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 classical 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. In spite of their fundamental importance in the N-body problem, relatively little is known about the global structure of these spaces. In particular, for non-zero angular momentum, the manifolds exhibit a variety of discontinuities and complexities at collinear configurations. These discontinuities are removed by the introduction of a blow-up construction, which is then exploited to provide a description of the integral manifolds, in the sense of a reduction formula for the homology groups of the manifolds. That reduction formula is then applied to investigate the topology of the integral manifolds for energies just below zero. For four masses, the homology groups for this energy range are fully determined; for arbitrary N, the homology with real coefficients is determined. Comparison to the values for positive energies confirms that for all masses, the topology of the integral manifold changes at zero energy.

“ N体问题”是指研究N点质量在其相互引力吸引下运动的动力学。对于这样的系统，经典的是要有质心，线性动量，角动量和能量的守恒量。这些守恒量的水平集\（{{mathfrak {M}}（c，h）\）由角动量c和能量h参数化，被称为积分流形。尽管它们在N中具有根本的重要性体问题，对这些空间的整体结构知之甚少。特别地，对于非零角动量，歧管在共线构造下表现出各种不连续性和复杂性。这些不连续性通过引入爆破结构来消除，然后从歧管的同源基团的简化公式的意义上讲，该爆炸结构可用于提供整体歧管的描述。然后，将该减少公式应用于调查积分歧管的拓扑，以了解能量刚好在零以下。对于四个质量，已完全确定了该能量范围的同源基团。对于任意N，确定具有实系数的同源性。与正能量值的比较证实，对于所有质量，积分歧管的拓扑都在零能量处发生变化。