Abstract. We study the number of Darboux charts needed to cover a closed connected symplectic manifold (M, ω) and effectively estimate this number from below and from above in terms of the Lusternik-Schnirelmann category of M and the Gromov width of (M, ω).

Abstract. Using a result of Singhof, we prove that cat(M × S m ) = cat M + 1 provided M is a connected closed PL manifold with dim M ≤ 2 cat M − 3 and S m is the m-sphere, m > 0.Let cat X denote the Lusternik-Schnirelmann category of X (normalized, i.e., cat S m = 1). There is a long standing Ganea conjecture that cat(X × S m ) = cat X + 1 for every connected finite CW -complex X and every m > 0, see However, it seems that nobody noted that Singhof's Theorem implies a stronger result. Namely, the following theorem is valid:Theorem. Let M be a connected closed PL manifold such thatProof. First, we prove that cat(M × T r ) = cat M + r where T r is the r-torus. We prove this by induction. For r = 1 this follows from (ii). Now, suppose that

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