Rank of the fundamental group of any component of a function space
HTML articles powered by AMS MathViewer
- by Gregory Lupton and Samuel Bruce Smith PDF
- Proc. Amer. Math. Soc. 135 (2007), 2649-2659 Request permission
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
- Armand Borel, Topology of Lie groups and characteristic classes, Bull. Amer. Math. Soc. 61 (1955), 397–432. MR 72426, DOI 10.1090/S0002-9904-1955-09936-1
- A. K. Bousfield and V. K. A. M. Gugenheim, On $\textrm {PL}$ de Rham theory and rational homotopy type, Mem. Amer. Math. Soc. 8 (1976), no. 179, ix+94. MR 425956, DOI 10.1090/memo/0179
- Edgar H. Brown Jr. and Robert H. Szczarba, On the rational homotopy type of function spaces, Trans. Amer. Math. Soc. 349 (1997), no. 12, 4931–4951. MR 1407482, DOI 10.1090/S0002-9947-97-01871-0
- U. Buijs and A. Murillo, The rational homotopy Lie algebra of function spaces, preprint.
- Yves Félix, Stephen Halperin, and Jean-Claude Thomas, Rational homotopy theory, Graduate Texts in Mathematics, vol. 205, Springer-Verlag, New York, 2001. MR 1802847, DOI 10.1007/978-1-4613-0105-9
- Pierre-Paul Grivel, Algèbres de Lie de dérivations de certaines algèbres pures, J. Pure Appl. Algebra 91 (1994), no. 1-3, 121–135 (French, with English summary). MR 1255925, DOI 10.1016/0022-4049(94)90137-6
- André Haefliger, Rational homotopy of the space of sections of a nilpotent bundle, Trans. Amer. Math. Soc. 273 (1982), no. 2, 609–620. MR 667163, DOI 10.1090/S0002-9947-1982-0667163-8
- Stephen Halperin, Finiteness in the minimal models of Sullivan, Trans. Amer. Math. Soc. 230 (1977), 173–199. MR 461508, DOI 10.1090/S0002-9947-1977-0461508-8
- Peter Hilton, Guido Mislin, and Joe Roitberg, Localization of nilpotent groups and spaces, North-Holland Mathematics Studies, No. 15, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975. MR 0478146
- 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.
- John Milnor, On spaces having the homotopy type of a $\textrm {CW}$-complex, Trans. Amer. Math. Soc. 90 (1959), 272–280. MR 100267, DOI 10.1090/S0002-9947-1959-0100267-4
- Edwin H. Spanier, Algebraic topology, Springer-Verlag, New York-Berlin, 1981. Corrected reprint. MR 666554
- R. Thom, L’homologie des espaces fonctionnels, Colloque de topologie algébrique, Louvain, 1956, Georges Thone, Liège; Masson & Cie, Paris, 1957, pp. 29–39 (French). MR 0089408
- Micheline Vigué-Poirrier, Sur l’homotopie rationnelle des espaces fonctionnels, Manuscripta Math. 56 (1986), no. 2, 177–191 (French, with English summary). MR 850369, DOI 10.1007/BF01172155
Additional Information
- Gregory Lupton
- Affiliation: Department of Mathematics, Cleveland State University, Cleveland, Ohio 44115
- MR Author ID: 259990
- Email: G.Lupton@csuohio.edu
- Samuel Bruce Smith
- Affiliation: Department of Mathematics, Saint Joseph’s University, Philadelphia, Pennsylvania 19131
- MR Author ID: 333158
- Email: smith@sju.edu
- 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
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 135 (2007), 2649-2659
- MSC (2000): Primary 55Q52, 55P62
- DOI: https://doi.org/10.1090/S0002-9939-07-08746-1
- MathSciNet review: 2302588