I think this question on math.stackexchange is more appropriate on mathoverflow. Correct me, if you don't think so.
The h-cobordism theorem is true in the topological and in the smooth category in dimensions $\ge 6$. (By "dimension", I mean the dimension of the ambient cobordism instead of the dimension of the boundary, as it is in the wikipedia article.)
The smooth case of dimension $\ge 6$ was first proven by Smale around 1962 (e.g. S. Smale, "On the structure of manifolds" Amer. J. Math., 84 (1962) pp. 387–399). But who did the topological case of dimension $\ge 6$ first, and can you please give me a reference? Did it follow from the work of Kirby and Siebenmann on topological manifolds at the end of the 60s?