Nielsen Coincidence Number Research Articles

Nielsen coincidence theory and coincidence-producing maps for manifolds with boundary

Nielsen coincidence theory is extended to manifolds with boundary. For X and Y compact connected oriented n-manifolds with boundary, and for maps ƒ : X → Y and g : ( X, ∂ X) → ( Y, ∂ Y), a coincidence index—which is a local version of Nakaoka's Lefschetz coincidence number—and a Nielsen coincidence number are defined and their properties explored. As an application, coincidence-producing maps g are characterized if Y is acyclic over the rationals and many new examples of coincidence-producing maps are constructed.

Lefschetz and Nielsen coincidence numbers on nilmanifolds and solvmanifolds

Suppose M 1, M 2 are compact, connected orientable manifolds of the same dimension. Then for all pairs of maps ƒ, g: M 1→ M 2, the Nielsen coincidence number N(ƒ, g) and the Lefschetz coincidence number L(ƒ, g) are measures of the number of coincidences of ƒ and g: points x∈ M 1 with f( x)= g( x). A manifold is a nilmanifold (solvmanifold) if it is a homogenous space of a nilpotent (solvable) Lie group. If M 1 and M 2 are compact connected orientable solvmanifolds, then N( ƒ, g)⩾| L( ƒ, g)| for all ƒ and g, with equality for all ƒ and g if M 2 is a nilmanifold.