Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Rank of the fundamental group of any component of a function space

Authors: Gregory Lupton and Samuel Bruce Smith
Journal: Proc. Amer. Math. Soc. 135 (2007), 2649-2659
MSC (2000): Primary 55Q52, 55P62
Published electronically: March 21, 2007
MathSciNet review: 2302588
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We compute the rank of the fundamental group of any connected component of the space $ \mathrm{map}(X, Y)$ for $ X$ and $ Y$ connected, nilpotent CW complexes of finite type with $ X$ finite. For the component corresponding to a general homotopy class $ f \colon X \to Y$, we give a formula directly computable from the Sullivan model for $ f$. For the component of the constant map, our formula retrieves a known expression for the rank in terms of classical invariants of $ X$ and $ Y$. When both $ X$ and $ Y$ are rationally elliptic spaces with positive Euler characteristic, we use our formula to determine the rank of the fundamental group of any component of $ \mathrm{map}(X, Y)$ explicitly in terms of the homomorphism induced by $ f$ on rational cohomology.

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

  • 1. A. Borel, Topology of Lie groups and characteristic classes, Bull. Amer. Math. Soc. 61 (1955), 397-432. MR 0072426 (17:282b)
  • 2. A. K. Bousfield and V. K. A. M. Gugenheim, On $ {\rm PL}$ de Rham theory and rational homotopy type, Mem. Amer. Math. Soc. 8 (1976), no. 179, ix+94. MR 0425956 (54:13906)
  • 3. E. H. Brown, Jr. and R. H. Szczarba, On the rational homotopy type of function spaces, Trans. Amer. Math. Soc. 349 (1997), no. 12, 4931-4951.MR 1407482 (98c:55015)
  • 4. U. Buijs and A. Murillo, The rational homotopy Lie algebra of function spaces, preprint.
  • 5. Y. Félix, S. Halperin, and J.-C. Thomas, Rational homotopy theory, Graduate Texts in Mathematics, vol. 205, Springer-Verlag, New York, 2001.MR 1802847 (2002d:55014)
  • 6. P.-P. Grivel, Algèbres de Lie de dérivations de certaines algèbres pures, J. Pure Appl. Algebra 91 (1994), no. 1-3, 121-135.MR 1255925 (95a:55024)
  • 7. A. Haefliger, Rational homotopy of the space of sections of a nilpotent bundle, Trans. Amer. Math. Soc. 273 (1982), no. 2, 609-620. MR 0667163 (84a:55010)
  • 8. S. Halperin, Finiteness in the minimal models of Sullivan, Trans. Amer. Math. Soc. 230 (1977), 173-199. MR 0461508 (57:1493)
  • 9. P. Hilton, G. Mislin, and J. Roitberg, Localization of nilpotent groups and spaces, North-Holland Publishing Co., Amsterdam, 1975, North-Holland Mathematics Studies, No. 15. MR 0478146 (57:17635)
  • 10. G. Lupton and S. B. Smith, Rationalized evaluation subgroups of a map I: Sullivan models, derivations, and $ G$-sequences, J. Pure Appl. Algebra, 209 (2007) 159-171.
  • 11. J. Milnor, On spaces having the homotopy type of $ {\rm CW}$-complex, Trans. Amer. Math. Soc. 90 (1959), 272-280.MR 0100267 (20:6700)
  • 12. E. H. Spanier, Algebraic topology, Springer-Verlag, New York, 1989. MR 0666554 (83i:55001)
  • 13. R. Thom, L'homologie des espaces fonctionnels, Colloque de topologie algébrique, Louvain, 1956, Georges Thone, Liège, 1957, pp. 29-39. MR 0089408 (19:669h)
  • 14. M. Vigué-Poirrier, Sur l'homotopie rationnelle des espaces fonctionnels, Manuscripta Math. 56 (1986), 177-191.MR 0850369 (87h:55009)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 55Q52, 55P62

Retrieve articles in all journals with MSC (2000): 55Q52, 55P62

Additional Information

Gregory Lupton
Affiliation: Department of Mathematics, Cleveland State University, Cleveland, Ohio 44115

Samuel Bruce Smith
Affiliation: Department of Mathematics, Saint Joseph’s University, Philadelphia, Pennsylvania 19131

Keywords: Function space, fundamental group, nilpotent space, nilpotent group, rank, rational homotopy, minimal models, derivations
Received by editor(s): February 1, 2006
Received by editor(s) in revised form: April 13, 2006
Published electronically: March 21, 2007
Communicated by: Paul Goerss
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society