The h-cobordism theorem does not cover boundary dimension four

The dimension hypothesis $n\ge 5$ (equivalently $\dim W\ge 6$) in the smooth h-cobordism theorem (The smooth simply connected h-cobordism theorem) cannot be relaxed to $n=4$. Two distinct obstructions are recorded here.

(i) The proof route stops working: the general-position argument that produces a clean embedded Whitney disk in a middle level of dimension $n$ requires both sheet dimensions at most $n-3$, or one of them at least $3$ in the borderline version, and in boundary dimension four the relevant ambient middle level is four-dimensional, where the smooth Whitney trick genuinely fails (The smooth Whitney trick fails in dimension four).

(ii) The conclusion itself fails in the smooth category in general: the s-cobordism theorem, and with it the triviality statement of the h-cobordism theorem, is known to be false for $n=\dim M_0=4$ in general by Donaldson's work (Lück §1.5, printed p. 21, citing Donaldson, Irrationality and the h-cobordism conjecture, J. Differential Geom. 26 (1987) 141--168), while in the topological category the corresponding statement for $n=4$ holds for good fundamental groups, in particular for the trivial group, by Freedman. For simply connected 4-manifolds the failure is documented explicitly: there are smooth orientable simply connected 4-manifolds that are all smoothly s-cobordant and homeomorphic but pairwise not diffeomorphic, so no h-cobordism between two of them is a product (Kasprowski--Powell--Ray, EMS Surv. Math. Sci. 9 (2022) 193--249, Example 1.13 and §5.8, where the first pair is due to Donaldson). Milnor's Concluding Remarks distinguish total dimension four (boundary dimension three), where the four-disk conjecture is discussed, from total dimension five (the boundary-dimension-four case here). The historical four-disk discussion is not a statement about this boundary-dimension-four range.

No smooth four-dimensional Poincaré or disk conclusion may be read off the theorem of this page: the smooth statement is genuinely restricted to boundary dimension at least five (h-Cobordism).