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

• Published in: Journal of fixed point theory and applications Vol. 26; no. 2
• Main Authors: Duan, Haibao, Graff, Grzegorz, Jezierski, Jerzy, Myszkowski, Adrian
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01-06-2024
• Subjects: Analysis; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; intersection forms; Periodic points; 57K41; fixed point index; 4-manifolds; smooth maps; Primary 37C25; Lefschetz numbers; 55M20
• ISSN: 1661-7738; 1661-7746
• DOI: 10.1007/s11784-024-01108-9

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
→
M
. This task is related to determining a topological invariant
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
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