Spaces whose rational homology and de Rham homotopy are both finite dimensional. (English) Zbl 0546.55015

Halperin, Stephen

Homotopie algébrique et algèbre locale, Journ. Luminy/France 1982, Astérisque 113-114 (1984).

Summary: [For the entire collection see Zbl 0535.00017.]
If a simply connected space S satisfies the hypothese of the title let n be the top integer for which \(H^ n(S;{\mathbb{Q}})\neq 0.\) It was known that \(\sum^{2n-1}_{i=n+1}\dim \pi_ i(S)\otimes {\mathbb{Q}}\leq 1;\) the case when equality holds is analyzed. We also show that \(\dim H^*(S;{\mathbb{Q}})\leq 2^ n,\) and derive a formula for the Lefschetz number of a map.

MSC:

55P62	Rational homotopy theory
55M20	Fixed points and coincidences in algebraic topology

Keywords:

spaces with finite dimensional rational homotopy; de Rham homotopy; Lefschetz number