For positive integers $k$, $n$, and $g$ with $k\geq2$, we give a closed-form expression for the $k$-th $\mathbb{ Z}_2$-zero-divisor cup length $\mathsf{ zcl}_k(SP^n(N_g))$ of the $n$-th symmetric product $SP^n(N_g)$ of the closed non-orientable surface $N_g$ of genus $g$. This allows us to estimate, and in some cases, completely determine, the $k$-th sequential topological complexity $\mathsf{ TC}_k(SP^n(N_g))$, as well as the Lusternik--Schnirelmann category of the homotopy cofiber of the $k$-th diagonal map $SP^n(N_g) \to (SP^n(N_g))^k$. Our results recover previously known facts for even-dimensional real projective spaces ($g=1$) and closed non-orientable surfaces ($n=1$). In addition, we show that, as $g$ grows, $\mathsf{ TC}_2(SP^n(N_g))$ behaves in a different way as all other invariants $\mathsf{ TC}_k(SP^n(N_g))$ do. Likewise, as $k$ grows, we describe an eventual maximal-possible linear growth of $\mathsf{ zcl}_k(SP^n(N_g))$, which allows us to prove the rationality conjecture of Farber and Oprea for the TC-generating function of $SP^n(N_g)$.
Categories: math.AT