Reidemeister Number in Lefschetz Fixed point theory

Abstract

Several interesting numbers such as the homotopy invariant numbers the Lefschets number L(f), the Nielsen number N(f), fixed point index i(X, f,U) and the Reidemeister number R(f) play important roles in the study of fixed point theorems. The Nielsen number gives more geometric information about fixed points than other numbers. However the Nielsen number is hard to compute in general. To compute the Nielsen number, Jiang related it to the Reidemeister number R(f ) of the induced homomorphism f  :  1(X)    1(X) when X is a lens space or an H-space (Jian type space). For such spaces, either N(f) = 0 or N(f) = R(f) the Reidemeister number of f  and if R(f) =   then N(f) = 0 which implies that f is homotopic to a fixed point free map. This is a review article to discuss how these numbers are related in fixed point theory.