Smooth exhaustion function - Imported Knowlpedia Content

Definition

Let $M$ be a smooth manifold. A smooth exhaustion function is a smooth map $\rho:M\to[0,\infty)$ that is proper. Equivalently, every sublevel set

M_c=\{x\in M:\rho(x)\le c\}

is a compact set, and the increasing family $(M_c)_{c\ge0}$ covers $M$ . Thus $\rho(x_j)\to\infty$ along every sequence that eventually leaves each compact subset of $M$ . The codomain and nonnegativity are conventional; a bounded-below proper smooth real-valued function carries the same exhaustion data after translation. This is a global condition.

Existence and construction

Every standard smooth manifold admits a smooth exhaustion function. One construction chooses a countable locally finite coordinate cover, compactly supported cutoff functions subordinate to it, and coefficients increasing rapidly enough that the resulting locally finite sum becomes proper. This is a smooth refinement of the fact that a second-countable manifold is $\sigma$ -compact; see Lee, chapter on smooth functions and partitions of unity.

Moreover, one can arrange additional properties, such as agreement with a prescribed function on a compact set, by modifying the construction with cutoff functions. Requiring all critical points to be nondegenerate is an extra Morse-theoretic refinement, not part of exhaustion.

Examples and non-examples

On $\mathbb R^n$ , the function $\rho(x)=\|x\|^2$ is a smooth exhaustion because its sublevel sets are closed balls. The height function $x\mapsto x_1$ is smooth and unbounded, but it is not an exhaustion: the inverse image of a compact interval is an unbounded slab and hence not compact.

On a compact manifold, every smooth map to $[0,\infty)$ is proper, so even a constant function is an exhaustion under the definition in the core.

Uses and scope

Exhaustions reduce arguments on a noncompact manifold to controlled compact stages. They are used to build compactly supported approximations, organize proper embeddings, and localize inductive constructions. A proper Morse function supplies both an exhaustion and handle data, but a general smooth exhaustion may have degenerate critical points or whole critical submanifolds.

References

John M. Lee, Introduction to Smooth Manifolds, 2nd ed., Graduate Texts in Mathematics 218, Springer, 2012. DOI record. Relevant: smooth partitions of unity, bump functions, and proper exhaustion constructions.
Morris W. Hirsch, Differential Topology, Graduate Texts in Mathematics 33, Springer, 1976. DOI record. Relevant: Chapter 6, Morse functions and proper functions on noncompact manifolds.