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In Smale's On the structure of manifolds paper there is his relative version of the h-cobordism theorem, specifically Theorem 3.1 (and 1.4). Roughly speaking this concerns the situation where one has an h-cobordism of pairs, and you want some compatibility between the ambient product structure and the submanifold product structure.

I find I don't think about this theorem often, but every once and a while I do need to use it. Each time I return to it, I take considerable time to process how it can be used -- somehow it is not worded in a way that plays well with my way of thinking about things. . . and there are a few distracting typos.

I imagine I am not the only person with this problem.

Are there other expositions of the relative h-cobordism theorem in the literature?

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    $\begingroup$ Here is a probably useless remark: Cor. 3.2 of the paper ought to follow from a space level statement. Given a compact submanifold $T\subset M$ of codimension zero which is a thickening of a positive codimensional submanifold $V\subset M$, there is a relative h-cobordism space $H(M,T)$. It ought to sit in a fiber sequence $CE(T,M) \to H(M,\text{rel} T) \to H(M)$, where $\text{CE}$ refers to concordance embeddings. When enough simple connectivity is around, both $CE(T,M)$ and $H(M)$ are connected (by the h-cobordism thm), implying that $H(M,T)$ will also be connected. $\endgroup$
    – John Klein
    Commented Jun 11, 2018 at 11:12

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