A proof of Ganea’s conjecture for rational spaces. (English) Zbl 0717.55014

Hess, Kathryn P.

Topology 30, No. 2, 205-214 (1991).

Ganea’s conjecture concerns the Lusternik-Schnirelmann category of spaces. It is easy to show that \(cat(S\times T)\leq cat(S)+cat(T)\). Both strict inequality and equality can occur, but the only known examples of strict inequality appear when S and T have homology torsion. This led Ganea to the conjecture that \(cat(T\times S^ n)=cat(T)+1\) for every finite complex T and \(n\geq 1\). The author gives a proof of Ganea’s conjecture whenever T is a rational simply connected space and \(n\geq 2\). The proof uses Sullivan’s minimal models. If (\(\Lambda\) X,d) denotes the minimal model of a space S, then \(cat_ 0(S)\), resp. \(Mcat_ 0(S)\), is the least integer n such that the minimal model of the projection \((\Lambda X,d)\to (\Lambda X,\Lambda^{>n}x,T)\) admits a retraction as differential graded algebra, resp. as differential graded (\(\Lambda\) X,d)-module. If S is a rational space, then \(cat(S)=cat_ 0(S)\). In general, \(cat_ 0(S)\leq cat(S)\). The first step in the proof is due to B. Jessup \((Mcat_ 0(S)+1=Mcat_ 0(S\times S^ n)),\) the second step, the main part of this paper, consists of proving that \(Mcat_ 0(S)=cat_ 0(S)\).

Reviewer: Y.Felix

MSC:

55P62	Rational homotopy theory
55M30	Lyusternik-Shnirel’man category of a space, topological complexity à la Farber, topological robotics (topological aspects)

Keywords:

differential graded module; Lusternik-Schnirelmann category; rational simply; connected space; Sullivan’s minimal models; differential graded algebra