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Degree of \(T\)-equivariant maps in \({\mathbb R}^n\). (English) Zbl 1121.47043

Jachymski, Jacek (ed.) et al., Fixed point theory and its applications. Proceedings of the international conference, Bȩdlewo, Poland, August 1–5, 2005. Warsaw: Polish Academy of Sciences, Institute of Mathematics. Banach Center Publications 77, 147-159 (2007).
The authors introduce a degree of \(T\)-equivariant maps in \(\mathbb R^n\), called the \(T\)-equivariant degree, where \(T : \mathbb R^n \to \mathbb R^n\) is an involution given by \(T(u,v)=(u,-v)\) for \((u,v)\in \mathbb R^n = \mathbb R^p \oplus \mathbb R^q\). As an equivariant version of the homotopy result due to A.Parusiński [Stud.Math.96, No.1, 73–80 (1990; Zbl 0714.57015)], it is shown that two \(T\)-equivariant gradient maps \(f,g : (B^n,S^{n-1}) \to (\mathbb R^n, \mathbb R^n \setminus\{0\})\) are \(T\)-homotopic iff they are gradient \(T\)-homotopic, where \(B^n\) denotes the open unit ball in \(\mathbb R^n\) and \(S^{n-1}\) the unit sphere, respectively.
The proof is based on the homotopy classifications of \(T\)-equivariant maps due to D.L.Ferrario [Topology 42, No.2, 447–465 (2003; Zbl 1017.55005)] and E.N.Dancer, K.Gȩba and S.M.Rybicki [Fundam.Math.185, No.1, 1–18 (2005; Zbl 1086.47031)].
For the entire collection see [Zbl 1112.47302].

MSC:

47H11 Degree theory for nonlinear operators
55P91 Equivariant homotopy theory in algebraic topology
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