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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?

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I think it would be desirable to wait at least a couple of days before reposting a Math.stackexchange question on MathOverflow. –  Ricardo Andrade Dec 29 '13 at 2:39
    
I agree with Ricardo. In general, I would wait at least 2 weeks to give people time to answer before considering moving it here. I have therefore voted to close. –  Andy Putman Dec 29 '13 at 4:09
    
(by the way, you are correct that it is due to Kirby-Siebenmann; in their book they proved that handlebody theory works in the topological category, which is exactly what is needed to make the classical proof go through -- see their book for more details). –  Andy Putman Dec 29 '13 at 4:10

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