$C_{pq}$-Injective Diagrams and a Combination Theorem for Minimal Models

TL;DR

This paper investigates injective diagrams of DGAs in equivariant rational homotopy theory for cyclic groups of order pq, establishing a combination theorem and constructing examples of formal spaces.

Contribution

It introduces a combination theorem for minimal models of injective diagrams over orbit categories of cyclic groups of order pq, advancing equivariant rational homotopy theory.

Findings

Necessary conditions for injectivity of diagrams over C_{pq}

A combination theorem for wedge diagrams over C_p and C_q

Construction of examples of C_{pq}-formal spaces

Abstract

We study diagrams of commutative differential graded algebras (DGAs) over the orbit category $\sO_G$ in the context of equivariant rational homotopy theory. For $G = C_{pq}$ with $p, q$ distinct primes, we give necessary conditions for injectivity. We prove a combination-type result: the equivariant wedge of injective diagrams over $\mathcal{O}_{C_p}$ and $\mathcal{O}_{C_q}$ with retract structure maps yields an injective diagram over $\mathcal{O}_{C_{pq}}$ with a level-wise minimal model. As an application, we construct examples of $C_{pq}$-formal spaces.