Prove Lemma 44.16 

Data from Lemma 44.16

The map ∂_*k : H_k(A/A') → H_k-1(A'), which we have just defined, is well defined, and is a homomorphism of H_k(A/A') into H_k-1(A').

Proof We leave this proof to the exercises (see Exercise 13).  Let i_*k be the map of Example 44.6. We now can construct the following diagram.

Transcribed Image Text:

Ə=k+¹ Hk(A¹) Ə×¹, Hk-1(A¹) ixh-1 ixh jak Hk(A) Hk-1(A) Hk(A/A') əxh-1 Hk-1(A/A')