Coincidences of maps into homogeneous spaces (Q1283498)

Using methods from his earlier article [Am. J. Math. 120, No. 1, 23-42 (1998; Zbl 0908.55002)] the author studies coincidences of maps from a closed connected orientable manifold \(X\) to the homogeneous space \(M\) of (left) cosets of a compact connected Lie group \(G\) by a finite subgroup \(K\) where \(\dim X=\dim M\). If \(f_1,f_2:X\to M\) then it is shown that the Lefschetz coincidence number satisfies \(L(f_1,f_2)=\frac{\deg\eta}{\deg q}\) where \(q\) and \(\eta\) are defined as follows: Let \(p:G\to M\) be the projection, let \(\Gamma:=f_{1\#}^{-1}(p_\#(\pi_1(G))\) and \(q:\tilde{X}\to X\) the finite cover corresponding to \(\Gamma\). Choose a lift \(\tilde{f}_1\) of \(f_1\) and define \(\eta:\tilde{X}\to G\) by \(\eta(\tilde{x})=[\tilde{f}_1(\tilde{x})]^{-1}\cdot f_1(q(\tilde{x}))\). Moreover, the author is able to prove that the coincidence classes of \(f_1\) and \(f_2\) have coincidence indices of the same sign. In particular, vanishing of the Lefschetz coincidence number implies vanishing of the Nielsen coincidence number. Finally, the author turns to the question of whether the results can be extended to \(H\)-spaces: Let \(X\) be as above and assume that \(M\) is a suitable manifold, i.e., there is point \(e\in M\) and a continuous map \(\theta_e\) from \(M\) into the group of homeomorphisms of \(M\) (with the compact-open topology) such that \(\theta_e(x)(x)=e\) and \(\theta_e(e)(x)=x\) whenever \(x\in M\). It is known that a suitable manifold is necessarily an \(H\)-manifold and that each \(x\in M\) possesses a unique inverse. If then \(f,g:X\to M\) it is shown that \(L(f,g)=\deg\phi\) where \(\phi(x):=[f(x)]^{-1}g(x)\).