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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?

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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.

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  • $\begingroup$ That makes sense. Thanks! $\endgroup$
    – Student005
    Commented Jun 8, 2022 at 18:43

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