Simplicial Homology

Abstract

“Homology groups” associated to a given simplicial complex K, constitute the first comprehensive topic of the subject of algebraic topology. In this chapter, we will explain how we can associate a sequence of abelian groups {H _n (K) : n ≥ 0} to a given simplicial complex K. These groups, called homology groups of the simplicial complex K, will have interesting functorial properties, viz., for each simplicial map f : K → L, there will be an induced group homomorphism (f_*) _n : H _n (K) → H _n (L) for each n ≥ 0 satisfying the following two properties:

• (i) If f : K → L and g : L → M are two simplicial maps, then for each n ≥ 0,

  $${\left( {{{\left( {g \circ f} \right)}_*}} \right)_n} = {\left( {{g_*}} \right)_n} \circ {\left( {{f_*}} \right)_n}:{H_n}\left( K \right) \to {H_n}\left( M \right).$$

• (ii) If I _K : K → K is the identity map, then for each n ≥ 0, the induced map ((I _K )_*) _n is the identity map on H _n (K).