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Is there a homology theory of posets which computes topological invariants (e.g., number of $k$-faces, etc.) of the associated Hasse diagrams (viewed as simplicial/cellular/singular complexes) as perhaps ranks of homology groups or values of Hilbert polynomials of nice rings (e.g., Stanley-Risner, Cohen-Macaulay, etc.)?

For example, the Hasse diagram of the Boolean lattice $B_{n}$ has $2^{n-k} \binom{n}{k}$ $k$-faces and therefore euler characteristic $\sum_k (-1)^k f_k = 1$. Is there a homology theory which relates the Boolean lattice to a contractible space? (Note the solid $n$-cube is homotopy equivalent to the $n$-ball.)

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Given a poset, $P$, one can form an associated simplicial complex called the order complex $S(P)$, by letting the underlying set of $P$ be the vertices, and the faces are the totally ordered subsets. Applying standard algebraic topology invariants, such as homology, to the complex $S(P)$ we can obtain invariants of $P$ in some sense.

If we let $P = 2^X$ for some set $X$ with the inclusion partial order, we see that there is a maximal element $X \in P$, and thus the geometric realization of $S(P)$ contracts to the vertex $X$.

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    What does the order complex say about the Hasse Diagram viewed as a polyhedral complex? If one is contractible, does it imply it is so for the other?2012-07-11
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You may look at [Poset Topology] and the related lecture notes by Michelle L. Wachs.