Relative Homology

Let A be a subspace of X Then for every q ≥ 0, S_q (A) is the submodule of S_q (X) consisting of linear combinations of singular q-simplexes Δ _q → X which actually map into A. We can then form the quotient module, and since the boundary operator sends S_q (A) into S _q–1(A), it induces a homomorphism ∂ ¯ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429502408/295d3acc-dfa3-4421-a2d3-a9c22ffb681c/content/eq266.tif"/> which makes the diagram https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429502408/295d3acc-dfa3-4421-a2d3-a9c22ffb681c/content/pg70_1.tif"/>