ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

6. Topology

1991 MS Classification: 55 Lemaire, Jean-Michel, CNRS et Université de Nice, France Rational Lusternik-Schnirelmann category of Poincaré complexes A CW complex X has rational Lusternik-Schnirelmann (resp. strong) category n — notation : cat_{0}(X) = n (resp. Cat_{0}(X) = n) — if its rationalization X_{0} cannot be covered by less than n + 1 open sets contractible in X_{0} (resp. in themselves). If the rational cohomology of X satisfies Poincaré duality, we show that cat_{0}(X) = Cat_{0}(X) = e_{0}(X), where the latter denotes the length of the (Milnor-Moore) filtration of H^{∗}(X; ℚ), induced by word length in the Sullivan minimal model. Both equalities are known to be false for arbitrary finite CW complexes. In addition, we observe that cat_{0}(X − x) = cat_{0}(X) − 1, i. e. the attachement of the top cell increases rational category by 1. These results are joint work with O. Cornéa, Y. Félix and S. Halperin. Reference: O. Cornéa, Y. Félix and J. -M. Lemaire, Rational category and cone-length of Poincaré complexes, Topology 37 No.4 (1998), 743–748.