The absolute degree and the Nielsen root number of compositions and Cartesian products of maps

Abstract

Brouwer's homological degree has the multiplicative property for the composition of maps. That is, if f :X→Y and g :Y→Z are maps between closed oriented manifolds X,Y,Z of the same dimension, then | deg(g∘f)|=| deg(f)|| deg(g)| . Hopf's absolute degree is defined for maps between all n-manifolds, whether orientable or not, and is equal to the absolute value of the Brouwer degree if the manifolds are orientable. It is shown that the absolute degree does not always have the multiplicative property for compositions, but that it does have this property for orientable maps, i.e., for maps that do not map any orientation-reversing loop to a contractible one. If at least one of f and g is not an orientable map, the absolute degree of the composition g∘f can still be calculated from the absolute degrees of f and g if additional information about these two maps and a “correction term” κ(f,g) that depends on the homomorphisms of the fundamental groups induced by f and g are included. Although the Nielsen root number is closely related to the absolute degree, the multiplicative property for compositions can fail to hold for it even if the manifolds are orientable, but it does hold after the insertion of the correction term κ(f,g). Other interpretations of this correction term are presented. Given maps f i :X i→Y i between n i -manifolds, for i=1,2, the Brouwer degree of their Cartesian product f 1×f 2 :X 1×X 2→Y 1×Y 2 has the multiplicative property | deg(f 1×f 2)|=| deg(f 1)|| deg(f 2)| . The results obtained concerning the multiplicative property for the composition of maps are used to investigate the multiplicative property for the Cartesian product of maps. We include an appendix on maps of aspherical spaces: Building on previous results of Brooks and Odenthal we show that if f :X→Y is a map of connected compact infrasolvmanifolds of the same dimension, then the Nielsen root number and absolute degree of f are equal.