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Homology groups measure the 'holes' in a space, which correspond to the number of $n$-dimensional cycles that do not bound an $(n+1)$-dimensional chain.

The Mayer-Vietoris sequence relates the homology groups of two spaces to the homology of their union and intersection, facilitating computation of complex spaces.

Derived functors extend the notion of a functor to chain complexes, enabling the translation of homological properties from one category to another.

Cohomology groups classify cohomology classes of cocycles and coboundaries, and are used to study the algebraic invariants of a topological space.

The tensor product of two chain complexes is defined pointwise and used to investigate the combined homological properties of both spaces.

The Universal Coefficient Theorem relates the homology of a space to its cohomology, providing a bridge between these two dual theories.

Chain maps are functions between chain complexes that preserve the structure of the complexes, and are used to compare different topological spaces homologically.

Spectral sequences provide a computational tool that condenses complex homological information into more manageable forms, often used in filtered topological spaces.

A chain complex is a sequence of abelian groups connected by boundary operators, with applications in calculating the homology groups of a space.