The plus construction with respect to subrings of the rationals

TL;DR

This paper constructs explicit models for plus constructions with respect to subrings of the rationals, exploring their properties and relations to nullification functors and classical plus constructions.

Contribution

It provides explicit models of universal acyclic spaces for subrings of the rationals and analyzes their properties and relations to classical plus constructions.

Findings

Acyclization functor and cellularization functor coincide.

The acyclization-plus construction fiber sequence is a cofiber sequence for simply connected spaces.

The fiber sequence property generally fails when the space is not simply connected.

Abstract

We construct explicit models of universal $H \mathbb{Z}[J^{-1}]$-acyclic spaces $\mathcal M$, for any subset $J$ of the prime numbers. The corresponding nullification functors provide thus plus construction functors for ordinary homology with $\mathbb{Z}[J^{-1}]$ coefficients. Motivated by classical results about Quillen's plus construction for integral homology, we prove that the $H \mathbb{Z}[J^{-1}]$-acyclization functor and the $\mathcal M$-cellularization functor coincide. We show that the acyclization-plus construction fiber sequence is always a cofiber sequence for simply connected spaces, but almost never so when the plus construction is not simply connected, unlike in the classical case.