In Wall's paper [Unknotting tori in codimension one and spheres in codimension two](https://www.maths.ed.ac.uk/%7Ev1ranick/papers/wallunknot.pdf), he states the following conjecture:



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> Any $h$-cobordism of $S^3 \times S^1$ to itself is diffeomorphic to $S^3 \times S^1 \times I$.
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What is the status of this conjecture?



Status of a conjecture of C.T.C. Wall?
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 $S^3 \times S^1$.


