Homotopy classes that are trivial mod \({\mathcal F}\) (Q5938059)

Let \(f: X\to Y\) be a map. Then by composition \(f\) induces the function \(f_*:[A, X]\to [A,Y]\) between the homotopy classes where \(A\) is any space. If \(A\) runs through the collection of spheres, then \(f_*\) is the induced homomorphism between the homotopy groups. In general, let \(F\) be a collection of topological spaces, then the homotopy class \([f]\in [X,Y]\) is called \(F\)-trivial if \(f_*= 0:[A, X]\to [A,Y]\) for all \(A\in F\). Denote by \(Z_F(X, Y)\subseteq [X,Y]\) the subset of all \(F\)-trivial homotopy classes in \([X; Y]\). If \(F= P\) the collection of all finite-dimensional complexes, then \(Z_P(X, Y)\) is exactly the collection of maps known under the name phantom maps. So one may think of \(Z_F(X, Y)\) as a generalization of phantom maps.NEWLINENEWLINENEWLINEThe authors first study general properties of \(Z_F(X, Y)\) sets. In particular, for collections: \(S=\text{the}\) collection of spheres; \(M=\text{the}\) the collection of Moore spaces; \(\Sigma=\) the collection of suspensions the following relation holds: \(Z_\Sigma(X, Y)\subseteq Z_M(X, Y)\subseteq Z_S(X, Y)\), and they show that there are finite complexes \(X\) and \(Y\) for which these inclusions are strict. If \(X= Y\), then the composition of \(F\)-trivial homotopy classes imposes a semigroup with zero structure on \(Z_F(X)= Z_F(X,X)\). It turns out that for a finite-dimensional complex \(X\), the semigroups \(Z_S(X)\), \(Z_M(X)\) and \(Z_\Sigma(X)\) are nilpotent. The authors then introduce and study the nilpotency of \(X\text{ mod }F\), denoted \(t_F(X)\) and defined as follows: If \(X\) is contractible, then \(t_F(X)= 0\), otherwise \(t_F(X)\) is the nilpotency of the semigroup \(Z_F(X)\). The nilpotency \(\text{mod }F\) is a new numerical invariant of homotopy type. Since \(S\subset M\subset\Sigma\) it holds \(0\leq t_\Sigma(X)\leq t_M(X)\leq t_S(X)\leq\infty\) for any space \(X\). But if \(X\) is of finite category, then \(t_\Sigma(X)\) is bounded by the least integer greater than or equal to \(\log_2(\text{cat}(X))\). For an arbitrary collection \(F\) the nilpotency \(\text{mod }F\) is related to another new numerical invariant \(\text{kl}_F(X)\), called \(F\)-killing length of \(X\) and the following relation holds: \(t_{F(X)}\leq \text{kl}_F(X)\). \(\text{kl}_F(X)\) is the least number of steps needed to go from \(X\) to a contractible space by successively attaching cones on wedges of spaces in \(F\). If \(F\) is closed under suspension, then \(\text{kl}_F(X)\leq \text{cl}_F(X)\) where \(\text{cl}_F(X)\) is called the \(F\)-cone length of \(X\) which is the least number of steps to go from a contractible space to \(X\). A sequence of calculations of \(Z_F(X)\) and \(t_F(X)\) is performed in the second part of the paper when \(F= S\), \(M\) or \(\Sigma\) and \(X\) is a projective space (real, complex or quaternionic). Here is a sample result for complex projective space:NEWLINENEWLINENEWLINE(a) \(Z_F(CP^n)= 0\) for each \(n\geq 1\) and each \(F= \Sigma\), \(M\) or \(S\);NEWLINENEWLINENEWLINE(b) \(Z_F(\Sigma^n CP^2)= 0\) for each \(n\geq 1\) and each \(F= \Sigma\), \(M\) or \(S\).NEWLINENEWLINENEWLINESome further calculations are given for \(\text{SU}(n)\) and \(\text{Sp}(n)\). A sample result:NEWLINENEWLINENEWLINE(a) For \(n\geq 5\), \(Z_\Sigma\neq 0\), and \(2\leq t_\Sigma(\text{SU}(n))\leq\) the least integer greater than or equal to \(\log_2(n)\);NEWLINENEWLINENEWLINE(b) For \(n\geq 2\), \(Z_\Sigma(\text{Sp}(n))\neq 0\) and \(2\leq t_\Sigma(\text{Sp}(n))\leq\) the least integer greater than or equal to \(2\log_2(n+ 1)\).NEWLINENEWLINENEWLINEThe paper concludes with a set of open problems.