Allan Wechsler wrote: ----- I have been trying to wrap my head around this corner of mathematics for my entire adult life, so I will be re-reading this post carefully in hopes of getting some insight. Unless I missed something, you don't actually define the homology and cohomology groups, do you? You are more saying, "These things exist, but I don't want to get into the details," right? (I think I could reconstruct the definition of the j-th homology group of a simplicial complex, but I'm quite sure I never knew the corresponding definition of the cohomology groups.) ----- I didn't want to get into the details *yet* until later. First I wanted to give a general overview. ----- If I remember correctly, two simplicial complexes that are isomorphic as topological spaces will always have identical homology and cohomology, so for a space X that can be represented as a simplicial complex, it makes sense to talk about the homology and cohomology of the space. But the reverse is not true -- two non-isomorphic spaces might have identical homology or cohomology. So the theory sometimes lets us prove that two spaces are different, but it can only suggest when they might be the same. ----- Yes, topologically equivalent spaces have isomorphic homology, and isomorphic cohomology rings, but having isomorphic homology and/or cohomology doesn't necessarily imply topological equivalence. It's easy to come up with examples of that, because two topological spaces that are merely "homotopy equivalent" always have the same homology and cohomology. Spaces X and Y are called homotopy equivalent when there are maps f : X —> Y and g : Y —> X such that the compositions g o f : X —> X and f o g : Y —> Y are each homotopic to the respective identity maps 1_X and 1_Y. For nice spaces like simplicial complexes, this just amounts to a series of continuous deformations willy nilly: X —> X_1 <— X_2 —> X_3 <— X_4 ... Y that ultimately turn X into Y. (For instance, a disk with a smaller disk removed from its interior is homotopy equivalent to a circle with three hairs sticking out from it.) ----- When you speak of complex surfaces, you mean specifically 2-dimensional spaces, right? Because I recall that the taxonomy for higher-dimensional surfaces is more ... fraught. ----- I don't believe I used the term "complex surfaces". But yes, the higher-dimensional generalization of a surface is an n-manifold: Where a small piece of a surface look like a small piece of the plane, a small piece of an n-manifold looks like a small piece of R^n. So a surface is a 2-dimensional manifold, or 2-manifold for short. —Dan