A sufficient condition for the realizability of the least number of periodic points of a smooth map

There are two algebraic lower bounds of the number of n -periodic points of a self-map f : M → M of a compact smooth manifold of dimension at least 3: N F n ( f ) = min { # Fix ( g n ) ; g ~ f ; g continuous } and N J D n ( f ) = min { # Fix ( g n ) ; g ~ f ; g smooth } . In general NJD n ( f ) may be much greater than NF n ( f ). In the simply connected case, the equality of the two numbers is equivalent to the sequence of Lefschetz numbers satisfying restrictions introduced by Chow, Mallet-Parret and Yorke ( 1983 ). The last condition is not sufficient in the non-simply connected case. Here we give some conditions which guarantee the equality when π 1 M = Z 2 .