In the paper *Homotopy properties of the poset of nontrivial $p$-subgroups of a group,  Adv. in Math., 28(1978)*, D.Quillen  proves a versiona   version of Vietoris-Begle theorem.

Recall that two a poset  $P$ we can associate a simplicial complex, its  *nerve* $|P|$. Any simplicial complex is, canonically, the  nerve of a poset, the poset of faces.

If $P, Q$ are posets, and $f: P\to Q$ is a monotone nondecreasing map, then its fibers are defined to be the subsets 

$$ f^{-1}(\leq y):=\bigl\{p \in P;\;\;f(p)\leq y\;\bigr\},\;\;y\in Q. $$

In the paper mentioned above Quillen proves    that if $f: P\to Q$ is monotone nondecreasing and the fibers are homotopically trivial (resp. acyclic) then $f$ induces a homotopy equivalence between the nerves (resp. an isomorphism between  the homology of the nerves).