Instanton construction of the mapping cone Thom-Smale complex

The wedge by a smooth closed $\mathrm{\ell }$-form induces the mapping cone de Rham cochain complex. This complex is quasi-isomorphic to the mapping cone Thom-Smale cochain complex. In this paper, we give a purely analytic instanton construction of the mapping cone Thom-Smale complex. More precisely, for a Morse function with the transversality condition on a closed oriented Riemannian manifold, we construct an instanton cochain complex using the eigenspaces of the mapping cone Laplacian deformed by the Morse function and two parameters. One parameter is inherited from the classical Witten deformation. The other parameter points to the cup product issue affecting the mapping cone situation. As the main result, we prove that our instanton complex is cochain isomorphic to the topologically constructed mapping cone Thom-Smale complex.