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Spaces of rational loops on a real projective space. (English) Zbl 0979.55008
Let \(\text{Rat}_n(m)\) denote the space of rational maps from \(\mathbb{C} P^1\) to \(\mathbb{C} P^m\) given by a polynomial of degree \(n\) that sends the point \(\infty\in \mathbb{C} P^1\) to a fixed point in \(\mathbb{C} P^m\). Let \((\Omega^2 \mathbb{C} P^m)_n\) be the component of the double loop space \(\Omega^2 \mathbb{C} P^m\) which parametrizes maps of degree \(n\). G. Segal [Acta Math. 143, 39-72 (1979; Zbl 0427.55006)], proved that the natural inclusion of \(\text{Rat}_n(m)\) in \((\Omega^2 \mathbb{C} P^m)_n\) is a homotopy equivalence up to dimension \(n(2m-1)\). Denote by \(\mathbb{R} \text{Rat}_n(m)\) the subspace of \(\text{Rat}_n(m)\) of maps which commute with complex conjugation and by rat\(_n(m)\) the closure of \(\mathbb{R} \text{Rat}_n(m)\) in \(\Omega \mathbb{R} P^m\). The author proves that the space rat\(_n(1)\) consists of \(n+1\) contractible components and if \(m>1\) the natural inclusion rat\(_n(m) \subset(\Omega \mathbb{R} P^m)_{n\bmod 2}\) is a homotopy equivalence up to dimension \(n(m-1)\). The proof goes along the line of Segal’s proof, however the technique involved (configuration spaces, action of the \(\pi_1(\mathbf{rat}_n(2)\)) on higher homotopy groups,…) seems to be of independent interest.

55P35 Loop spaces
26C15 Real rational functions
55P10 Homotopy equivalences in algebraic topology
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[1] C. P. Boyer, B. M. Mann, J. C. Hurtubise, and R. J. Milgram, The topology of the space of rational maps into generalized flag manifolds, Acta Math. 173 (1994), no. 1, 61 – 101. · Zbl 0844.57037 · doi:10.1007/BF02392569 · doi.org
[2] C. Boyer, J. Hurtubise - R. J. Milgram, Stability theorems for spaces of rational curves. Preprint, 1999, math.AG/9903099 · Zbl 1110.58303
[3] R. W. Brockett, Some geometric questions in the theory of linear systems, Proceedings of the IEEE Conference on Decision and Control including the 14th Symposium on Adaptive Processes (Houston, Tex., 1975) Inst. Electr. Electron. Engrs., New York, 1975, pp. 71 – 76. Roger W. Brockett, Some geometric questions in the theory of linear systems, IEEE Trans. Automatic Control AC-21 (1976), no. 4, 449 – 455.
[4] Martin A. Guest, The topology of the space of rational curves on a toric variety, Acta Math. 174 (1995), no. 1, 119 – 145. · Zbl 0826.14035 · doi:10.1007/BF02392803 · doi.org
[5] M. A. Guest, A. Kozlowski - K. Yamaguchi, Spaces of polynomials with roots of bounded multiplicity. Fund. Math. 161 (1999), 93-117. · Zbl 1016.55004
[6] Morris W. Hirsch, Differential topology, Springer-Verlag, New York-Heidelberg, 1976. Graduate Texts in Mathematics, No. 33. · Zbl 0356.57001
[7] L. Kronecker, Über Systeme von Funktionen mehrerer Variabeln. Monatsberichte Berl. Acad. (1869), 159-193 and 688-698. · JFM 02.0203.02
[8] A. Kozlowski - K. Yamaguchi, Topology of complements of discriminants and resultants. J. Math. Soc. Japan 52 (2000), 949-959. CMP 2000:16 · Zbl 0974.55002
[9] S. Lefschetz, Topology. AMS Colloquium Publications, New York, 1930. · JFM 56.0491.08
[10] Dusa McDuff, Configuration spaces of positive and negative particles, Topology 14 (1975), 91 – 107. · Zbl 0296.57001 · doi:10.1016/0040-9383(75)90038-5 · doi.org
[11] J. Milnor, Morse theory, Based on lecture notes by M. Spivak and R. Wells. Annals of Mathematics Studies, No. 51, Princeton University Press, Princeton, N.J., 1963. · Zbl 0108.10401
[12] Graeme Segal, The topology of spaces of rational functions, Acta Math. 143 (1979), no. 1-2, 39 – 72. · Zbl 0427.55006 · doi:10.1007/BF02392088 · doi.org
[13] Victor A. Vassiliev, Invariants of ornaments, Singularities and bifurcations, Adv. Soviet Math., vol. 21, Amer. Math. Soc., Providence, RI, 1994, pp. 225 – 262. · Zbl 0854.57011
[14] Victor A. Vassiliev, Topology of discriminants and their complements, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) Birkhäuser, Basel, 1995, pp. 209 – 226. · Zbl 0852.55003
[15] V. A. Vassiliev, Complements of discriminants of smooth maps: topology and applications, Translations of Mathematical Monographs, vol. 98, American Mathematical Society, Providence, RI, 1992. Translated from the Russian by B. Goldfarb.
[16] K. Yamaguchi, Complements of resultants and homotopy types, J. Math. Kyoto Univ. 39 (1999), 675-684. CMP 2000:08 · Zbl 0957.55006
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