Whitehead products in function spaces: Quillen model formulae. (English) Zbl 1193.55005

Lupton, Gregory; Smith, Samuel Bruce

J. Math. Soc. Japan 62, No. 1, 49-81 (2010).

Let \(f: X \to Y\) be a based map of based simply connected CW complexes with \(X\) a finite complex. Denote by Map\((X,Y;f)\) the path component containing \(f\) of the space of basepoint-free continuous functions from \(X\) to \(Y\), and by Map\(_*(X,Y;f)\) the corresponding component in the space of basepoint-preserving functions. In this paper the authors give a process to compute Whitehead products in the rational homotopy groups of Map\((X,Y;f)\) and Map\(_*(X,Y;f)\) in terms of the Quillen minimal model of \(f\). The computation follows from a purely algebraic development in the setting of chain complexes of differential graded Lie algebras. Several applications are given. Denote by \(Wh_Q(Z)\) the rational Whitehead length of a space \(Z\), i.e., the length of the longest non-zero Whitehead bracket in \(\pi_*(Z)\otimes\mathbb Q\). Lupton and Smith prove that
\[ Wh_Q(\text{Map}_*(X,Y;f)) \leq cl_0(X)\,, \]
where \(cl_0(X)\) denotes the rational cone length of \(X\). They also compare the Whitehead length of a space \(Y\) to the Whitehead length of Map\((X,Y;f)\). In particular they give a map \(f : S^3\to Y\) with \(Wh_Q(Y)= 1\) and \(Wh_Q(\text{Map}(S^3,Y;f))\geq 2\).. The space \(Y\) is the total space of the pullback of the path space fibration on \(K(\mathbb Z,6)\) along the map \(g : S^2\times S^3\times S^3\to K(\mathbb Z,6)\) corresponding to the fundamental class of \(S^2\times S^3\times S^3\). The map \(f: S^3\to Y\) is the injection of a factor \(S^3\).

Reviewer: Yves Félix (Louvain-La-Neuve)

MSC:

55P62	Rational homotopy theory
55Q15	Whitehead products and generalizations

Keywords:

Whitehead product; function space; Quillen minimal model; coformal space; Whitehead length

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