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

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Companion: A marked tensor obstruction to four-dimensional disk embedding evidence for this event

The four-dimensional disc embedding conjecture and whether the free group is good evidence for this event

VibeMathed record: The four-dimensional disc embedding conjecture and whether the free group is good evidence for this event

The four-dimensional disc embedding conjecture and whether the free group is good parent of this event

Freedman-Quinn, Topology of 4-manifolds (1990) evidence for this event

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. parent of this event

Kim-Orson-Park-Ray, Open problems, in The Disc Embedding Theorem (2021) evidence for this event