Editing Mayer–Vietoris sequence (section)

===Analogy with the Seifert–van Kampen theorem===
There is an analogy between the Mayer–Vietoris sequence (especially for homology groups of dimension 1) and the [[Seifert–van Kampen theorem]].<ref name="Hatcher 2002 150"/><ref>{{harvnb|Massey|1984|p=240}}</ref> Whenever <math>A\cap B</math> is [[path-connected]], the reduced Mayer–Vietoris sequence yields the isomorphism

:<math>H_1(X) \cong (H_1(A)\oplus H_1(B))/\text{Ker} (k_* - l_*)</math>

where, by exactness,

:<math>\text{Ker} (k_* - l_*) \cong \text{Im} (i_*, j_*).</math>

This is precisely the [[Commutator subgroup#Abelianization|abelianized]] statement of the Seifert–van Kampen theorem. Compare with the fact that <math>H_1(X)</math> is the abelianization of the [[fundamental group]] <math>\pi_1(X)</math> when <math>X</math> is path-connected.<ref>{{harvnb|Hatcher|2002|loc=Theorem 2A.1, p. 166}}</ref>