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**A rational splitting of a based mapping space.**
*(English)*
Zbl 1097.55010

Let \(X\) be a finite, connected CW complex. Let \(X \bigcup_\alpha e^{k+1}\) denote the adjunction space corresponding to an attaching map \(\alpha \colon S^k \to X\). The authors give conditions under which the fibration sequence on spaces of based maps
\[
\Omega^{k+1}Y = \mathcal{F}_*(S^{k+1}, Y) \to \mathcal{F}_*(X \bigcup_\alpha e^{k+1}, Y) \to \mathcal{F}_*(X, Y)
\]
corresponding to the cofibration sequence \(X \to X \bigcup_\alpha e^{k+1} \to S^{k+1}\) splits after rationalization. Precisely, the authors prove this fibration splits rationally if \((1)\) the rational connectivity of \(Y\) is at least as large as the dimension of the adjuction space and \((2)\) the element \(\alpha\) is decomposable in \(\pi_*(\Omega X) \otimes \mathbb Q\) of bracket length strictly larger than the longest bracket in \(\pi_*(\Omega Y) \otimes \mathbb Q.\) As a consequence, they obtain that if \(X\) is a finite, simply connected complex with dimension no larger than the connectivity of \(Y\) and with attaching maps all of bracket length greater than the longest bracket in the rational homotopy of \(Y,\) then \(\mathcal{F}_*(X, Y)\) is a rational \(H\)-space.

The authors obtain the main result by studying a model for the based mapping space \(\mathcal{F}_*(X, Y)\) which can be made explicit when the dimension of \(X\) is no larger than the connectivity of \(Y.\) In this case, the first author has constructed a Sullivan model of the form \((\mathbb Q[V \otimes C_*(L, d_L)], \delta)\) where \(V\) is a space of generators for the Sullivan minimal model of \(Y\), \((L, d_L)\) is a Quillen minimal model for \(X\) and \(C_*(L, d_L)\) is Quillen’s functor from the category of differential graded Lie algebras to differential graded co-algebras. The differential \(\delta\) is expressed in terms of the product in the Sullivan minimal model of \(Y\) and the coproduct in the \(C_*(L, d_L)\). This model is adapted from the work of E. H. Brown jun. and R. H. Szczarba [Trans. Am. Math. Soc. 349, 4931–4951 (1997; Zbl 0927.55012)]. The paper contains several nice applications of the main result and examples showing the necessity of the hypotheses.

The authors obtain the main result by studying a model for the based mapping space \(\mathcal{F}_*(X, Y)\) which can be made explicit when the dimension of \(X\) is no larger than the connectivity of \(Y.\) In this case, the first author has constructed a Sullivan model of the form \((\mathbb Q[V \otimes C_*(L, d_L)], \delta)\) where \(V\) is a space of generators for the Sullivan minimal model of \(Y\), \((L, d_L)\) is a Quillen minimal model for \(X\) and \(C_*(L, d_L)\) is Quillen’s functor from the category of differential graded Lie algebras to differential graded co-algebras. The differential \(\delta\) is expressed in terms of the product in the Sullivan minimal model of \(Y\) and the coproduct in the \(C_*(L, d_L)\). This model is adapted from the work of E. H. Brown jun. and R. H. Szczarba [Trans. Am. Math. Soc. 349, 4931–4951 (1997; Zbl 0927.55012)]. The paper contains several nice applications of the main result and examples showing the necessity of the hypotheses.

Reviewer: Samuel Smith (Philadelphia)

### MSC:

55P62 | Rational homotopy theory |

### Citations:

Zbl 0927.55012
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\textit{K. Kuribayashi} and \textit{T. Yamaguchi}, Algebr. Geom. Topol. 6, 309--327 (2006; Zbl 1097.55010)

### References:

[1] | I Berstein, T Ganea, Homotopical nilpotency, Illinois J. Math. 5 (1961) 99 · Zbl 0096.17602 |

[2] | E H Brown Jr., R H Szczarba, On the rational homotopy type of function spaces, Trans. Amer. Math. Soc. 349 (1997) 4931 · Zbl 0927.55012 |

[3] | Y Félix, S Halperin, J C Thomas, Differential graded algebras in topology, North-Holland (1995) 829 · Zbl 0868.55016 |

[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 · Zbl 0396.55010 |

[6] | P Hilton, G Mislin, J Roitberg, Localization of nilpotent groups and spaces, North-Holland Publishing Co. (1975) · Zbl 0323.55016 |

[7] | S Kaji, On the nilpotency of rational \(H\)-spaces, J. Math. Soc. Japan 57 (2005) 1153 · Zbl 1085.55007 |

[8] | Y Kotani, Note on the rational cohomology of the function space of based maps, Homology Homotopy Appl. 6 (2004) 341 · Zbl 1065.55005 |

[9] | K Kuribayashi, A rational model for the evaluation map, to appear in Georgian Mathematical Journal 13 (2006) · Zbl 1097.18004 |

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