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We present a classification of the possible regular, spherically symmetricsolutions of the Einstein-Yang-Mills system which is based on a bundletheoretical analysis for arbitrary gauge groups. It is shown that such solitonsmust be of magnetic type, at least if the magnetic Yang-Mills charge vanishes.Explicit expressions for the Chern-Simons numbers of these selfgravitatingYang-Mills solitons are derived, which involve only properties of irreducibleroot systems and some information about the asymptotics of the solutions. Itturns out, as an example, that the Chern-Simons numbers are alwayshalf-integers or integers for the gauge groups $SU(n)$. Possible physicalimplications of these results, which are based on analogies with the unstablesphaleron solution of the electroweak theory, are briefly indicated.Comment: 31 pages, Latex, report ZU-TH 23/93, figure can be obtained from the  author

Self-gravitating Yang–Mills solitons and their Chern–Simons numbers