Concerning the topological interpretation of Mobius numbers

Today I asked in class whether one could extract more (topological) information from \mu(P). I had the feeling that there had to be a connection between this and the Betti numbers of the order complex (those who have dealt with Euler characteristics before might have thought of this already):

I have not thought about it for too long, but I am wondering about what the topology could tell us about the combinatorics of the poset . . .


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