On the homology of the integral manifolds in the planar \(N\)-body problem (Q2730716)

In the planar \(N\)-body problem, dynamics takes place on a \((4N-7)\)-dimensional manifold, the reduced integral manifold \(m_R\). The topology of this manifold depends only on masses \(M=(m_1,\ldots,m_N)\) and on the product \(\nu = -hc^2\) involving the energy \(h\) and the angular momentum \(c\) integrals. Building on Smale's topological analysis, the author describes the homology of \(m_R(M, \nu)\). In particular, the homology groups of \(m_R(M, \nu)\) are computed for all \(M\) when \(\nu \) is large, and for all \(\nu \) for four equal masses.