Divisor spaces on punctured Riemann surfaces

Kallel, Sadok

doi:10.1090/S0002-9947-98-02032-7

Divisor spaces on punctured Riemann surfaces
HTML articles powered by AMS MathViewer

by Sadok Kallel

Trans. Amer. Math. Soc. 350 (1998), 135-164

DOI: https://doi.org/10.1090/S0002-9947-98-02032-7

PDF | Request permission

Abstract:

In this paper, we study the topology of spaces of $n$-tuples of positive divisors on (punctured) Riemann surfaces which have no points in common (the divisor spaces). These spaces arise in connection with spaces of based holomorphic maps from Riemann surfaces to complex projective spaces. We find that there are Eilenberg-Moore type spectral sequences converging to their homology. These spectral sequences collapse at the $E^2$ term, and we essentially obtain complete homology calculations. We recover for instance results of F. Cohen, R. Cohen, B. Mann and J. Milgram, The topology of rational functions and divisors of surfaces, Acta Math. 166 (1991), 163–221. We also study the homotopy type of certain mapping spaces obtained as a suitable direct limit of the divisor spaces. These mapping spaces, first considered by G. Segal, were studied in a special case by F. Cohen, R. Cohen, B. Mann and J. Milgram, who conjectured that they split. In this paper, we show that the splitting does occur provided we invert the prime two.

References

L. Andrew Campbell and Arno van den Essen, Jacobian pairs, $D$-resultants, and automorphisms of the plane, J. Pure Appl. Algebra 104 (1995), no. 1, 9–18. MR 1359687, DOI 10.1016/0022-4049(95)00116-E
C.-F. Bödigheimer and F. R. Cohen, Rational cohomology of configuration spaces of surfaces, Algebraic topology and transformation groups (Göttingen, 1987) Lecture Notes in Math., vol. 1361, Springer, Berlin, 1988, pp. 7–13. MR 979504, DOI 10.1007/BFb0083031
C.-F. Bödigheimer, F. R. Cohen, and R. J. Milgram, Truncated symmetric products and configuration spaces, Math. Z. 214 (1993), no. 2, 179–216. MR 1240884, DOI 10.1007/BF02572399
Glen E. Bredon, Equivariant cohomology theories, Lecture Notes in Mathematics, No. 34, Springer-Verlag, Berlin-New York, 1967. MR 0214062, DOI 10.1007/BFb0082690
Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335
F. R. Cohen, A course in some aspects of classical homotopy theory, Algebraic topology (Seattle, Wash., 1985) Lecture Notes in Math., vol. 1286, Springer, Berlin, 1987, pp. 1–92. MR 922923, DOI 10.1007/BFb0078738
F. R. Cohen, R. L. Cohen, B. M. Mann, and R. J. Milgram, The topology of rational functions and divisors of surfaces, Acta Math. 166 (1991), no. 3-4, 163–221. MR 1097023, DOI 10.1007/BF02398886
Ralph L. Cohen and Don H. Shimamoto, Rational functions, labelled configurations, and Hilbert schemes, J. London Math. Soc. (2) 43 (1991), no. 3, 509–528. MR 1113390, DOI 10.1112/jlms/s2-43.3.509
Albrecht Dold, Homology of symmetric products and other functors of complexes, Ann. of Math. (2) 68 (1958), 54–80. MR 97057, DOI 10.2307/1970043
Albrecht Dold and René Thom, Quasifaserungen und unendliche symmetrische Produkte, Ann. of Math. (2) 67 (1958), 239–281 (German). MR 97062, DOI 10.2307/1970005
V. Arnol′d, Problems on singularities and dynamical systems, Developments in mathematics: the Moscow school, Chapman & Hall, London, 1993, pp. 251–274. MR 1264427
Marvin J. Greenberg and John R. Harper, Algebraic topology, Mathematics Lecture Note Series, vol. 58, Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1981. A first course. MR 643101
S. Kallel, R.J. Milgram, “The geometry of the space of holomorphic maps from a Riemann sphere to complex projective space”, preprint 1995.
John McCleary, User’s guide to spectral sequences, Mathematics Lecture Series, vol. 12, Publish or Perish, Inc., Wilmington, DE, 1985. MR 820463
R. James Milgram, The bar construction and abelian $H$-spaces, Illinois J. Math. 11 (1967), 242–250. MR 208595
R. James Milgram, The homology of symmetric products, Trans. Amer. Math. Soc. 138 (1969), 251–265. MR 242149, DOI 10.1090/S0002-9947-1969-0242149-X
R.J. Milgram, lecture notes.
Graeme Segal, The topology of spaces of rational functions, Acta Math. 143 (1979), no. 1-2, 39–72. MR 533892, DOI 10.1007/BF02392088
E. Spanier, Infinite symmetric products, function spaces, and duality, Ann. of Math. (2) 69 (1959), 142–198[ erratum, 733. MR 105106, DOI 10.2307/1970099