I am reading Kirby's paper paper "Akbulut's corks and h-cobordisms of smooth simply connected 4-manifolds" and I have a question about how to actually prove the cork theorem from the results in the paper. Specifically, I want to show that any pair $M_0, M_1$ of homeomorphic but not diffeomorphic closed simply connected $4$-manifolds is related by a cork twist. A cork is a pair $(C,\tau)$ of a contractible $4$-manifold $C$ together with a diffeomorphism $\tau$ of its boundary (sometimes taken to be an involution) which does not extend as a diffeomorphism over all of $C$. The cork theorem then says that for any such $M_0, M_1$ we can find $C \subseteq M_0$ s.t $M_1 \underset{\text{sm}}{\cong} M_0 \setminus \mathrm{int}(C) \cup_\tau C$ (cutting out $C$ and gluing it back in along $\tau$ is the cork twist).

From Kirby's paper we know that any $h$-cobordism between $M_0$ and $M_1$ splits into a product $h$-cobordism and a contractible sub-$h$-cobordism $A$. We also know that the ends of $A$ can be taken to be diffeomorphic via an involution, and that $A$ itself is diffeomorphic to $B^5$.

One idea, then, is the following: $$M_1 = (M_1 \setminus \mathrm{int} A_1) \cup_\partial A_1 \underset{\text{sm}}{\cong} (M_0 \setminus \mathrm{int} A_0) \cup_\partial A_1 \underset{\text{sm}}{\cong} (M_0 \setminus \mathrm{int} A_0) \cup_\tau A_0$$

where the first diffeomorphism replaces $M_1 \setminus \mathrm{int} A_1$ with $M_0 \setminus \mathrm{int} A_0$ using the product $h$-cobordism, and the second diffeomorphism replaces $A_1$ with $A_0$ using the diffeomorphism between the two ends of $A$ that restricts to an involution.

The trouble is that the diffeomorphism restricted to the boundary (which we label $\tau$) clearly extends over the contractible itself, so it cannot be our cork diffeomorphism. But I don't see any other way to get an involution.