The four-dimensional disc embedding conjecture and whether the free group is good

Theorem 1.1: there is a compact oriented smooth 4-manifold $M$ with immersed proper discs $f_1,\dots,f_k$ with embedded boundary circles and framed (even smoothly embedded) spheres $g_i$ satisfying $\lambda(f_i,g_j)=\delta_{ij}$, $\lambda(g_i,g_j)=0$, $\widetilde\mu(g_i)=0$, whose boundary circles bound no pairwise disjoint locally flat discs, in any relative homotopy class. Corollary 1.2: $F_2$, and every group containing it, is not good in the Freedman-Quinn sense. Corollary 1.3: the unrestricted topological surgery and s-cobordism assertions cannot both hold. The marked tensor companion proves the framing-preserving version. Not shown: which of surgery or s-cobordism fails, or that the Borromean rings are not A-B slice.

Theorem 1.1: there is a compact oriented smooth 4-manifold $M$ with immersed proper discs $f_1,\dots,f_k$ with embedded boundary circles and framed (even smoothly embedded) spheres $g_i$ satisfying $\lambda(f_i,g_j)=\delta_{ij}$, $\lambda(g_i,g_j)=0$, $\widetilde\mu(g_i)=0$, whose boundary circles bound no pairwise disjoint locally flat discs, in any relative homotopy class. Corollary 1.2: $F_2$, and every group containing it, is not good in the Freedman-Quinn sense. Corollary 1.3: the unrestricted topological surgery and s-cobordism assertions cannot both hold. The marked tensor companion proves the framing-preserving version. Not shown: which of surgery or s-cobordism fails, or that the Borromean rings are not A-B slice.