The Hopf-Wiener extension is a Toeplitz-type extension for the group C*-algebra of $\mathbb{R}$. It is also constructed from a bordism with cylindrical end, namely, a complete manifold with half-cylinder attached to the boundary.

On such a manifold we can develop the Callias index theorem, an index theorem for the Dirac operator with a suitable potential. I shall talk about the relation between the Callias index and the connection map in the 6-term exact sequence defined from a bordism extension.