Instanton construction of the mapping cone Thom-Smale complex

The wedge by a smooth closed $\ell$-form induces the mapping cone de Rham cochain complex. This complex is quasi-isomorphic to the mapping cone Thom-Smale cochain complex. We give an instanton construction of the mapping cone Thom-Smale complex in this paper. 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. As the main result, we prove that our instanton complex is cochain isomorphic to the topologically constructed mapping cone Thom-Smale complex.

💡 Research Summary

The paper investigates the interaction between a closed ℓ‑form ω on a closed oriented Riemannian manifold M and a Morse–Smale pair (f,g). The wedge product with ω defines a mapping‑cone de Rham complex (1.1). Earlier work showed that this analytic complex is quasi‑isomorphic to a topologically constructed mapping‑cone Thom‑Smale complex, but the construction relied on a mixture of analytic and topological tools. The central question posed by the author is whether one can build a purely analytic counterpart of the mapping‑cone Thom‑Smale complex using only eigenspaces of a suitably deformed Laplacian.

To answer this, the author introduces a two‑parameter deformation of the mapping‑cone differential: \