Homological stability

In mathematics, homological stability is any of a number of theorems asserting that the group homology of a series of groups ${\displaystyle G_{1}\subset G_{2}\subset \cdots }$ is stable, i.e.,

${\displaystyle H_{i}(G_{n})}$

is independent of n when n is large enough (depending on i). The smallest n such that the maps ${\displaystyle H_{i}(G_{n})\to H_{i}(G_{n+1})}$ is an isomorphism is referred to as the stable range. The concept of homological stability was pioneered by Daniel Quillen whose proof technique has been adapted in various situations.^[1]