Equivariant coincidence Nielsen numbers
Let \(W\) be a finite group acting on closed \(n\)-manifolds \(X\) and \(Y\). This paper studies coincidences of \(W\)-maps \(f,g: X\to Y\), that is, solutions to \(f(x)= g(x)\). Nielsen coincidence theory for maps between closed manifolds of the same dimension began with work of Schirmer in the 1950's. A Nielsen fixed point theory for equivariant maps was introduced by Wong in 1991. Thus it was natural to extend equivariant Nielsen theory to the coincidence setting and Fagundes published a paper on this subject in 1997. The present paper improves substantially our understanding of the coincidence theory of equivariant maps. Note that if \(x\in X\) is a coincidence point of \(f\) and \(g\), then every point in the \(W\)-orbit of \(x\) is also a coincidence point, thus one may also consider coincidence orbits. Let \(M_WO(f,g)\) denote the minimum number of coincidence orbits among all \(W\)-maps \(\phi,\psi: X\to Y\) that are homotopic through equivariant maps to \(f\) and \(g\), respectively. Under reasonable hypotheses, the authors are able to define a Nielsen number that is a \(W\)-homotopy invariant lower bound for \(M_WO(f, g)\). Now let \(H\) be a subgroup of \(W\) and let \(X^H\) denote the subspace of \(X\) consisting of points that are fixed by all elements of \(H\). Then the restrictions \(f^H,g^H: X^H\to Y^H\) are \(WH\)-maps for \(WH\) the Weyl group of \(H\). For \(X^{(H)}\) the union of all \(X^K\), where \(K\) is conjugate to \(H\), there are the maps \(f^{(H)}, g^{(H)}: X^{(H)}\to Y^{(H)}\). Under appropriate hypotheses, the authors define Nielsen numbers that give \(W\)-homotopy invariant lower bounds for \(M_WO(f^{(H)}, g^{(H)})\) as well as for the corresponding concept regarding the actual number of coincidence. A significant feature of the present theory is that it eliminates a compactness restriction on the homotopies in Fagundes' theory, a requirement that appears in parts of Wong's work as well. This paper also presents an example that is important to the development of equivariant Nielsen theory. It is of a closed 12-dimensional \(W\)-manifold \(X\) where \(W\) is the product of the group of order two and the symmetric group on three letters. Equivariant maps \(f, g: X\to X\) are defined whose coincidence theory, up to \(W\)-homotopy, can only be adequately bounded using the equivariant Nielsen numbers introduced in the paper. This is notable because, for all previous examples in equivariant Nielsen theory, the already existing relative Nielsen numbers or Nielsen periodic point numbers would have given the same information as that obtained by Nielsen numbers defined for the purpose of equivariant theory.
scientific article; zbMATH DE number 1913874
scientific article