On the sequential topological complexity of group homomorphisms (Q6611775)

For a fibration \( p : E\rightarrow B\), the sectional category of \(p\), denoted by \(\mathrm{secat}(p)\), is the smallest integer \(k\) such that \(B\) can be covered by \(k + 1\) open sets \(U_0, \dots, U_k\) such that each admits a partial section \(s_i : U_i \rightarrow E\) of \(p\). In fact, this is the invariant ``genus'' defined by \textit{A. S. Shvarts} [Transl., Ser. 2, Am. Math. Soc. 55, 49--140 (1962; Zbl 0178.26202); translation from Tr. Moskov. Mat. Obshch. 10, 217--272 (1961), 11, 99--126 (1961)].\N\NIn [\textit{Y. B. Rudyak}, Topology Appl. 157, No. 5, 916--920 (2010; Zbl 1187.55001)], the sequential topological complexity of a path-connected space \(X\) is defined as \(TC_r(X) = \mathrm{secat}(\Delta_0)\), where \(r\geq 2\) and \(\Delta_0 : X^{J_r}\rightarrow X^r\) is some specific fibration.\N\NThe author considers an equivalent definition for the sequential topological complexity of a path connected space \(X\). This equivalent version of the definition is inspired by \textit{M. Farber}'s definition of topological complexity [Discrete Comput. Geom. 29, No. 2, 211--221 (2003; Zbl 1038.68130)]. This equivalent version of the definition uses sequential motion planners. For a path connected space \(X\), a sequential motion planner on a subset \(U\subset X^r\) is a map \(s : U\rightarrow PX\) such that \(s(x_0, x_1, \dots , x_{r-1})(\frac{j}{r-1}) = x_j\) for all \(j = 0,\dots r - 1\). The sequential topological complexity of \( X\), denoted by \(TC_r(X)\), is the minimal number \(k\) such that \(X_r\) is covered by \(k + 1\) open sets \(U_0,\dots , U_k\) on which there are sequential motion planners. If no such \(k\) exists, we set \(TC_r(X) = \infty\). Instead of a space one can also consider the sequential topological complexity of a map.\N\NThe sequential topological complexity of a map \(f : X\rightarrow Y\) is \(TC_r(f) := \mathrm{secat}((f^r)^*\Delta_0^Y)\) where \(f^r=f\times \dots \times f:X^r\rightarrow Y^r\) and \((f^r)^*\Delta_0^Y)\) is the pull-back fibration of the base point fibration \(\Delta_0: PY\rightarrow Y.\) The sequential topological complexity of a map \(f : X\rightarrow Y\) is a homotopy invariant.\N\NAs in the case of a space, the author gives a new definition of the sequential topological complexity of a map in terms of sequential motion planners. The author defines that for a map \( f : X\rightarrow Y\), a sequential \(f\)-motion planner on a subset \(U\subset X^r\) is a map \(f_U : U\rightarrow PY\) such that \[f_U(\overline{x})( \frac{j}{r-1}) =f_U(x_0, x_1, \dots , x_{r-1})( \frac{j}{r-1} ) = f(x_j)\] for all \(j = 0, \dots , r- 1\). The (pullback) sequential topological complexity of a map \(f\), denoted \(TC_r(f)\), is the minimal number \(k\) such that \(X^r\) is covered by \(k+1\) open sets \(U_0, \dots , U_k\) on which there are sequential \(f\)-motion planners. If no such \(k\) exists, we set \(TC_r(f) = \infty\).\N\NThe author proves that this definition is equivalent to the previous definition and this is a more general definition as for \(r=2\) this is \textit{J. Scott}'s topological complexity for a map [Topology Appl. 314, Article ID 108094, 25 p. (2022; Zbl 1494.55005)] and if \(f\) is the identity on \(X\), it coincides with Rudyak's sequential topological complexity of the space \(X\) [op. cit.].\N\NThe author gives upper-lower bounds for the sequential topological complexity of a map between path connected normal spaces, and for a map between \(CW\)-complexes, and proves some properties for this homotopy invariant.\N\NThe author considers topological complexity of group homomorphisms and gives a formula for the sequential topological complexity of a group homomorphism between finitely generated abelian groups. He also calculates this invariant for the epimorphisms between non-trivial free groups.\N\NThe sequential topological complexity of homomorphisms between discrete groups is known for a few classes of groups. The author gives upper bounds and lower bounds for the sequential topological complexity of homomorphisms between surface groups, finitely generated torsion free nilpotent groups and for a specific epimorphism between torsion free almost nilpotent groups.\N\NFinally, the author gives a characterization of the cohomological dimension of epimorphisms between fundamental groups of closed surfaces.\N\NThe calculations of the sequential topological complexity of homomorphisms between some specific groups is rather complicated. The author ends the paper with two nice questions for this invariant and cohomological dimension of group homomorphisms in the case of geometrically finite groups.