We show that closed $\widetilde{\mathbb{SL}} \times \mathbb{E}^n$-manifolds are topologically rigid if $n\geq 2$, and are rigid up to $s$-cobordism, if $n=1$.

We study the topological $4$-dimensional surgery problem for a closed connected orientable topological $4$-manifold $X$ with vanishing second homotopy and $\pi_1(X)\cong A * F(r)$, where $A$ has one end and $F(r)$ is the free group of rank $r\ge 1$. Our result is related to a theorem of Krushkal and Lee, and depends on the validity of the Novikov conjecture for such fundamental groups.