Combinatorial scheme of finding minimal number of periodic points for smooth self-maps of simply connected manifolds

Let M be a closed smooth connected and simply connected manifold of dimension m at least 3, and let r be a fixed natural number. The topological invariant \({{D^m_r} [f]}\) , defined by the authors in [Forum Math. 21 (2009), 491–509], is equal to the minimal number of r-periodic points in the smooth homotopy class of f, a given self-map of M. In this paper, we present a general combinatorial scheme of computing \({{D^m_r} [f]}\) for arbitrary dimension m ≥ 4. Using this approach we calculate the invariant in case r is a product of different odd primes. We also obtain an estimate for \({{D^m_r} [f]}\) from below and above for some other natural numbers r.