On the rationality conjecture of some finite CW-complexes

TL;DR

This paper proves the rationality conjecture for certain highly connected finite CW-complexes with specific cohomology properties, providing a minimal cell structure and advancing understanding in algebraic topology.

Contribution

It establishes the rationality conjecture for a class of CW-complexes with unique spherical cohomology classes and describes their minimal cell structures.

Findings

Proved the rationality conjecture for ($r-1$)-connected $kr$-dimensional CW-complexes.

Characterized the minimal cell structure of these complexes.

Demonstrated the cohomology as a truncated polynomial algebra.

Abstract

In this paper, we establish the rationality conjecture raised in \cite{FKS} for any $(r-1)$-connected ($r\geq 2$) $kr$-dimensional CW-complex $X$ ($k\geq 2$) having a unique spherical cohomology class $u\in \tilde{H}^r(X, \mathbb{Z})$ such that $u^k\not =0$. %which is nilpotent with order of nilpotency equal to $k+1$. Next, we illustrate (topologically) our result by giving the minimal cell structure of such a CW-complex whose cohomology is a truncated polynomial algebra.