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

Recall that to 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).

If $P, Q$ are the posets of faces of simplicial sets, then their nerves coincide  with the simplicial sets they were obtained from. A simplicial map $f$ between the simplicial sets induces a monotone map between the corresponding posets of faces, and Quillen's theorem  can be rephrased as saying that if the preimage of any face is contractible, then $f$  is  a homotopy equivalence.

For more details see this [nice survey of Anders Bjorner.][1]


  [1]: https://people.kth.se/~bjorner/files/TopMeth.pdf