Geodesic
A rational splitting of a based mapping space
Kuribayashi, Katsuhiko1 ; Yamaguchi, Toshihiro2
1Department of Mathematical Sciences, Faculty of Science, Shinshu University, Matsumoto, Nagano 390-8621, Japan
2Department of Mathematics Education, Faculty of Education, Kochi University, Kochi 780-8520, Japan
Algebraic and Geometric Topology, Tome 6 (2006) no. 1, pp. 309-327
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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 ∪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 ℱ∗(X ∪αek+1,Y ) has the rational homotopy type of the product space ℱ∗(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 ℱ∗(X,Y ) can be decomposed rationally as the product of iterated loop spaces.

DOI : 10.2140/agt.2006.6.309
Keywords: mapping space, $d_1$–depth, bracket length, Whitehead length

Kuribayashi, Katsuhiko  1   ; Yamaguchi, Toshihiro  2

1 Department of Mathematical Sciences, Faculty of Science, Shinshu University, Matsumoto, Nagano 390-8621, Japan
2 Department of Mathematics Education, Faculty of Education, Kochi University, Kochi 780-8520, Japan 
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Kuribayashi, Katsuhiko; Yamaguchi, Toshihiro. A rational splitting of a based mapping space. Algebraic and Geometric Topology, Tome 6 (2006) no. 1, pp. 309-327. doi: 10.2140/agt.2006.6.309

[1] I Berstein, T Ganea, Homotopical nilpotency, Illinois J. Math. 5 (1961) 99

[2] E H Brown Jr., R H Szczarba, On the rational homotopy type of function spaces, Trans. Amer. Math. Soc. 349 (1997) 4931

[3] Y Félix, S Halperin, J C Thomas, Differential graded algebras in topology, from: "Handbook of algebraic topology", North-Holland (1995) 829

[4] Y Félix, S Halperin, J C Thomas, Rational homotopy theory, Graduate Texts in Mathematics 205, Springer (2001)

[5] J B Friedlander, S Halperin, An arithmetic characterization of the rational homotopy groups of certain spaces, Invent. Math. 53 (1979) 117

[6] P Hilton, G Mislin, J Roitberg, Localization of nilpotent groups and spaces, North-Holland Publishing Co. (1975)

[7] S Kaji, On the nilpotency of rational $H$-spaces, J. Math. Soc. Japan 57 (2005) 1153

[8] Y Kotani, Note on the rational cohomology of the function space of based maps, Homology Homotopy Appl. 6 (2004) 341

[9] K Kuribayashi, A rational model for the evaluation map, to appear in Georgian Mathematical Journal 13 (2006)

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