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On the cohomology of elliptic coformal spaces

Youssef Rami

TL;DR

The paper resolves cohomological properties of rationally elliptic coformal spaces by establishing an algebraic framework that proves Lupton's conjecture for this broad class and deduces Hilali's conjecture. It leverages Sullivan minimal models with quadratic differentials, the connection between spherical cohomology and the $V\cap\ker d$ content, and the rational Toomer invariant $e_0$ to bound cohomology in graded components. The central technical tool is a bigraded Gysin exact sequence obtained by splitting off a generator, enabling an inductive analysis that shows $\dim H^*_k(\Lambda V,d)\ge 2$ for all $1\le k\le e_0-1$ and vanishing beyond $e_0$, with a sharp exception describing truncated polynomial algebras on two generators. Consequently, Lupton's conjecture holds for all coformal elliptic Sullivan algebras and Hilali's conjecture holds for all elliptic coformal spaces, significantly extending known cases and clarifying the role of spherical classes.

Abstract

In this article, we pursue the study begun in \cite{Lup02} on the cohomology of rationally elliptic coformal spaces. Consequently, we complete, for such spaces, the proof of Lupton's conjecture and deduce Hilali's.

On the cohomology of elliptic coformal spaces

TL;DR

The paper resolves cohomological properties of rationally elliptic coformal spaces by establishing an algebraic framework that proves Lupton's conjecture for this broad class and deduces Hilali's conjecture. It leverages Sullivan minimal models with quadratic differentials, the connection between spherical cohomology and the content, and the rational Toomer invariant to bound cohomology in graded components. The central technical tool is a bigraded Gysin exact sequence obtained by splitting off a generator, enabling an inductive analysis that shows for all and vanishing beyond , with a sharp exception describing truncated polynomial algebras on two generators. Consequently, Lupton's conjecture holds for all coformal elliptic Sullivan algebras and Hilali's conjecture holds for all elliptic coformal spaces, significantly extending known cases and clarifying the role of spherical classes.

Abstract

In this article, we pursue the study begun in \cite{Lup02} on the cohomology of rationally elliptic coformal spaces. Consequently, we complete, for such spaces, the proof of Lupton's conjecture and deduce Hilali's.

Paper Structure

This paper contains 13 sections, 10 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Main tools
  3. Sullivan minimal models
  4. Sphyrical cohomology classes
  5. Rational Toomer invariant
  6. Algebraic statements of our results
  7. Proof of Theorem \ref{['alg-ver']}
  8. The bigraded Gysing exact sequence and preparatory results
  9. Proof of Theorem \ref{['alg-ver']}
  10. First step: $\dim H^*_1(\Lambda V, d) \geq 2$
  11. Second step: $\dim H^*_2(\Lambda V,d)\geq 2$
  12. Suppose $m_1>2r$:
  13. General step: $\dim H_k^*(\Lambda V,d)\geq 2$

Key Result

Theorem 1.1

Let $X$ be a coformal elliptic space whose graded algebra $H^*(X, \mathbb{Q})$ is generated by at least two generators, and all its spherical classes have even degrees. Then,

Theorems & Definitions (14)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 3.1
  • Conjecture 3.2
  • Theorem 3.3
  • Corollary 3.4
  • Corollary 3.5
  • Lemma 4.2
  • proof
  • ...and 4 more