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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
DOI: https://doi.org/10.1090/S0002-9939-07-08746-1
Published electronically: March 21, 2007
MathSciNet review: 2302588
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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
Email: G.Lupton@csuohio.edu

Samuel Bruce Smith
Affiliation: Department of Mathematics, Saint Joseph’s University, Philadelphia, Pennsylvania 19131
Email: smith@sju.edu

DOI: https://doi.org/10.1090/S0002-9939-07-08746-1
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.

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