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Real Bott manifold structure of $n$-dimensional Klein bottle and its rational Betti numbers

Navnath Daundkar, Priyavrat Deshpande

TL;DR

This work studies an $n$-dimensional analogue of the Klein bottle realized as a moduli space of planar polygons and establishes its structure as a real Bott manifold, including an explicit Bott matrix and a small-cover interpretation. Using toric-topology tools such as small covers and the Suciu–Trevisan formula, it derives the rational Betti numbers for the $K_n$ and extends the analysis to two additional polygon-space families arising from two other long genetic codes, providing explicit Betti-number formulas. The paper also demonstrates orientability criteria and non-symplecticity for these spaces, and gives cohomology descriptions via matroidal data. Overall, it integrates real Bott manifold theory with polygon spaces to yield exact topological invariants and a toric-topology perspective on these moduli spaces, with potential implications for understanding their geometric and combinatorial structure.

Abstract

Donald Davis initiated the study of an $n$-dimensional analogue of the Klein bottle. This generalized Klein bottle occurs as a moduli space of planar polygons for a certain choice of side lengths. In this paper, we show that the $n$-dimensional Klein bottle is a real Bott manifold and determine the corresponding Bott matrix. We determine the small cover structure on two other classes of moduli spaces of planar polygons. As an application, we compute the rational Betti numbers of these spaces using a formula, due to Suciu and Trevisan.

Real Bott manifold structure of $n$-dimensional Klein bottle and its rational Betti numbers

TL;DR

This work studies an -dimensional analogue of the Klein bottle realized as a moduli space of planar polygons and establishes its structure as a real Bott manifold, including an explicit Bott matrix and a small-cover interpretation. Using toric-topology tools such as small covers and the Suciu–Trevisan formula, it derives the rational Betti numbers for the and extends the analysis to two additional polygon-space families arising from two other long genetic codes, providing explicit Betti-number formulas. The paper also demonstrates orientability criteria and non-symplecticity for these spaces, and gives cohomology descriptions via matroidal data. Overall, it integrates real Bott manifold theory with polygon spaces to yield exact topological invariants and a toric-topology perspective on these moduli spaces, with potential implications for understanding their geometric and combinatorial structure.

Abstract

Donald Davis initiated the study of an -dimensional analogue of the Klein bottle. This generalized Klein bottle occurs as a moduli space of planar polygons for a certain choice of side lengths. In this paper, we show that the -dimensional Klein bottle is a real Bott manifold and determine the corresponding Bott matrix. We determine the small cover structure on two other classes of moduli spaces of planar polygons. As an application, we compute the rational Betti numbers of these spaces using a formula, due to Suciu and Trevisan.
Paper Structure (6 sections, 23 theorems, 54 equations)

This paper contains 6 sections, 23 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Realizing Lg as the real Bott manifold and its rational Betti numbers
  3. The case of other two long genetic codes
  4. The small cover Lg
  5. Betti numbers of Lg
  6. Declarations

Key Result

Theorem 1.3

Let $\alpha$ be a length vector with the genetic code $\langle\{1,2,\dots,n-1,n+3\}\rangle$. Then $\overline{\mathrm{M}}_{\alpha}\cong K_n.$

Theorems & Definitions (54)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5
  • proof
  • ...and 44 more