# Understanding the correlation between the Van Kampen theorem and Mayer Vietoris sequences from the Hatcher Algebraic Topology text

I am trying to understand a sentence from Hatcher's section 2.2 on Mayer Vietoris sequences. It states that:

"Mayer-Vietoris sequences can be viewed as analogs of the van Kampen theorem since if $$A\cap B$$ is path-connected, the $$H_1$$ terms of the reduced Mayer-Vietoris sequence yield an isomorphism $$H_1(X)\approx \left(H_1(A)\oplus H_1(B)\right)/Im \:\Phi.$$ This is exactly the abelianized statement of the van Kampen theorem, and $$H_1$$ is the abelianization of $$\pi_1$$ for path-connected spaces..."

Does this mean that the term $$H_2(X)$$ is $$0$$, which would imply that there is a short exact sequence $$0 \rightarrow \tilde{H_1}(A\cap B) \rightarrow \tilde{H_1}(A)\oplus \tilde{H_1}(B) \rightarrow \tilde{H_1}(X) \rightarrow 0$$ ?

Or else, how may the statement be explained?

I think you are slightly misreading the statement: there is no assertion of injectivity of the homomorphism $$\widetilde H_1(A \cap B) \xrightarrow{\Phi} \widetilde H_1(A) \oplus \widetilde H_1(B)$$ What he writes is that $$\widetilde H_1(X)$$ is the quotient of $$\widetilde H_1(A) \oplus \widetilde H_1(B)$$ by the image of $$\Phi$$, which is true by exactness if you had left off the $$0 \to$$ at the beginning of your exact sequence.