Algebraic periods and minimal number of periodic points for smooth self-maps of 1-connected 4-manifolds with definite intersection forms (Q6566222)

In this paper, the authors study the problem of finding the minimal number of \(r\)-perodic points in the smooth homotopy class of a given map \(f:M\rightarrow M\). To establish the mutual relations between the continuous and the smooth category is one of the most important challenges in modern periodic point theory.\N\N\textbf{Definition} [\textit{G. Graff} et al., Forum Math. 21, No. 3, 491--509 (2009; Zbl 1173.37014)]: Let \(\{L(f^n)\}_{n|r}\) be a finite sequence of Lefschetz numbers. We decompose \(\{L(f^n)\}_{n|r}\) into the sum of \(DD^m(1|r)\) sequences \(\{c_i\}_{n|r}\) for \(i=1,\dots, s.\) Thus, for each \(n|r\), the following equality holds: \[ L(f^n)=c_1(n)+\cdots+c_s(n) \] Each such decomposition determines the number \(s\). We define the number \(D_r^m[f]\) as the smallest \(s\) which can be obtained in this way.\N\NActually, the invariant \(D_r^m[f]\) is equal to the minimal number of \(r\)-periodic points in the smooth homotopy class of \(f\).\N\N\textbf{Theorem} [\textit{G. Graff} et al., J. Fixed Point Theory Appl. 13, No. 1, 63--84 (2013; Zbl 1276.55005)]:\N\NLet \(M\) be a closed smooth and simply connected manifold of dimension \(m\geq 4\) and \(r\in \mathbb{N}\) a fixed number. Then \[ D_r^m[f]=min\{\sharp\mbox{Fix}(g^r): g\mbox{ is a smoothly homotopic to }f\} \]\N\NThe authors compute the invariants \(D_r^4[f]\) for the self-maps of closed \(1\)-connected smooth \(4\)-manifolds with definite intersection forms. The main results are as follows:\N\NOne is about the self-maps of closed smooth \(1\)-connected \(4\)-manifolds:\N\N\textbf{Theorem 26}: Let \(f\) be a smooth self-map of a connected sum of \(m\) copies of complex projective planes with degree \(k\) and \(r=p_1\cdots p_s\) be a product of different odd prime numbers.\N\NAssume that one of the following conditions holds:\N\begin{itemize}\N\item[(i)] \(k\geq 6(m+1)^2\)\N\item[(ii)] \(k>1\) is arbitrary and \(m=2\)\N\end{itemize}\NThen the value of \(D_r^4\) mod reg\(_1\) depends only on \(s\) and is equal: \[ D_r^4[f] \mbox{ mod reg}_1=\frac{2^s+(-1)^{s+1}}{3}=h(s). \] If\N\begin{itemize}\N\item[(iii)] \(k>1\) and \(m>1\) are arbitrary and \(i>2\log_{\frac{2}{3}k}(2(1+m))=:r_0,\) \N\end{itemize}\Nthen\N\[ \frac{2^{s^{'}}-1}{3}+\lceil\frac{\sharp G(r_0)}{3}\rceil\leq D_r^4[f] \mbox{ mod reg}_1 \leq \frac{2^s+(-1)^{s+1}}{3} \] where \(s^{'}\) denotes the number of elements in \(\{p_1,\dots, p_s\}\) which are \(\geq r_0\), \(G(r_0)=\{\beta| r:\beta\geq r_0 \mbox{ and }\exists_{\alpha|r,\alpha\ne 1}\alpha<r_0 \mbox{ and }\alpha|\beta\}\), \(\lceil a\rceil\) stands for the least integer greater than or equal to a number \(a\).\N\NAnother main result is about \(D_r^4[f]\) mod reg\(_1\) for self-maps of \(\mathbb{C}P^2\sharp \mathbb{C}P^2\) and small values of \(r\).\N\NThe authors give an algorithm and obtain the values of \(D_r^4[f]\) mod reg\(_1\) for \(M=\{a_i: r=\prod_{i=1}^k p_i^{a_i}\}\) in Section 8.2.