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# Minimization of the number of periodic points for smooth self-maps of closed simply-connected 4-manifolds

• Let $M$ be a smooth closed simply-connected $m$-dimensional manifold, $f$ be a smooth self-map of $M$ and $r$ be a given natural number. The invariant $D^m_r [f]$ de fined by the authors in [Forum Math. 21 (2009)] is equal to the minimum of #Fix($g^r$) over all maps $g$ smoothly homotopic to $f$. In this paper we calculate the invariant $D^4_r [f]$ for the class of smooth self-maps of 4-manifolds with fast grow of Lefschetz numbers and for $r$ being a product of di erent primes.
Mathematics Subject Classification: Primary: 37C25, 55M20; Secondary: 37C05.

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