Mapping class groups of simply connected high-dimensional manifolds need not be arithmetic

TL;DR

This paper clarifies Sullivan's result on the arithmetic nature of mapping class groups of simply connected high-dimensional manifolds, showing that under standard definitions, these groups are not always residually finite, thus refining the understanding of their structure.

Contribution

It demonstrates that Sullivan's statement about the arithmeticity of these groups does not imply residual finiteness, providing a nuanced interpretation of the original theorem.

Findings

Mapping class group of a certain manifold is not residually finite

Sullivan's result is clarified with a new perspective

Standard definitions affect the interpretation of arithmeticity

Abstract

It is well known that Sullivan showed that the mapping class group of a simply connected high-dimensional manifold is commensurable with an arithmetic group, but the meaning of "commensurable" in this statement seems to be less well known. We explain why this result fails with the now standard definition of commensurability by exhibiting a manifold whose mapping class group is not residually finite. We do not suggest any problem with Sullivan's result: rather we provide a gloss for it.