Journal of the Mathematical Society of Japan

The homotopy of spaces of maps between real projective spaces


Full-text: Open access


We study the homotopy groups of spaces of continuous maps between real projective spaces and we generalize the results given in [5], [8] and [12]. In particular, we determine the rational homotopy types of these spaces and compute their fundamental groups explicitly.

Article information

J. Math. Soc. Japan, Volume 58, Number 4 (2006), 1163-1184.

First available in Project Euclid: 21 May 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55P15: Classification of homotopy type 55P62: Rational homotopy theory
Secondary: 55P10: Homotopy equivalences

rational homotopy equivalence group action fundamental group


YAMAGUCHI, Kohhei. The homotopy of spaces of maps between real projective spaces. J. Math. Soc. Japan 58 (2006), no. 4, 1163--1184. doi:10.2969/jmsj/1179759542.

Export citation


  • F. R. Cohen, J. C. Moore and J. A. Neisendorfer, The double suspension and exponents of the homotopy groups of spheres, Ann. Math., 110 (1979), 549–565.
  • V. L. Hansen, On the space of maps of a closed surface into the $2$-sphere, Math. Scand., 35 (1974), 140–158.
  • A. Kozlowski and K. Yamaguchi, Spaces of holomorphic maps between complex projective spaces of degree one, Topology Appl., 132 (2003), 139–145.
  • M. Mimura, G. Nishida and H. Toda, Localization of CW complexes and its applications, J. Math. Soc. Japan, 23 (1971), 593–624.
  • W. Meier and R. Strebel, Homotopy group of acyclic spaces, Quart. J. Math. Oxford, 32 (1981), 81–95.
  • M. Mimura and H. Toda, Topology of Lie groups, I, II, Translation of Math. Monographs, 91, Amer. Math. Soc. Providence, 1991.
  • J. M. Møller, On spaces of maps between complex projective spaces, Proc. Amer. Math. Soc., 91 (1984), 471–476.
  • S. Sasao, The homotopy of $\mbox{Map}(\CP^m,\CP^n)$, J. London Math., 8 (1974), 193–197.
  • J. P. Serre, Groupes d'homotopie et classes de groupes abéliens, Ann. Math., 58 (1953), 258–294.
  • H. Toda, Composition methods in homotopy groups of spheres, Annals of Math. Studies, 49, Princeton Univ. Press, 1962.
  • G. W. Whitehead, On products in homotopy groups, Ann. Math., 47 (1946), 460–475.
  • K. Yamaguchi, The topology of spaces of maps between real projective spaces, J. Math. Kyoto Univ., 43 (2003), 503–507.
  • A. Kozlowski and K. Yamaguchi, Spaces of algebraic maps between real projective spaces, preprint.