ScalingStacks

Proposition 3.4. Let $i\colon W\to W^{\prime}$ be the inclusion of a four-manifold $W$ into $W^{\prime}$. Then we have a natural map

$$i_{*}\colon\mathcal{S}_{0}^{N}(W;\emptyset)\to\mathcal{S}_{0}^{N}(W^{\prime},\emptyset).$$

If $W^{\prime}$ is the result of a $k$-handle attachment to $W$, then $i_{*}$ is a surjection for $k=3$ and an isomorphism for $k=4$.