Product Property of Equivariant Degree Under the Action of a Compact Abelian Lie Group
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Product Property of Equivariant Degree Under the Action of a Compact Abelian Lie Group Bartosz Kamedulski1 Received: 30 April 2020 / Accepted: 16 November 2020 © Sociedade Brasileira de Matemática 2020
Abstract We study local equivariant maps on real finite dimensional orthogonal representations of a compact abelian Lie group G. Equivariant degree degG 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 degG . For that purpose, we use the otopy classification and distinguish a special kind of map in each class. Keywords Equivariant degree · Product property · Burnside ring Mathematics Subject Classification 55P91 · 54C35
1 Introduction A continuous map f : Df → Rn is called local when Df is an open subset of Rn and the set of zeros f−1 (0) is compact. The Brouwer degree is well-defined for such maps. A product of two local maps f and f˜ is also local and its degree can be evaluated ˜ It is a well-known characteristic of the degree, by multiplying the degrees of f and f. called product property (Brown 1993, Prop. 8.7). In this paper, we prove that analogous property holds for equivariant degree degG , where G is a compact abelian Lie group, that is: ˜ = degG f · degG f, ˜ degG (f × f) where the maps f, f˜ are local and equivariant. What is worth noting is that while Brouwer degree of a given map is an integer, the values of degG are elements of the Burnside ring of G, where multiplication is non-trivial.
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Bartosz Kamedulski [email protected] Institute of Mathematics, Faculty of Mathematics, Physics and Informatics, University of Gda´nsk, Wita Stwosza 57, 80-308 Gda´nsk, Poland
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B. Kamedulski
In López Garza and Rybicki (2010), an analogous result is presented for a different topological invariant ∇G -deg, defined on the class of equivariant gradient maps and with elements in the Euler ring U(G). The authors discuss product formula in the case when G is an n-dimensional torus. A more general result for degG (product property for a compact Lie group, not necessarily abelian) has already been obtained in Ge˛ba et al. (1994), but the proof provided by the authors was very formal and a bit sketchy in some parts. A complete, step-by-step proof of the case where G is finite can be found in Bartłomiejczyk et al. (2019). The goal of this paper is to present a clear and complete proof of the abelian case, not necessarily finite. The proof in the general setting is in further perspective. The paper is divided into several sections. Section 2 contains basic definitions. In Sect. 3 we recall neccessary theorems from the vector bundle theory. Then we describe the equivariant degree degG along with its basic properties (Sect. 4). Our main result is stated in Sect. 5. In Sect. 6 we introduce standard and polystandard maps and then study their properties. These are used in Sect. 7, which contains the proof of our main result. The final Sect. 8 contains additional commentary on certain technical det
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