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Bounds on Systoles and Homotopy Complexity

Tsachik Gelander, Paul Vollrath

Abstract

We give an elementary proof of a weak variant of the homotopy type conjecture from [Gel04]. The proof relies on Belolipetskys estimate on the systole in terms of the volume which is a consequence of Prasads volume formula. In particular we extend the estimate established in [B20] for hyperbolic manifolds to general arithmetic manifolds.

Bounds on Systoles and Homotopy Complexity

Abstract

We give an elementary proof of a weak variant of the homotopy type conjecture from [Gel04]. The proof relies on Belolipetskys estimate on the systole in terms of the volume which is a consequence of Prasads volume formula. In particular we extend the estimate established in [B20] for hyperbolic manifolds to general arithmetic manifolds.

Paper Structure

This paper contains 5 sections, 12 theorems, 16 equations.

Table of Contents

  1. Lower bound on the systole
  2. Embedding of $G/K$
  3. Degree of Definition and Dobrowolski's bound
  4. The homotopy type conjecture
  5. Bounds on torsion

Key Result

Proposition 1.1

For any $g\in G$ the following holds:

Theorems & Definitions (17)

  • Proposition 1.1
  • proof
  • Corollary 1.2
  • Theorem 1.3
  • Proposition 1.4
  • Theorem 1.5
  • proof
  • Definition 2.1
  • Definition 2.2
  • Conjecture 2.3
  • ...and 7 more