Consider the cotangent bundle $T^*S^2$ equipped with the Eguchi-Hanson metric. Since the metric is ALE and the cone at infinity is $\mathbb R^4/\mathbb Z_2$, we can define a compact manifold $D$ which is the contraction of $T^*S^2$ such that $\partial D=\mathbb RP^3$.
Now we have the following long exact sequence for $(D,\partial D)$, $$ H_{2}(\partial D) \rightarrow H_{2}(D) \rightarrow H_{2}(D,\partial D) \rightarrow H_{1}(\partial D) \rightarrow H_{1}(D). $$ Since $H_2(\partial D)=H_2(\mathbb RP^3)=0$ and $H_1(D)=H_1(T^*S^2)=0$, we have $$ 0 \rightarrow H_{2}(D) \rightarrow H_{2}(D,\partial D) \rightarrow H_{1}(\partial D)\rightarrow 0, $$ and hence $$ H_{2}(D,\partial D)=H_{2}(D) \oplus H_1(\partial D)=\mathbb Z \oplus \mathbb Z_2. $$ However, by the Lefschetz duality, $H_2(D,\partial D)=H^2(D)=\mathbb Z$.
Where did I make a mistake in this argument?