Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Spaces of rational loops on a real projective space


Author: Jacob Mostovoy
Journal: Trans. Amer. Math. Soc. 353 (2001), 1959-1970
MSC (2000): Primary 26C15, 55P35
Published electronically: January 3, 2001
MathSciNet review: 1813601
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract:

We show that the loop spaces on real projective spaces are topologically approximated by the spaces of rational maps $\mathbf{RP}^{1}\rightarrow \mathbf{RP}^{n}$. As a byproduct of our constructions we obtain an interpretation of the Kronecker characteristic (degree) of an ornament via particle spaces.


References [Enhancements On Off] (What's this?)

  • 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. MR 1294670, 10.1007/BF02392569
  • 2. C. Boyer, J. Hurtubise - R. J. Milgram, Stability theorems for spaces of rational curves. Preprint, 1999, math.AG/9903099
  • 3. Roger W. Brockett, Some geometric questions in the theory of linear systems, IEEE Trans. Automatic Control AC-21 (1976), no. 4, 449–455. MR 0469386
  • 4. Martin A. Guest, The topology of the space of rational curves on a toric variety, Acta Math. 174 (1995), no. 1, 119–145. MR 1310847, 10.1007/BF02392803
  • 5. M. A. Guest, A. Kozlowski - K. Yamaguchi, Spaces of polynomials with roots of bounded multiplicity. Fund. Math. 161 (1999), 93-117.
  • 6. Morris W. Hirsch, Differential topology, Springer-Verlag, New York-Heidelberg, 1976. Graduate Texts in Mathematics, No. 33. MR 0448362
  • 7. L. Kronecker, Über Systeme von Funktionen mehrerer Variabeln. Monatsberichte Berl. Acad. (1869), 159-193 and 688-698.
  • 8. A. Kozlowski - K. Yamaguchi, Topology of complements of discriminants and resultants. J. Math. Soc. Japan 52 (2000), 949-959. CMP 2000:16
  • 9. S. Lefschetz, Topology. AMS Colloquium Publications, New York, 1930.
  • 10. Dusa McDuff, Configuration spaces of positive and negative particles, Topology 14 (1975), 91–107. MR 0358766
  • 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. MR 0163331
  • 12. Graeme Segal, The topology of spaces of rational functions, Acta Math. 143 (1979), no. 1-2, 39–72. MR 533892, 10.1007/BF02392088
  • 13. Victor A. Vassiliev, Invariants of ornaments, Singularities and bifurcations, Adv. Soviet Math., vol. 21, Amer. Math. Soc., Providence, RI, 1994, pp. 225–262. MR 1310605
  • 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. MR 1403923
  • 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. MR 1168473
  • 16. K. Yamaguchi, Complements of resultants and homotopy types, J. Math. Kyoto Univ. 39 (1999), 675-684. CMP 2000:08

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 26C15, 55P35

Retrieve articles in all journals with MSC (2000): 26C15, 55P35


Additional Information

Jacob Mostovoy
Affiliation: Instituto de Matemáticas (Unidad Cuernavaca), Universidad Nacional Autónoma de México, A.P. 273-3, C.P. 62251, Cuernavaca, Morelos, México
Email: jacob@matcuer.unam.mx

DOI: http://dx.doi.org/10.1090/S0002-9947-01-02644-7
Keywords: Loop space, rational map, ornament, Kronecker characteristic
Received by editor(s): June 2, 1998
Received by editor(s) in revised form: October 18, 1999
Published electronically: January 3, 2001
Article copyright: © Copyright 2001 American Mathematical Society