A few examples of $p$-good and $p$-bad classifying spaces

TL;DR

This paper provides examples of classifying spaces with varying $p$-goodness and $p$-badness, exploring the influence of Sylow $p$-subgroups and homological structures on their homotopy types.

Contribution

It introduces new examples of classifying spaces with different $p$-goodness properties and analyzes how Sylow $p$-subgroups affect their homotopy types.

Findings

Examples of spaces with different $p$-good and $p$-bad properties

Conditions under which wedges of classifying spaces are good or bad

Relations between $R$-homological structures and homotopy types

Abstract

We give examples of spaces which are good and bad at different primes in the sense of Bousfield and Kan in any arbitrary combination and investigate which impact the existence of a Sylow $p$-subgroup has on the homotopy type on the classifying space and under which conditions the homotopy type of wedges of classifying spaces is good or bad for a solid ring $R$. We give results relating to various other $R$-homological structures and a collection of examples.