[Math/Geometry] Instanton Construction of the Mapping Cone: Bridging Analysis and Topology

TL;DR

In modern differential geometry, the transition from smooth analysis (de Rham cohomology) to discrete topology (Morse theory) is famously bridged by the Witten Deformation. This paper extends this bridge to the Mapping Cone setting. The author constructs an "instanton cochain complex" using the eigenspaces of a deformed Laplacian and proves it is isomorphic to the topological mapping cone Thom-Smale complex. This provides a purely analytic way to calculate filtered cohomology and symplectic invariants.

Background: The Mapping Cone Challenge

The mapping cone de Rham complex $d^{\omega }$ is a sophisticated structure where the exterior derivative is twisted by a closed $\ell$-form $\omega$. It is defined as: $(\alpha ,\beta )\mapsto (d\alpha +\omega \wedge \beta ,(-1{)}^{\ell -1}d\beta )$ While this complex has deep roots in symplectic geometry (computing filtered cohomology), its "Morse" counterpart—the Thom-Smale complex—was primarily understood through topological cup products. The missing link was a purely analytic construction: can we see the topology of the mapping cone just by looking at the low-energy spectrum of a Laplacian?

The difficulty lies in the fact that $\omega \wedge$ acts as a perturbation that destroys the neat eigenspace structure of the standard Witten Laplacian.

Methodology: The Two-Parameter Deformation

To solve this, the author introduces a dual-parameter deformation $d_{ST}^{\omega }$. Here, $T$ drives the forms toward the critical points of a Morse function $f$ (the classical Witten approach), while $S$ is a scaling factor for $\omega$ that allows the analysis to bypass the "compatibility" issues between $\omega$ and the Morse-Smale pair $(f,g)$.

1. The Deformed Operator

The core operator used is the "Mapping Cone Laplacian" ${\mathbb{D}}_{ST}^2$, where: ${\mathbb{D}}_{ST}=\left[\begin{array}{cc}d+d^{\ast }+T\hat{c}(df)& S^{-1}\omega \\ S^{-1}{\omega }^{\ast }& (-1{)}^{\ell -1}(d+d^{\ast }+T\hat{c}(df))\end{array}\right]$

2. Spectral Isolation

The author proves that for sufficiently large $S$ and $T$, the spectrum of ${\mathbb{D}}_{ST}^2$ splits. There is a "gap" that separates the low-energy eigenvalues (near 0) from the high-energy eigenvalues (near $T$). The image above represents the definition of the instanton complex $F_{ST}^k$ as the sum of eigenspaces in the $[0,1]$ eigenvalue range.

The Main Result: Cochain Isomorphism

The breakthrough of this paper is Theorem 1.5. By using orthogonal projections and carefully estimating Sobolev norms, the author demonstrates that the mapping ${\Phi }_{ST}^{\omega }:F_{ST}^{k}o{C}^k(f,g)\oplus C^{k-\ell +1}(f,g)$ is a cochain isomorphism.

Why does this matter? It means that the intricate trajectories between critical points (topological flow lines) and the integration of $\omega$ over these manifolds are perfectly captured by the analytic eigenforms of the deformed Laplacian.

Experiments and Analytical Evidence

The author validates the theory by reproducing the Mapping Cone Morse Inequalities.

Morse Inequality Refinement:

The paper provides a refined analytic expression for the rank $R_k$ of the complex: This identity links the number of critical points (${\mu }_j$) to the dimensions of the mapping cone cohomology ($b_j^{\omega }$), proving that the analytic construction is consistent with the global topology of the manifold.

Cohomology Decomposition

A significant corollary (1.8) shows that the resulting cohomology $H^k(F)$ can be decomposed into a kernel and a cokernel of the cup product map. This mirrors the algebraic behavior of mapping cones in category theory, now verified melalui spectral analysis.

Critical Insight & Future Outlook

The requirement that $S>e^{C_{0}T}$ is the technical "secret sauce." It ensures that the Morse-driven localization (which happens at rate $T$) is not overwhelmed by the $\omega$ perturbation.

Limitations: The current work assumes a closed oriented manifold. Future research could explore:

• Group Actions: Can this be extended to Equivariant Morse theory?
• Symplectic Stability: How does the instanton complex behave under deformations of the symplectic form?

This paper represents a robust milestone in Spectral Geometry, proving that even complex "cone" structures in algebraic topology have a natural, harmonic home in analysis.

Disclaimer: This summary is based on the paper "Instanton construction of the mapping cone Thom-Smale complex" by Hao Zhuang (2026).