Degree theory for equivariant maps. I

Ize, J.; Massabò, I.; Vignoli, A.

doi:10.1090/S0002-9947-1989-0935940-8

Degree theory for equivariant maps. I
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by J. Ize, I. Massabò and A. Vignoli

Trans. Amer. Math. Soc. 315 (1989), 433-510

DOI: https://doi.org/10.1090/S0002-9947-1989-0935940-8

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Abstract:

A degree theory for equivariant maps is constructed in a simple geometrical way. This degree has all the basic properties of the usual degree theories and takes its values in the equivariant homotopy groups of spheres. For the case of a semifree ${S^1}$-action, a complete computation of these groups is given, the range of the equivariant degree is determined, and the general ${S^1}$-action is reduced to that special case. Among the applications one recovers and unifies both the degree for autonomous differential equations defined by Fuller [F] and the ${S^1}$-degree for gradient maps introduced by Dancer [Da]. Also, a simple but very useful formula of Nirenberg [N] is generalized (see Theorem 4.4(ii)).

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