A rational splitting of a based mapping space

Kuribayashi, Katsuhiko; Yamaguchi, Toshihiro

doi:10.2140/agt.2006.6.309

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A rational splitting of a based mapping space

Katsuhiko Kuribayashi and Toshihiro Yamaguchi

Algebraic & Geometric Topology 6 (2006) 309–327

DOI: 10.2140/agt.2006.6.309

arXiv: 0902.4876

Abstract

Let $ℱ_{*} (X, Y)$ 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 $X \cup_{α} e^{k + 1}$ 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 $α : S^{k} \to X$ is greater than the Whitehead length $WL (Y)$ of $Y$ , then $ℱ_{*} (X \cup_{α} e^{k + 1}, Y)$ has the rational homotopy type of the product space $ℱ_{*} (X, Y) \times Ω^{k + 1} Y$ . 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 $dim X$ , then the mapping space $ℱ_{*} (X, Y)$ 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

Volume 6, issue 1 (2006)