On the sequential topological complexity and the LS-category of the cofiber of higher diagonals for symmetric products of non-orientable surfaces

Abstract

For positive integers $k$, $n$, and $g$ with $k\ge 2$, we give a closed-form expression for the $k$-th ${\mathbb{Z}}_2$-zero-divisor cup length ${\mathsf{zcl}}_k(S{P}^n(N_g))$ of the $n$-th symmetric product $S{P}^n(N_g)$ 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 ${\mathsf{TC}}_k(S{P}^n(N_g))$, as well as the Lusternik--Schnirelmann category of the homotopy cofiber of the $k$-th diagonal map $S{P}^n(N_g)\to (S{P}^n(N_g){)}^k$. 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, ${\mathsf{TC}}_2(S{P}^n(N_g))$ behaves in a different way as all other invariants ${\mathsf{TC}}_k(S{P}^n(N_g))$ do. Likewise, as $k$ grows, we describe an eventual maximal-possible linear growth of ${\mathsf{zcl}}_k(S{P}^n(N_g))$, which allows us to prove the rationality conjecture of Farber and Oprea for the TC-generating function of $S{P}^n(N_g)$.

29 pages. Comments welcome!