Let $X$ be a connected, compact, smooth manifold of dimension $n$ with a non-empty boundary $\partial X$. Define the boundary homotopy type of $X$ as the homotopy type of the pair $(X, \partial X)$.
Consider the space $B(X)$ of all continuous maps $f: \partial X \to X$ that can be extended to a homotopy equivalence between $X$ and a compact manifold $Y$ with $\partial Y \cong \partial X$. Denote the set of homotopy classes of such maps by $[\partial X, X]$.
For a fixed manifold $X$, does there exists a continuous function $f: [\partial X, X] \to \mathbb{R}$ satisfying the following properties:
- Continuity: $f$ is continuous with respect to the compact-open topology on $[\partial X, X]$.
- Energy Minimization: For any homotopy class represented by $g: \partial X \to X$, the value $f(g)$ quantifies a notion of "energy" based on a Morse function defined on $X$, specifically restricted to $\partial X$.
- Topological Equivalence: If $\partial X$ is simply connected, then $f$ factors through the space of homotopy classes of maps from $\partial X$ to the cone $C(X)$ over $X$.
My current research focuses on establishing a rigorous correspondence between the topological invariants of $\partial X$ and the algebraic structure inherent in $[\partial X, X]$. I believe that a deeper comprehension of this relationship could enable me to provide an explicit characterization of the function $f$ by exploiting the critical points of an appropriately chosen Morse function on $X$. In particular, it may be fruitful to investigate whether the number or type of critical points can be directly linked to specific topological invariants of $\partial X$, thereby furnishing a concrete geometric interpretation of the "energy" quantifier associated with each homotopy class in $[\partial X, X]$.
