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

Abstract

AbstractLet 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 → 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