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The main result states that if f : X → X f:X \to X is any map on a k k -dimensional torus X X , then the Nielsen number and Lefschetz number of f f are related by the formula N ( f ) = | L ( f ) | N(f) = |L(f)| . Thus, on the torus, the Lefschetz number gives information, not just on the existence of fixed points, but on the number of fixed points as well. No other compact Lie group has this property. The main result, when applied to certain types of maps on compact Lie groups, produces new information on the fixed point theory of such maps.

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The Nielsen root number N (f ; c) of a map f : M → N at a point c ∈ N is a homotopy invariant lower bound for the number of roots at c, that is, for the cardinality of f −1 (c). There is a formula for calculating N (f ; c) if M and N are closed oriented manifolds of the same dimension. We extend the calculation of N(f; c) to manifolds that are not orientable, and also to manifolds that have non-empty boundaries and are not compact, provided that the map f is boundary-preserving and proper. Because of its connection with degree theory, we introduce the transverse Nielsen root number for maps transverse to c, obtain computational results for it in the same setting, and prove that the two Nielsen root numbers are sharp lower bounds in dimensions other than 2. We apply these extended root theory results to the degree theory for maps of not necessarily orientable manifolds introduced by Hopf in 1930. Thus we re-establish, in a new and modern treatment, the relationship of Hopf 's Absolutgrad and the geometric degree with homotopy invariants of Nielsen root theory, a relationship that is present in Hopf 's work but not in subsequent re-examinations of Hopf 's degree theory.