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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).

For more details see this nice survey of Anders Bjorner.

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.

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 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).

For more details see this nice survey of Anders Bjorner.

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).

For more details see this nice survey of Anders Bjorner.

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).

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 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).

For more details see this nice survey of Anders Bjorner.