A pseudofree $S^1$-action is a smooth $S^1$-action on a smooth $(2n+1)$-manifold such that the action is free except for finitely many exceptional orbits with isotropy $\Bbb Z_{a_1},\dots,\Bbb Z_{a_k}$ where $a_1,\dots,a_k$ are pairwise relatively prime. Suppose we have a pseudofree $S^1$-action on a 5-manifold $Q$. Then $X=Q/S^1$ is a 4-manifold with isolated singularities whose neighborhoods are cones on lens spaces $L(a_i,b_i)$ corresponding to the exceptional orbits in $Q$. Set $D(X)=X-\bigcup_{i=1}^k \text{int}(cL(a_i,b_i))$. The $S^1$-action over $D(X)$ is free; hence it is classified by an Euler class $e\in H^2(D(X);\Bbb Z)$. Since the tubular neighborhood of an exceptional orbit $E_i$ with isotropy $\Bbb Z_{a_i}$ in $Q$ is $D^4\times_{\Bbb Z_{a_i}} S^1$ which is diffeomorphic to $D^4\times S^1$, the part of $Q$ over each $L(a_i,b_i)$ is just $S^3\times S^1$. From this we see that $i^*e$ is a unit in $\Bbb Z_{\alpha}$ $(\alpha=a_1\cdots a_k)$, where $0\to H^2(D(X),\partial D(X);\Bbb Z)\to H^2(D(X);\Bbb Z)\xrightarrow{i^*} H^2(\partial D(X);\Bbb Z)=\Bbb Z_\alpha$.
These are from the first two paragraphs of section 2 of the paper https://cpb-us-e2.wpmucdn.com/faculty.sites.uci.edu/dist/3/246/files/2011/03/23_PseudoFreeOrbifolds.pdf. I can see that the part of $Q$ over each $L(a_i,b_i)$ is $S^3\times S^1$, but how does this imply that $i^*e $ is a unit in $\Bbb Z_\alpha$?