Hey everyone,
is there a good way to determine whether a map of (the topological realizations of) posets is a fibration without explicitely proving that it has the homotopy lifting property?
Robert
MathOverflow is a question and answer site for professional mathematicians. It only takes a minute to sign up.
Sign up to join this communityHey everyone,
is there a good way to determine whether a map of (the topological realizations of) posets is a fibration without explicitely proving that it has the homotopy lifting property?
Robert
What you're looking for might be Quillen's Theorem B. Roughly speaking, it says that if all the "combinatorial homotopy fibers" are homotopy equivalent (via base change along morphisms in the poset), then the combinatorial homotopy fibers are weakly equivalent to the honest homotopy fibers. It originally appeared in Higher Algebraic K-theory, I (Lecture Notes in Math 341).
I just though about this for another minute and it seems like I posted the first part of my question too early. If $f^{-1}(q)$ is contractible, then $f^{-1}(Q_{\leq q})$ is homotopy equivalent to a poset with maximal element, namely the contracted fiber over $q$. Thus, $f^{-1}(Q_{\leq q})$ is contractible.
However, I do not know an answer to the second problem yet.