On the Euler-Poincar\'{e} characteristics of a simply connected rationally elliptic CW-complex

TL;DR

This paper introduces new numerical invariants for simply connected rationally elliptic CW-complexes, linking their cohomology and homotopy Euler-Poincaré characteristics through Whitehead exact sequences in algebraic models.

Contribution

It defines the invariants _X and _X using Whitehead sequences in Quillen and Sullivan models, establishing a novel relationship with Euler-Poincare9 characteristics.

Findings

Introduction of _X and _X invariants

Relation between these invariants and Euler-Poincare9 characteristics

New insights into the structure of rationally elliptic CW-complexes

Abstract

For a simply connected rationally elliptic CW-complex $X$, we show that the cohomology and the homotopy Euler-Poincar\'e characteristics are related to two new numerical invariants namely $\eta_{X}$ and $\rho_{X}$ which we define using the Whitehead exact sequences of the Quillen and the Sullivan models of $X$.