Volume 6, issue 1 (2006)

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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
A rational splitting of a based mapping space

Katsuhiko Kuribayashi and Toshihiro Yamaguchi

Algebraic & Geometric Topology 6 (2006) 309–327

arXiv: 0902.4876

Abstract

Let (X,Y ) be the space of base-point-preserving maps from a connected finite CW complex X to a connected space Y . Consider a CW complex of the form X αek+1 and a space Y whose connectivity exceeds the dimension of the adjunction space. Using a Quillen–Sullivan mixed type model for a based mapping space, we prove that, if the bracket length of the attaching map α : Sk X is greater than the Whitehead length WL(Y ) of Y , then (X αek+1,Y ) has the rational homotopy type of the product space (X,Y ) × Ωk+1Y . This result yields that if the bracket lengths of all the attaching maps constructing a finite CW complex X are greater than WL(Y ) and the connectivity of Y is greater than or equal to dimX, then the mapping space (X,Y ) can be decomposed rationally as the product of iterated loop spaces.

Keywords
mapping space, $d_1$–depth, bracket length, Whitehead length
Mathematical Subject Classification 2000
Primary: 55P62
Secondary: 54C35
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Publication
Received: 19 July 2005
Revised: 14 February 2006
Accepted: 14 February 2006
Published: 7 March 2006
Authors
Katsuhiko Kuribayashi
Department of Mathematical Sciences
Faculty of Science
Shinshu University
Matsumoto
Nagano 390-8621
Japan
Toshihiro Yamaguchi
Department of Mathematics Education
Faculty of Education
Kochi University
Kochi 780-8520
Japan