Coloring maps of period three. (Q700645)

A color of a map \(f:X \to X\) is an open subset \(B\) of \(X\) such that \(f(B) \cap B=\emptyset\). A coloring of \((X,f)\) is a finite cover of \(X\) consisting of colors. The color number, col\((X,f)\), is the minimal cardinality of a coloring of \((X,f)\). The color and the coloring number can also be defined by using closed sets. The authors study fixed point free maps \(\sigma :X \to X\) of order 3. A subset \(B\) of \(X\) is said to be of first type if there is a color \(C\) of \((X,f)\) such that \(B=C \cup \sigma(C) \cup \sigma^2(C)\). The genus, gen\((X,\sigma)\), is the least number \(k\) such that \(X\) can be written as a union of \(k\) sets of first type. The authors carry out a detailed study of a particular example of a space with a free action of order 3: it is the ``\((n-1)\)-dimensional \(Y\)-sphere'', \(S^{n-1}_Y\), the combinatorial boundary of the \(n\)-fold product of a standard tripod \(Y\) in the plane. They prove a number of results relating the color number, the genus and the existence of equivariant maps \(X \to S^{n-1}_Y\). In particular, they show that col\((S^n_Y)=n+3\) and gen\((S^n_Y)=n+1\). Some of these results are analogous to the Ljusternik-Schnirelman theorems for involutions. Periodic actions of prime periods greater than 3 are also discussed. Reviewer's remark: The paper is written in a careful and elegant style as are other papers of the authors. The \(Y\)-sphere is a nice example of a space with a free action of order three -- but there are certainly many other examples too. The \(Y\)-sphere, or its analogues for other periods, are subsets of \((D^2)^n \cong S^{2n-1}\), where \(D^2\) is a 2-disk. That space admits a free circle action, and thus also free actions of all periods. There is a little misprint in formula (2) on page 264: \(\gamma2\) should read \(\gamma^2\).