This paper is in the area of discrete mathematics. It describes a self map of a free group, referred to as a palindromization map (Pal). Various properties of Pal are proved: braid groups, profinite topology, and cohomological interpretations.

Section 0 briefly explains Pal with background literature. The free group, denoted by F₂, is generated by a two-letter alphabet and involves right iterated palindromic closures. Section 1 describes a class of automorphisms on F₂ and proves a lemma. Section 2 explains group homomorphisms involving braid group B₃ and automorphisms on F₂. Section 3 describes an equation of Pal and its interpretation in the language of Serre’s non-Abelian cohomology; the appendix presents Serre’s non-Abelian cohomology. Section 4 proves the main contributions of the paper: the connection of Pal with F₂ and the extension to right iterated palindromic closure. Section 5 presents the conditions for Pals to be equal. Section 7 proves a conjugation property involving image Pal(F₂). In addition, a theorem is presented that states that a subset of Pal(F₂) is closed in F₂ for the profinite topology. Background for the proof is developed in Section 5.

Understanding this paper requires a strong background and interest in discrete mathematical structures.