Čech cohomology of infinite projective spaces, flag manifolds, and related spaces

TL;DR

The paper computes the Čech cohomology rings of the countable product of infinite complex projective spaces and of the infinite flag manifold, using an inverse-limit framework and a general colimit-compatibility result for cohomology of countable CW complexes. It develops a Künneth-type statement for Čech cohomology, along with tautness and continuity properties, to show that $H^*((\mathbb{C}P^\infty)^{\times \mathbb{N}}) \cong \varinjlim_n R[t_1,\dots,t_n] = R[t_1,t_2,\dots]$. Furthermore, it proves an analogous result for the infinite flag manifold by constructing a compatible system of homotopy equivalences between finite flag manifolds and products of projective spaces, establishing $Fl(\mathbb{C}^\infty) \simeq (\mathbb{C}P^\infty)^{\times \mathbb{N}}$ in a cohomological sense and hence $H^*Fl(\mathbb{C}^\infty) \cong R[t_1,t_2,\dots]$. The work provides a rigorous basis for understanding the cohomology of these infinite-dimensional spaces and related classifying-space phenomena.

Abstract

We compute the Čech cohomology ring of a countable product of infinite projective spaces, and that of an infinite flag manifold. The method of our first result in fact computes the cohomology ring of a countably infinite product of paracompact Hausdorff spaces, under some mild assumptions.

Theorems & Definitions (12)

• Theorem A
• Corollary
• Proposition 1
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof
• Theorem B