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

Jesús González, Ekansh Jauhari

Abstract

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 $SP^n(N_g)$.

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 , , and with , we give a closed-form expression for the -th -zero-divisor cup length of the -th symmetric product of the closed non-orientable surface of genus . This allows us to estimate, and in some cases, completely determine, the -th sequential topological complexity , as well as the Lusternik--Schnirelmann category of the homotopy cofiber of the -th diagonal map . Our results recover previously known facts for even-dimensional real projective spaces () and closed non-orientable surfaces (). In addition, we show that, as grows, behaves in a different way as all other invariants do. Likewise, as grows, we describe an eventual maximal-possible linear growth of , which allows us to prove the rationality conjecture of Farber and Oprea for the TC-generating function of .
Paper Structure (17 sections, 23 theorems, 97 equations)

This paper contains 17 sections, 23 theorems, 97 equations.

Table of Contents

  1. Introduction and main results
  2. Preliminaries and related work
  3. Symmetric products of closed non-orientable surfaces
  4. Zero-divisor cup length and sequential topological complexity
  5. LS-category of the homotopy cofiber of the diagonal map
  6. Comparison from computations for $SP^n(M_g)$ and $P^{2n}$
  7. Consequences and observations
  8. Topological complexity and the rationality conjecture
  9. Cohomological aspects
  10. The cohomology of $SP^n(N_g)$
  11. The capacity function and the cohomology of products of $SP^n(N_g)$
  12. $k$-th zero-divisors
  13. The case of $k=2$
  14. The case of odd $k\ge 3$
  15. General preparation
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $2^e\le n< 2^{e+1}$ for some $e\ge 0$. Then for $g\ge 1$, we have that

Theorems & Definitions (58)

  • Theorem 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Theorem 2.1
  • Theorem 2.2
  • proof
  • Example 3.1
  • ...and 48 more