Smooth four-dimensional surgery is not covered by the high-dimensional program

Every construction on this page is stated for a supplied framed embedded sphere and proves only what follows from that datum (p-surgery kills the represented pi-p class below the middle dimension, Surgery on a normal map preserves its normal bordism class); the existence statements quoted below the middle use the representation of classes by embedded spheres and the smoothing of intersections, which need dimension hypotheses.

In smooth dimension four the general high-dimensional Whitney argument is unavailable: an immersed Whitney disk cannot in general be replaced by a clean embedded disk disjoint from the other sheets, so a kernel class need not be representable by a framed embedded sphere (The framing obstruction lives in the normal bundle of the surgery sphere) and the middle-dimensional argument of the high-dimensional programme stops (Middle-dimensional surgery has an intersection-form obstruction). The source's dimension conditions are explicit: the Whitney trick is available when the two complementary dimensions are at least three, or when one is two and the other at least three with a fundamental-group condition, and the cited general embedded Whitney disk construction is guaranteed under those hypotheses in ambient dimension at least five; the condition $\dim M\ge 5$ in the middle dimension is exactly what makes the self-intersection criterion of the preceding remark available.

Consequently this page supplies no general smooth four-dimensional surgery existence or classification theorem based on cancelling middle-dimensional intersections. Its conditional constructions and vanishing conclusions still apply when their displayed hypotheses hold. In particular, $m=4$, $p=1$, $q=3$ satisfies $p\le q-2$, so surgery along a supplied framed embedded circle kills its represented ${\pi }_1$-class by the killing lemma cited above. The missing general Whitney argument concerns the existence of suitable embedded representatives and clean disks, rather than the performance of a surgery once its datum is supplied. The following Whitney-trick and h-cobordism pages retain their own dimension hypotheses. Nothing here addresses the four-dimensional theory by other methods (the failure of a general Whitney-move guarantee in that dimension is a technical boundary, not a claim that no theory exists).