Background & definitions: I am studying algebraic topology, in particular (co)homology groups and Mayer-Vietoris sequences.
Important terms:
Algebraic invariant = property of a topo. space preserved under algebraic transformations (e.g. transformation given by a general linear group).
Mayer-Vietoris-sequence in this context, I work with the sequence for reduced homology groups (which is a bit nicer). 
From the intro of Wikipedia article:
In mathematics, particularly algebraic topology and homology theory, the Mayer–Vietoris sequence is an algebraic tool to help compute algebraic invariants of topological spaces, known as their homology and cohomology groups.
My question: How are (co)homology groups algebraic invariants? Why not topological invariants instead? I want to know what properties of a topo. space X (co)homology groups preserve.
I am gratfeul for any insgiths, also possibly from category-theoretic view ((co)homology as functors etc.)