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$\mathbb{Z}_p$-torus actions on positively curved manifolds

Muhammad Abdullah, Catherine Searle

TL;DR

This work addresses the classification problem for closed manifolds of positive sectional curvature admitting an effective, isometric $\mathbb{Z}_p^r$-action with a fixed point. The authors fuse geometric/topological methods (notably Wilking’s Connectedness Lemma, fixed-point theory, and periodicity results) with finite-length $q$-ary coding bounds (the Elias-Bassalygo bound) to derive explicit, dimension- and prime-dependent lower bounds on the rank $r$ that guarantee a homotopy-type classification. For primes in $\{3,5,7,11,13,17,19,23,29\}$ and sufficiently large dimension, they show $M^n$ is homotopy equivalent to one of the standard spaces: $S^n$, $\mathbb{R}P^n$, $\mathbb{C}P^{n/2}$, or a lens space, improving the known bound in the $p=3$ case to $r>9n/32$ for $n\ge 1908$ and sharpening bounds relative to prior $3n/8$ results. A finite-length $q$-ary Elias-Bassalygo bound is developed and deployed, with independent interest for translating fixed-point-geometry problems into rigid codimension constraints via isotropy representations.

Abstract

In this article, we consider closed, positively curved $n$-manifolds admitting an isometric and effective $\mathbb{Z}_p^r$-action with a fixed point, with $p$ a prime, classifying such manifolds up to homotopy equivalence when $r$ satisfies the lower bound given in our main theorem and $p \in \{3,5,7,11,13,17,19,23,29\}$. In particular, for $p=3$ and $n\geq 1908$, we show that $r>9n/32$. The lower bound we find for this range of primes represents an improvement over the approximately $3n/8$ bound found by Fang and Rong in even dimensions and by Ghazawneh in odd dimensions. To obtain this result, we derive a finite-length Elias-Bassalygo bound for $q$-ary codes that is of independent interest.

$\mathbb{Z}_p$-torus actions on positively curved manifolds

TL;DR

This work addresses the classification problem for closed manifolds of positive sectional curvature admitting an effective, isometric -action with a fixed point. The authors fuse geometric/topological methods (notably Wilking’s Connectedness Lemma, fixed-point theory, and periodicity results) with finite-length -ary coding bounds (the Elias-Bassalygo bound) to derive explicit, dimension- and prime-dependent lower bounds on the rank that guarantee a homotopy-type classification. For primes in and sufficiently large dimension, they show is homotopy equivalent to one of the standard spaces: , , , or a lens space, improving the known bound in the case to for and sharpening bounds relative to prior results. A finite-length -ary Elias-Bassalygo bound is developed and deployed, with independent interest for translating fixed-point-geometry problems into rigid codimension constraints via isotropy representations.

Abstract

In this article, we consider closed, positively curved -manifolds admitting an isometric and effective -action with a fixed point, with a prime, classifying such manifolds up to homotopy equivalence when satisfies the lower bound given in our main theorem and . In particular, for and , we show that . The lower bound we find for this range of primes represents an improvement over the approximately bound found by Fang and Rong in even dimensions and by Ghazawneh in odd dimensions. To obtain this result, we derive a finite-length Elias-Bassalygo bound for -ary codes that is of independent interest.

Paper Structure

This paper contains 8 sections, 16 theorems, 75 equations.

Table of Contents

  1. Introduction
  2. Organization
  3. Acknowledgments
  4. Error Correcting Codes
  5. The Finite-Length Elias-Bassalygo Bound.
  6. The Topology and Geometry of Positive Curvature
  7. Bounds for --tori in Positive Curvature
  8. The Proof of Theorem A

Key Result

Theorem A

Assume $\mathbb{Z}_p^r$ acts effectively and isometrically with a fixed point on a closed, positively curved manifold $M^n$, where $p$ is a prime, $3\leq p\leq 29$, and $n \geq 16$. Suppose where $f_i(p)$, given in Display constants_f, are functions depending on $\log_p$ terms and the entropy and Johnson functions, given in Definition entropy. Then $M$ is homotopy equivalent to $S^n$, $\mathbb{R}

Theorems & Definitions (32)

  • Theorem A
  • Definition 2.1
  • Definition 2.2: $\bf{q}$-ary entropy function and the Johnson radius
  • Lemma 2.3
  • proof
  • Theorem 2.4: Finite-Length Elias-Bassalygo Bound
  • proof
  • Corollary 2.5: Continuous Form of Theorem \ref{['EB(p)']}
  • proof
  • Corollary 2.6
  • ...and 22 more