Product property of equivariant degree under the action of a compact abelian Lie group (Q2052811)

Let \(V\) and \(W\) be real finite dimensional orthogonal representations of a compact abelian Lie group \(G.\) A map \(f\) in \(V\) is called an equivariant local map if its domain \(D_f\) is an open \(G\)-invariant subset; \(f^{-1}(0)\) is compact; \(f(gx) = gf(x)\) for each \(x \in D_f\) and \(g \in G.\) The space of all equivariant local maps in \(V\) is denoted by \(\mathcal{C}_G(V).\) The main result is the following product property of the equivariant degree (see, [\textit{P. Bartłomiejczyk}, Colloq. Math. 147, No. 2, 315--324 (2017; Zbl 1371.55009)]). Let \(f \in \mathcal{C}_G(V),\) \(\tilde{f} \in \mathcal{C}_G(W).\) Then \(f \times \tilde{f} \in \mathcal{C}_G(V \times W)\) and \[\mathrm{deg}_G(f \times \tilde{f}) = \mathrm{deg}_G f \cdot \mathrm{deg}_G \tilde{f}.\]