Let $X:=\{(x,y,z)\in \mathbb{R}^3: (x^2+y^2-1)(x^2+z^2-9)=0\}$. How do I compute $H_k(X)$?
First note that $X$ is homotopic to the following space by deformation retraction.
Now, to have a Mayer-Vietoris sequence, chop $X$ into a excisive couple as follows:
Note that $A,B$ are now homotopic to $S^1$ and $\alpha,\beta,\gamma$ are the generators for $\tilde H_1(-)$ for each space.
Now consider the reduced Mayer-Vietoris sequence as follows:
From this, we conclude that $\tilde H_2(A\cup B)=0$ and $\tilde H_1(A\cup B)\cong \mathbb{Z}\oplus \mathbb{Z}/(1,1)\cong \mathbb{Z}$
So we have $H_1(X)=\mathbb{Z},H_2(X)=0$.
Since $X$ is obviously path connected, we have $H_0(X)=\mathbb{Z}$. Since for each $A,B,A\cap B$, $H_k(-)=0$ for $k\geq 2$, we have $H_k(X)=0$ for $k\geq 2$ by Mayer-Vietoris sequence.
Is my argument correct?