[Paper Review] A proof of Morse's theorem about the cancellation of critical points

François Laudenbach|arXiv (Cornell University)|Jul 9, 2013

Advanced Topology and Set Theory5 references3 citations

TL;DR

This paper provides a geometric proof of Morse's cancellation theorem for a pair of non-degenerate critical points of indices $k$ and $k+1$ on a closed manifold. By reducing the problem to a one-dimensional model and using pseudo-gradient flows, the authors construct an explicit deformation of the Morse function that cancels the critical points while preserving the gradient structure, showing the cancellation is possible under transverse intersection and orbit conditions of unstable and stable manifolds.

ABSTRACT

In this note, we give a proof of the famous theorem of M. Morse dealing with the cancellation of a pair of non-degenerate critical points of a smooth function. Our proof consists of a reduction to the one-dimensional case where the question becomes easy to answer.

Motivation & Objective

• To provide a geometric, visible proof of Morse's cancellation theorem for critical points of consecutive indices.
• To make the deformation of the Morse function explicit by specifying the support of the perturbation.
• To reduce the general cancellation problem to the one-dimensional case where it becomes tractable.
• To clarify the role of pseudo-gradients and the transversality condition in enabling cancellation.
• To recover and generalize Milnor's and Smale's results with a more transparent, geometric construction.

Proposed method

• Reduce the cancellation problem to a one-dimensional model by analyzing the flow lines between critical points of indices $k$ and $k+1$.
• Use a pseudo-gradient vector field $X$ satisfying Lyapunov and non-degeneracy conditions to define stable and unstable manifolds.
• Construct a tubular neighborhood $\widetilde{W}$ of the connecting orbit $\ell$ between $p$ and $q$, fibered over a curve $A$ in $M$.
• Define a family of Morse functions $f_u$ on fibers $\widetilde{D}_u$ of the fibration, each with a single critical point of index $k$.
• Apply Lemma 2 to smoothly decrease the critical value of $f_u$ while preserving the pseudo-gradient structure.
• Extend the deformation over the entire neighborhood $U$ to cancel the pair $(p,q)$, keeping the function Morse except at $t=1/2$, where it has a cubic singularity.

Experimental results

Research questions

• RQ1 Under what geometric conditions can two critical points of consecutive indices be canceled in a Morse function on a closed manifold?
• RQ2 How can the cancellation process be made geometrically visible and supported explicitly in the manifold?
• RQ3 What role does the pseudo-gradient structure play in enabling the cancellation of critical points?
• RQ4 Can the general cancellation problem be reduced to a one-dimensional model for simpler analysis?
• RQ5 How does the transversality of $W^u(p) \cap W^s(q)$ and the orbit structure affect the possibility of cancellation?

Key findings

• The cancellation of a pair of critical points of indices $k$ and $k+1$ is possible if their unstable and stable manifolds intersect transversely in a single orbit and satisfy a global crossing condition with respect to a regular level set.
• The deformation of the Morse function is supported in a neighborhood $U$ of the closure of $W^u(p) \cap \{f \geq f(q) - \varepsilon\}$, ensuring no change outside this region.
• At $t=1/2$, the function $f_t$ develops a cubic singularity, which is the intermediate stage of the cancellation process.
• The final function $f_1$ has no critical points in $U$, confirming the successful cancellation of the pair $(p,q)$.
• The proof establishes that the cancellation is compatible with the pseudo-gradient structure, preserving the flow dynamics throughout the deformation.
• The result generalizes Milnor’s and Smale’s criteria and provides a constructive, geometric realization of the cancellation process.