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In Wall's paper Unknotting tori in codimension one and spheres in codimension two, he states the following conjecture:

Any $h$-cobordism of $S^3 \times S^1$ to itself is diffeomorphic to $S^3 \times S^1 \times I$.

What is the status of this conjecture?

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The conjecture has been solved. This is Theorem 16.1 in C.T.C. Wall. (1999). Surgery on Compact Manifolds, Second edition. Mathematical Surveys and Monographs, Vol. 69. Here, two proofs of this theorem are given: the first one is an application of a more general method presented in the book, while the second one relies on some geometrical properties of $S^3 \times S^1$.

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