On relations between gradient and classical equivariant homotopy groups of spheres
Let \(V\) be a finite-dimensional representation of a compact Lie group \(G\) and let \(f : U \to V\) denote a \(G\)-equivariant map on an open \(G\)-invariant subset \(U \subset \mathbb{R}^k\oplus V\) such that \(f^{-1}(0)\) is compact. If \(f\) satisfies the equality \(f(x, v)=\nabla_v\varphi(x, v)\) where \(\varphi : U \to \mathbb{R}\) is a \(G\)-invariant \(C^1\)-function, then it is called a \(G\)-equivariant gradient map. This map \(f\) extends continuously as a map \(\hat{f} : S^{k+V} \to S^V\) between their one-point compactifications. This paper is concerned with the problem of whether or not any two equivariant gradient maps belonging to the same stable equivariant homotopy class are stably homotopic in the category of gradient equivariant maps. The authors examine this problem by enlarging the category of equivariant maps considered there and in fact the authors prove that under a certain condition its assertion holds true for orthogonal equivariant maps. Here by a \(G\)-equivariant orthogonal map is meant a \(G\)-map \(f : U \to V\) satisfying the condition \(f(x, v) \bot T_vGv\). Let \(S_e^{k+V}\) denote a subset of \(S^{k+V}\) on which \(G\) acts freely. Put \(A^{k+V}=S^{k+V}\setminus S_e^{k+V}\) and write \([S^{k+V}, A^{k+V}; S^V, *]_G^\bot\) for the set of relative homotopy classes of orthogonal equivariant maps. Then the principal conclusion of this paper is that if \(q=\dim G \leq \dim V-k-2\), then there is a natural bijection from \([S^{k+V}, A^{k+V}; S^V, *]_G^\bot\) to the set \([S^{q+k+V}, A^{q+k+V}; S^V, *]_G^*\) of relative homotopy classes of equivariant maps.
scientific article; zbMATH DE number 6162209