The degree of maps of free \(G\)-manifolds (Q926602)

Let \(G\) be a compact Lie group of dimension \(k\) acting freely on a closed connected smooth manifold \(M\) of dimension \(n\), and let \(\varphi_M:M/G\to BG\) be a classifying map. The first author [ibid. 1, 111--121 (2007; Zbl 1124.55002)] defined the \(G\)-class group \(\Gamma(G,M)=\text{im}\varphi^*_M\subset H^{n-k}(M/G;\mathbb{Z})\). If \(f:M\to N\) is an equivariant map between free, closed, connected, smooth \(G\)-manifolds of dimension \(n\) and \(a\) is the index of \(\Gamma(G,M)\) in \(H^{n-k}(M/G;\mathbb{Z})\) and \(b\) the index of \(\Gamma(G,N)\) in \(H^{n-k}(N/G;\mathbb{Z})\) then the first author [op. cit.] proved that \(a=b\cdot\deg f\). In the present article the authors consider a more general situation: \(G\) and \(H\) are connected compact Lie groups acting freely on manifolds \(M\) and \(N\) of the same dimension \(k\), \(u:G\to H\) is an epimorphism and \(f:M\to N\) satisfies \(f(\lambda x)=u(\lambda)f(x)\). Assume in addition that \(G\) and \(H\) are oriented and that \(M\) and \(N\) are compact, connected, and orientable. Let \(\overline{\varphi}:M/G\to BG\) and \(\overline{\psi}:N/H\to BH\) be classifying maps for \(M\) and \(N\). Denote by \(Bu:BG\to BH\) the map induced by \(u\), let \(K\) be the kernel of \(u\), and let \(\tau:H^n(M/K)\to H^{n-k}(M/G)\) and \(\tau:H^nN\to H^{n-k}(N/H)\) be transfer maps (the first author [op. cit.] showed that these transfer maps are isomorphisms). Denote the index of \(\overline{\varphi}^*(Bu)^*(H^{n-k}(BH))\) in \(H^{n-k}(M/G)\) by \(a\) and let \(b\) be the index of \(\overline{\psi}^*(H^{n-k}(BH))\) in \(H^{n-k}(N/H)\). Let \(f\) and \(u\) be as above then the authors show that \(a\cdot\deg q=b\cdot\deg f\) where \(q:M\to M/K\) is the obvious covering map.