Abstract
This chapter opens with homology and cohomology theories which play a key role in algebraic topology. Homology and cohomology groups are also topological invariants like homotopy groups and Euler characteristic. Homology (cohomology) theory is a sequence of covariant (contravariant) functors from the category of chain (cochain) complexes to the category of abelian groups (modules). A key feature of these functors is their homotopy invariance in the sense that homotopic maps induce the same homomorphism in homology (cohomology). In particular, topological spaces of the same homotopy type have isomorphic homology (cohomolgy) groups.
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