this chapter introduces the concept of homology theory, which is of fundamental importance in algebraic topology. A homology theory involves a sequence of covariant functors H n to the category of abelian groups, and we shall define homology theories on two categories-the singular homology theory on the category of topological pairs and the simplicial homology theory on the category of simplicial pairs. The former is topologically invariant by definition and is formally easier to work with, while the latter is easier to visualize geometrically and by definition is effectively computable for finite simplicial complexes. The two theories are related by the basic result that the singular homology of a polyhedron is isomorphic to the simplicial homology of any of its triangulating simplicial complexes.
KeywordsExact Sequence Simplicial Complex Chain Complex Homology Group Short Exact Sequence
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