Rational category of the space of sections of a nilpotent bundle
Let X be a finite nilpotent CW-complex and Y be a nilpotent space; denote by \(Y^ X\) the space of continuous maps of X into Y. The author proves that the Lusternik-Schnirelmann category (LS-category) of \(Y^ X\) is often infinite. Conditions for finiteness deal only with rational invariants and are therefore computable. The main theorem contains as corollary a previous result of \textit{E. Fadell} and \textit{S. Husseini} [Proc. Am. Math. Soc. 107, No.2, 527-536 (1989; Zbl 0676.58019)]: If Y is a 1-connected space of finite LS-category, with \(\tilde H(Y;{\mathbb{Q}})\neq 0\), then the free loop space of Y has finite LS-category. The proof uses the Haefliger's model for a space of sections [\textit{A. Haefliger}, Trans. Am. Math. Soc. 273, 609-620 (1982; Zbl 0508.55019)] and a determination of Gottlieb's elements whose number is less or equal to the LS-category [\textit{Y. Félix} and \textit{S. Halperin}, Trans. Am. Math. Soc. 273, 1-37 (1982; Zbl 0508.55004)].
scientific article; zbMATH DE number 4178122