One of my favorite examples illustrating the power of (co)homology is Sperner Lemma; see the very nice short paper of D.I.A. Cohen *On the Sperner Lemma*,  J. Combin. Theory, 2(1967), 585-587, or [these notes][1].


  [1]: http://www3.nd.edu/~lnicolae/Local-global.pdf