[Paper Review] Extending functions from a neighborhood of the sphere to the ball

Valentin Seigneur|arXiv (Cornell University)|May 17, 2018

Homotopy and Cohomology in Algebraic Topology2 references3 citations

TL;DR

This paper provides an algebraic necessary condition for extending a smooth function germ defined near the sphere $\mathbb{S}^n$ to a non-critical function on the ball, using the Morse chain complex with $\mathbb{Z}$-coefficients and a group $G(\tilde{f})$ of graded isomorphisms. For $n \geq 6$, under specific index and labeling constraints, the extension is possible if and only if $\det(\partial_{++,k+1}) \equiv \pm 1 \mod d_1(\partial_{+-,k+1})$, where $d_1$ is the gcd of the entries in the $\partial_{+-,k+1}$ block.

ABSTRACT

In this article, we are interested in the problem of extending the germ of a smooth function $ ilde{f}$ defined along the standard sphere of dimension $n$ to a function defined on the ball which has no critical points. The article gives a necessary condition using the Morse chain complex associated to the function $f$, restriction of $ ilde{f}$ to the sphere $\mathbb{S}^n$, which is assumed to be a Morse function.

Motivation & Objective

• To determine necessary and sufficient conditions for extending a smooth function germ defined along the sphere $\mathbb{S}^n$ to a non-critical function on the closed ball $\mathbb{D}^{n+1}$.
• To generalize prior combinatorial results by Barannikov and others by introducing an algebraic condition based on the Morse complex with $\mathbb{Z}$-coefficients.
• To characterize when such an extension exists under constraints on critical point indices and normal derivatives (labels $\pm$).
• To establish a computable criterion involving the determinant of the $\partial_{++,k+1}$ block modulo the gcd of entries in $\partial_{+-,k+1}$ for $n \geq 6$.
• To show that the necessary condition from Theorem 3.1 becomes sufficient under additional topological constraints on critical point indices and ordering of critical values.

Proposed method

• Define the Morse germ $\tilde{f}$ as a germ along $\partial M = \mathbb{S}^n$ with no critical points, and classify critical points of $f = \tilde{f}|_{\mathbb{S}^n}$ by the sign of the normal derivative: $\mathcal{C}^+$ (positive) and $\mathcal{C}^-$ (negative).
• Construct the Morse chain complex with $\mathbb{Z}$-coefficients, decomposing the boundary operator $\partial_k$ into blocks $\partial_{\ell_1\ell_2,k}$ for $\ell_1, \ell_2 \in \{+, -\}$.
• Introduce the group $G(\tilde{f})$ of graded isomorphisms that preserve the decomposition by label and index, acting via handle slides on the critical points.
• Prove that a non-critical extension exists only if there exists $M \in G(\tilde{f})$ such that $M\partial M^{-1}$ has zero $(-,+)$ block and the $++$ block defines a chain complex with trivial homology except in degree $n$, where it is $\mathbb{Z}$.
• For $n \geq 6$, under the assumption of only one maximum, one minimum, and critical points of indices $k$ and $k+1$ with $2 \leq k \leq n-2$, show that the extension exists iff $\det(\partial_{++,k+1}) \equiv \pm 1 \mod d_1(\partial_{+-,k+1})$, where $d_1$ is the gcd of entries in $\partial_{+-,k+1}$.
• Use Cerf theory and generic paths of functions to construct a homotopic Morse function $f_1$ with ordered critical values and a canonical handle slide structure, enabling the derivation of the determinant condition.

Experimental results

Research questions

• RQ1 Under what algebraic conditions on the Morse complex with $\mathbb{Z}$-coefficients can a non-critical germ along $\mathbb{S}^n$ be extended to the ball?
• RQ2 Is the necessary condition from Theorem 3.1 sufficient for non-critical extension when $n \geq 6$?
• RQ3 Can the extension problem be reduced to a computable arithmetic condition on the boundary operator blocks?
• RQ4 How do the labels $\pm$ (based on normal derivative sign) and the ordering of critical values affect the existence of a non-critical extension?
• RQ5 What role does the group $G(\tilde{f})$ of graded isomorphisms play in transforming the Morse complex to a form admitting a non-critical extension?

Key findings

• The necessary condition from Theorem 3.1 — that there exists $M \in G(\tilde{f})$ such that $M\partial M^{-1}$ has zero $(-, +)$ block and the $++$ block defines a chain complex with $\mathbb{Z}$ in degree $n$ and trivial homology elsewhere — is not sufficient in general.
• For $n \geq 6$, if $f$ has only one local maximum, one local minimum, and critical points of indices $k$ and $k+1$ with $2 \leq k \leq n-2$, then the germ $\tilde{f}$ extends non-critically if and only if $\det(\partial_{++,k+1}) \equiv \pm 1 \mod d_1(\partial_{+-,k+1})$, where $d_1$ is the greatest common divisor of the entries in $\partial_{+-,k+1}$.
• The condition $\det(\partial_{++,k+1}) \equiv \pm 1 \mod d_1(\partial_{+-,k+1})$ is equivalent to the surjectivity of the matrix $\begin{pmatrix}\partial_{++,k+1} & \partial_{+-,k+1}\end{pmatrix}$, which ensures the existence of a matrix $N_{k+1}$ such that $\partial_{++,k+1} + \partial_{+-,k+1}N_{k+1}$ is invertible.
• The proof constructs a function $f_1$ via handle slides that orders critical values and simplifies the boundary operator, ultimately realizing the $\mathbb{Z}$-homology condition required for non-critical extension.
• The group $G(\tilde{f})$ allows for the transformation of the Morse complex via graded isomorphisms, and its structure is determined by the order of critical values and the labeling of critical points by $\pm$.
• When all $+$-labeled critical points of index $k$ and $k+1$ have higher critical values than $-$-labeled ones, the condition becomes computable and fully algebraic, enabling algorithmic verification of non-critical extendability.