On the equivariant Hopf theorem
Hopf's classification theorem states that the homotopy set \([X, S^n]\) of maps from a compact \(n\)-dimensional manifold \(X\) to \(S^n\) is in one-to-one correspondence with \(H^n(X;\mathbb{Z})\). The aim of this paper is to give an equivariant version of this theorem for a finite group \(G\). Let \(X\) be a compact locally smooth \(G\)-manifold and \(S\) an orthogonal \(G\)-sphere such that for fixed manifolds \(X^H\) and \(S^H\dim X^H\leq \dim S^H\) holds and \(S^K\nsubseteq S^H\) if \(H\nsubseteq K\) for isotropy subgroups \(H\) and \(K\) for \(X\). An essential point for classifying \(G\)-maps here is to use equivariant obstruction theory. In order to apply this method the author introduces an unstable equivariant degree \(\widetilde{d}_G\). For each isotropy group \(H\) let \(X_H\subset X^H\) denote the subspace consisting of points \(x\in X\) with isotropy \(H\) and let \(C\) be a component of \(X_H\). Then the above two conditions on \(X\) and \(S\) allow to choose a point \(y_H\in S^H\) with minimal isotropy type. Moreover, for any \(G\)-map \(f :X\to G\) one finds \((f^H)^{-1}y_H\) to be compact in \(C\) whenever \(\dim C= \dim S^H\). So assign to \(\widetilde{d}_G(f)(C)\) the Brouwer degree of \(f^H|C\) in this case and zero otherwise. Then, to put it simply, \(\widetilde{d}_G(f)\) is defined as a collection of \(d_G(f)(C)s\). This unstable degree is rather fine as compared with the stable one due to tom Dieck and others. If \(X^H\) is orientable for every isotropy group \(H\), then these two degrees are equivalent in a class of certain groups named 2-split groups (Proposition 5.2). Let \(p_H: X_H\to X_H/W_G H\) be the projection onto the quotient by the Weyl group of \(H\) in \(G\). Let \(\overline{C}=p_H(C)\) where \(C\) is as above. This is called the full-dimensional chamber of \(X/G\) (with isotropy type \((H)\)) whenever \(\dim\overline{C} >0\) and \(\dim \overline{C} = \dim S^H\). To state the main results of this paper one needs further the notion of discordant and concordant chambers, a notion too complicated to be repeated here. Using this terminology without hesitation the author's equivariant classification theorem may be stated as follows: The unstable equivariant degree \(\widetilde{d}_G\) classifies \(G\)-maps in \([X, S]_G\) if and only if every full-dimensional chamber in \(X/G\) is concordant with \(S\) (Corollary 6.5). This is an immediate corollary of the main theorem (Theorem 6.4) in which the number of distinct \(G\)-homotopy classes of maps with the same unstable degree is computed. Here for \(\widetilde{d}_G\) to classify \(G\)-maps means that \(\widetilde{d}_G\) classifies two maps which coincide on the 0-dimensional equivariant strata up to \(G\)-homotopy. In the final section many examples are given which make the paper well readable. Moreover, they help the readers to verify the assumptions on \(X\) and \(S\) and show how computations can be carried out.
scientific article; zbMATH DE number 1880022