II. Ser.7, 577–583 (1974) · Zbl 0275.55019 · doi:10.1112/jlms/s2-7.4.577
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.