The \(N\)-body problem refers to the study of the dynamics of a system of \(N\) point masses, moving under their mutual gravitational attraction. For such a system, it is a classical fact that there are conserved quantities of center of mass, linear momentum, angular momentum and energy. The level sets \({\mathfrak M}(c, h)\) of these conserved quantities are parameterized by the angular momentum \(c\) and the energy \(h\), and are known as the integral manifolds. These are algebraic sets, whose topology varies with the angular momentum \(\vec{{\mathbf c}}\) and energy \(h\). The spatial integral manifolds \({\mathfrak M}(c, h)\) have dimension \(6N -10\), while the planar integral manifolds \({\mathfrak m}(c, h)\) have dimension \(4N - 6\) and the collinear manifolds have dimension \(2N - 3\).
In addition to the integrals, the \(N\)-body problem also admits a symmetry. The potential function, equations of motion and integrals are all invariant under rotation. Thus, the planar integral manifold admits an \(S O_2\) action. The spatial problem is a little different. In the presence of a non-zero angular momentum, the angular momentum vector provides a preferred direction, and there is an \(S O_2\) symmetry of rotations about that vector. On the other hand, with zero angular momentum, there is no preferred direction, so the spatial integral manifold admits an \(S O_3\) symmetry. The resulting quotient spaces \({\mathfrak M}_R(c, h)= {\mathfrak M}(c, h)/S O_k\) (with \(k=3\) if \(c=0\) and \(k=2\) otherwise) and \({\mathfrak m}_R(c, h) = {\mathfrak m}(c, h)/S O_2\) are refered to as the reduced integral manifolds. Since the dynamics of the \(N\)-body is invariant under these rotations, the reduced integral manifolds have well-defined dynamics. They are, in some sense, the smallest spaces on which the complete dynamics of the \(N\)-body problem is displayed: with non-zero angular momentum, \(\dim ({\mathfrak M}_R(c, h))= 6N-11\) and \(\dim ({\mathfrak m}_R(c, h))= 4N-7\).
Poincaré’s work on the restricted three-body problem is where one of the fundamental concepts of dynamical systems was first raised: that the topology of a space influences the dynamics that can occur in it. It is natural in the \(N\)-body problem to understand the topology of integral manifolds and reduced integral manifolds. In particular, to understand how that topology varies with the parameters \(h\) and \(c\). Building on previous work by Smale, Cabral, Easton, Albouy and others, this paper returns to the investigation of the global structure of the integral manifolds of the classical Newtonian \(N\)-body problem. In particular, a survey of the efforts to understand the topology of these spaces can be found in [C. K. McCord et al., The integral manifolds of the three body problem. Providence, RI: American Mathematical Society (AMS) (1998; Zbl 0904.70007)]. An obvious difficulty threads through that body of work: how do we “understand” the topology of a high-dimensional space? The author claims that unless that space has a very simple structure (for example, a product of spheres), it is unlikely to be able to explicitly characterize its topology.
Following Poincaré, the author’s strategy is to use algebraic topology, particularly, the theory of homology, as a tool to describe the integral manifolds. The overall goal of the paper under review is three-fold. It goes on to compute the homology.

1.

From the algebraic equations that define the integral manifold, compute the homology groups of spaces.

2.

By detecting changes in the homology, detect changes in the topology of the manifolds.

3.

By relating the homology to the dynamics, detect the existence or non-existence of dynamical features in the manifolds.

The current paper adds four objectives for understanding the topological structure of integral maifolds.

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To generalize and refine the program deployed in [loc. cit.], in order to identify the bifurcation values of the energy on the angular momentum surface (for non-zero angular momentum) and describe in some sense the integral manifolds. “Describe” means “compute the homology groups”, and one of the goals is to provide a formula for the homology groups of the integral manifolds and reduced integral manifolds in terms of the homology of spaces associated with a function defined on the configuration space.

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To address the complexities and discontinuities in this process that are associated with the behavior of the manifolds near collinear configurations. It is found that the central part of this aspect corresponds to an extension of the blow-up construction introduced in [H. Cabral and the author, J. Dyn. Differ. Equations 14, No. 2, 259–293 (2002; Zbl 1032.70008)]

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To begin the application of those techniques to the four-body problem with equal masses, building on the geometric analysis carried out by Q. Wang [Contemp. Math. 292, 239–266 (2002; Zbl 1020.70004)]. To that end, the homology groups of the integral manifolds are computed for energies just below zero. The author sees that in this instance, the same calculation applies to any set of four masses, allowing him to calculate \(H_*({\mathfrak M}(c, h); {\mathbb Z})\) and \(H_*({\mathfrak M}_R(c, h); {\mathbb Z})\) for the energy range \(-\dfrac{m_1^3m_2^3}{2c^2(m_1+m_2)}< h < 0\), where the masses are ordered so that \(m_1\leq m_2 \leq m_3 \leq m_4\). The program theoretically has no issues computing the integral manifold homology for four equal masses at any energy level.

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To build on the approaches of Q. Wang [Contemp. Math. 292, 239–266 (2002; Zbl 1020.70004)] and the author [Ergodic Theory Dyn. Syst. 21, No. 3, 861–883 (2001; Zbl 1028.70008)], to extract as much homological information as possible when only partial information is available. This is illustrated by extending the calculations for energy just below zero from four masses to \(N\) masses. The author calculates the Poincaré polynomials (e.g., the characteristic-zero homology) rather than the integer homology. While the size of the first negative energy range depends on the specific mass values, the homology groups obtained are independent of the mass values, and depend only on the number of particles.

In a lengthy manuscript, the article is split into eight sections that are focused on these objectives. Thanks to the way the article is organized, the reader can follow with ease the steps that were taken to arrive at each result.