Given a smooth manifold M, let c(M) denote the minimal number of critical points of a smooth real-valued function on M. Let (M, ω) be a closed symplectic manifold, and let ϕ : M → M be a Hamiltonian symplectomorphism of M. The well-known Arnold conjecture claims that the number Fix ϕ of fixed points of ϕ is at least c(M), i.e. Fix ϕ ≥ c(M). Floer and Hofer proved that, for manifolds M with π2(M) = 0, the number Fix ϕ is at least the cup-length of M. So, since c(M) is at least the cup-length of M, we have here a weak form of the Arnold conjecture. Based on Floer–Hofer analytical results, and using some additional topological arguments, we prove that, for manifolds M2n with π2(M) = 0, c(M) = 2n + 1 and Fix ϕ ≥ 2n + 1. In particular, the original Arnold conjecture holds for such manifolds.