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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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