# Varying definitions of cohomology

So I know that given a chain complex we can define the $d$-th cohomology by taking $\ker{d}/\mathrm{im}_{d+1}$. But I don't know how this corresponds to the idea of holes in topological spaces (maybe this is homology, I'm a tad confused).

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One can compute (co)homology of different complexes. In particular, for any topological space one can define it's singular complex (see Eric's answer for an idea how it's done) which in some sense indeed counts holes. But the idea of (co)homology is more general. –  Grigory M Jul 24 '10 at 15:28
I couldn't really do better than Eric's answer, and like Grigory says, cohomology is more general. So instead I want to mention a case where cohomology doesn't do this: Sheaf Cohomology of Algebraic Groups, classify dominant Vector Bundles. This is called the Borel-Weil-Bott Theorem and has some nice ramifications for Representation Theory and Algebraic Geometry. –  BBischof Jul 24 '10 at 19:48