Abstract
We study local equivariant maps on real finite dimensional orthogonal representations of a compact abelian Lie group G. Equivariant degree \(\deg _G\) is an invariant applied to determine whether a given map has zeros. The goal of this paper is to present a complete, straightforward proof of the product property of \(\deg _G\). For that purpose, we use the otopy classification and distinguish a special kind of map in each class.
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References
Bartłomiejczyk, P.: On the space of equivariant local maps. Topol. Methods Nonlinear Anal. 45(1), 233–246 (2015). https://doi.org/10.12775/TMNA.2015.012
Bartłomiejczyk, P.: A Hopf type theorem for equivariant local maps. Colloq. Math. 147(2), 315–324 (2017). https://doi.org/10.4064/cm6457-8-2016
Bartłomiejczyk, P., Gęba, K., Izydorek, M.: Otopy classes of equivariant local maps. J. Fixed Point Theory Appl. 7(1), 145–160 (2010). https://doi.org/10.1007/s11784-010-0013-0
Bartłomiejczyk, P., Kamedulski, B., Nowak-Przygodzki, P.: Degree product formula in the case of a finite group action. N. Y. J. Math. 25, 362–373 (2019)
Brown, R.F.: A Topological Introduction to Nonlinear Analysis. Birkhäuser, Basel (1993). https://doi.org/10.1007/978-1-4757-1209-4
Gęba, K., Krawcewicz, W., Wu, J.: An equivariant degree with applications to symmetric bifurcation problems. Part I: Construction of the degree. Proc. Lond. Math. Soc. 69, 377–398 (1994). https://doi.org/10.1112/plms/s3-69.2.377
Guillemin, V., Pollack, A.: Differential Topology. Prentice-Hall, Upper Saddle River (1974). https://doi.org/10.1090/chel/370
López Garza, G., Rybicki, S.: Equivariant bifurcation index. Nonlinear Anal. TMA 73(7), 2779–2791 (2010). https://doi.org/10.1016/j.na.2010.06.001
tom Dieck, T.: Transformation Groups, De Gruyter Studies in Mathematics, vol. 8. Springer, Berlin (1987). https://doi.org/10.1515/9783110858372
tom Dieck, T.: Transformation Groups and Representation Theory. Lecture Notes in Mathematics, vol. 766. Springer, Berlin (1979). https://doi.org/10.1007/BFb0085965
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Kamedulski, B. Product Property of Equivariant Degree Under the Action of a Compact Abelian Lie Group. Bull Braz Math Soc, New Series 52, 879–891 (2021). https://doi.org/10.1007/s00574-020-00236-3
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DOI: https://doi.org/10.1007/s00574-020-00236-3