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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

Degree theory for equivariant maps. I


Authors: J. Ize, I. Massabò and A. Vignoli
Journal: Trans. Amer. Math. Soc. 315 (1989), 433-510
MSC: Primary 58E07; Secondary 47H15, 58C30
MathSciNet review: 935940
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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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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1989-0935940-8
PII: S 0002-9947(1989)0935940-8
Keywords: Equivariant topological degree, $ {S^1}$-homotopy groups of spheres, Fuller's degree
Article copyright: © Copyright 1989 American Mathematical Society