Homology lens spaces in topological 4-manifolds

Abstract

For a closed $4$-manifold $X^4$ and closed $3$-manifold $M^3$ we investigate the smallest integer $n$ (perhaps $n=\infty$) such that $M^3$ embeds in $\#_nX^4$, the connected sum of $n$ copies of $X^4$. It is proven that any lens space (or homology lens space) embeds topologically locally flatly in $\#_2({\mathbf C}P^2\#\ \overline {{\mathbf C}P}^2)$, in $\#_4 S^2\times S^2$ and in $\#_8 \mathbf{C}P^2$.