Algebraic periods and minimal number of periodic points for smooth self-maps of \(\textbf{1}\)-connected \(\textbf{4}\)-manifolds with definite intersection forms

Let M be a closed 1-connected smooth 4-manifolds, and let r be a non-negative integer. We study the problem of finding minimal number of r-periodic points in the smooth homotopy class of a given map \(f:M \rightarrow M\) $f:M\to M$ . This task is related to determining a topological invariant \(D^4_r[f]\) $D_r^4[f]$ , defined in Graff and Jezierski (Forum Math 21(3):491–509, 2009), expressed in terms of Lefschetz numbers of iterations and local fixed point indices of iterations. Previously, the invariant was computed for self-maps of some 3-manifolds. In this paper, we compute the invariants \(D^4_r[f]\) $D_r^4[f]$ for the self-maps of closed 1-connected smooth 4-manifolds with definite intersection forms (i.e., connected sums of complex projective planes). We also present some efficient algorithmic approach to investigate that problem