On the sequential topological complexity and the LS-category of the cofiber of higher diagonals for symmetric products of non-orientable surfaces
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910054048858112 |
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| author | González, Jesús Jauhari, Ekansh |
| author_facet | González, Jesús Jauhari, Ekansh |
| contents | 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 $SP^n(N_g)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_14788 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the sequential topological complexity and the LS-category of the cofiber of higher diagonals for symmetric products of non-orientable surfaces González, Jesús Jauhari, Ekansh Algebraic Topology 55S15, 55M30 (Primary) 57N65, 70B15 (Secondary) 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 $SP^n(N_g)$. |
| title | On the sequential topological complexity and the LS-category of the cofiber of higher diagonals for symmetric products of non-orientable surfaces |
| topic | Algebraic Topology 55S15, 55M30 (Primary) 57N65, 70B15 (Secondary) |
| url | https://arxiv.org/abs/2603.14788 |