According to (Desbrun, Kanso, and Tong 2008), the exterior derivative operator \(d\) is the adjoint of the boundary operator \(\partial\). This is based on the Stokes’ theorem. Let \(\omega\) be a \(k\)-form and \(\sigma\) be a simplicial chain. Then we have

\[\int_{\sigma} d\omega = \int_{\partial\sigma} \omega.\]

This can be rewritten as \(\left[ d\omega, \sigma \right] = \left[ \omega, \partial\sigma \right]\), where the brackets \([\cdot,\cdot]\) group two entities comprising a differential form and a simplicial chain via integration. And the integration is also a linear operator. According to the concept of adjoint operator in functional analysis, \(d\) is the adjoint of \(\partial\).

We already know that the boundary of boundary is an empty set, i.e. \(\partial \circ \partial = \emptyset\). Then we have

\[\left[ \omega, \partial\circ\partial\sigma \right] = \left[ d\omega, \partial\sigma \right] = \left[ d\circ d\omega, \sigma \right] \equiv 0.\]

This just brings about the exactness of the exterior derivative operator, which means the coboundary of coboundary also vanishes.

Desbrun, Mathieu, Eva Kanso, and Yiying Tong. 2008. “Discrete Differential Forms for Computational Modeling.” In Discrete Differential Geometry, edited by Alexander I. Bobenko, John M. Sullivan, Peter Schröder, and Günter M. Ziegler, 287–324. Oberwolfach Seminars. Basel: Birkhäuser. https://doi.org/10.1007/978-3-7643-8621-4_16.