= Homology of two disks meeting inside a sphere
Attach two disks to a simply connected space $A$ along disjoint null-homotopic boundary circles, and additionally identify one interior point of each disk. Assume the attachments are <CW complexes> or admit compatible triangulations. The extra interior identification contributes a copy of $\mathbb Z$ to the first <homology group>, in addition to the two new second-homology classes. Indeed, after the first attachment $Y=A\cup D_1$, the intersection with $D_2$ has two components, $S^1\sqcup\{p\}$. In the <Mayer–Vietoris sequence>, the map $H_0(S^1\sqcup\{p\})\to H_0(Y)\oplus H_0(D_2)$ sends $(m,n)$ to $(m+n,-m-n)$, whose <kernel> is $\mathbb Z$. If $H_1(A)=0$, exactness gives $H_1(Y\cup D_2)=\mathbb Z$. The degree-two sequence is $0\to H_2(Y)\to H_2(Y\cup D_2)\to\mathbb Z\to0$, which splits because $\mathbb Z$ is free. For $A=S^3$, the integral <homology groups> are $\mathbb Z,\mathbb Z,\mathbb Z^2,\mathbb Z$ in degrees zero through three.
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