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).