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Higher topological complexity of Seifert fibered manifolds

Navnath Daundkar, Rekha Santhanam, Soumyadip Thandar

TL;DR

The paper investigates the higher topological complexity $TC_r$ of orientable, aspherical Seifert fibered manifolds (those that are $K(G,1)$ with infinite $\pi_1$) and refines lower bounds through the introduction of higher TC$_r$-weights. By developing a framework based on stable cohomology operations and zero-divisor technology, it derives sharp cohomological bounds and applies them to Seifert manifolds, obtaining $TC_r$ values in the range ${3r-1,3r,3r+1}$ and identifying conditions under which exact values such as $TC_r(M_O)=3r$ or $TC_r(M_O)=3r$ arise. The work also establishes a precise wedge result, showing $TC_r$ of the wedge of finitely many closed, orientable, aspherical $3$-manifolds is $3r+1$, and provides bounds for connected sums that illuminate the interplay between topological complexity, group cohomology, and 3-manifold topology. Overall, the article advances sharp, class-specific computations of higher TC by blending cohomological weights, mod-$p$ cohomology, and group-theoretic methods, with broader implications for topological robotics in three-manifold settings.

Abstract

In this article, we investigate the higher topological complexity of oriented Seifert fibered manifolds that are Eilenberg--MacLane spaces $K(G,1)$ with infinite fundamental group $G$. We first refine the cohomological lower bounds for higher topological complexity by introducing the notion of higher topological complexity weights. As an application, we show that the $r^{\text{th}}$ topological complexity of these manifolds lies in $\{3r-1, 3r, 3r+1\}$, and characterize large families where the value is $3r$ or $3r+1$. Additionally, we establish a sufficient condition for higher topological complexity to be exactly $3r$ when the base surface is orientable and aspherical. Finally, we show that the higher topological complexity of the wedge of finitely many closed, orientable, aspherical $3$-manifolds is exactly $3r+1$.

Higher topological complexity of Seifert fibered manifolds

TL;DR

The paper investigates the higher topological complexity of orientable, aspherical Seifert fibered manifolds (those that are with infinite ) and refines lower bounds through the introduction of higher TC-weights. By developing a framework based on stable cohomology operations and zero-divisor technology, it derives sharp cohomological bounds and applies them to Seifert manifolds, obtaining values in the range and identifying conditions under which exact values such as or arise. The work also establishes a precise wedge result, showing of the wedge of finitely many closed, orientable, aspherical -manifolds is , and provides bounds for connected sums that illuminate the interplay between topological complexity, group cohomology, and 3-manifold topology. Overall, the article advances sharp, class-specific computations of higher TC by blending cohomological weights, mod- cohomology, and group-theoretic methods, with broader implications for topological robotics in three-manifold settings.

Abstract

In this article, we investigate the higher topological complexity of oriented Seifert fibered manifolds that are Eilenberg--MacLane spaces with infinite fundamental group . We first refine the cohomological lower bounds for higher topological complexity by introducing the notion of higher topological complexity weights. As an application, we show that the topological complexity of these manifolds lies in , and characterize large families where the value is or . Additionally, we establish a sufficient condition for higher topological complexity to be exactly when the base surface is orientable and aspherical. Finally, we show that the higher topological complexity of the wedge of finitely many closed, orientable, aspherical -manifolds is exactly .
Paper Structure (7 sections, 24 theorems, 54 equations)

This paper contains 7 sections, 24 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Weights of cohomology classes
  4. Cohomology of Seifert fibered manifolds
  5. Higher cohomology weights
  6. Higher topological complexity of Seifert fibered manifolds
  7. Wedge and connected sum of some 3-manifolds and concluding remarks

Key Result

Proposition 2.2

Let $p:E\to B$ be a fibration and $u_i \in H^{d_i}(B; G_i)$ be cohomology classes, $i = 1,\dots,l$, such that their cup-product $\prod_{i=1}^l u_i\in H^d(B; \otimes_{i=1}^l G_i)$ is non-zero, where $d = \sum _{i=1}^ld_i$. Then

Theorems & Definitions (46)

  • Definition 2.1
  • Proposition 2.2: GrantSymm
  • Lemma 2.3: GrantCweights
  • Definition 2.4
  • Theorem 2.5: GrantCweights
  • Proposition 2.6: CoringSiefert
  • Remark 2.7
  • Proposition 2.8: CoringSiefert
  • Proposition 2.9: CoringSiefert
  • Definition 3.1
  • ...and 36 more