Proper smooth map - Imported Knowlpedia Content

Definition

Let $M$ and $N$ be smooth manifolds. A smooth map $f:M\to N$ is proper if $f^{-1}(K)$ is a compact set in $M$ for every compact subset $K\subseteq N$ . Properness is a global topological condition on the underlying continuous map; it does not impose a rank condition on its differential. In particular, every fiber $f^{-1}(y)$ is compact because a singleton in a manifold is compact.

Equivalent characterizations

For manifolds, properness is equivalent to the sequence criterion: whenever a sequence $(x_j)$ in $M$ has $f(x_j)$ converging in $N$ , the sequence $(x_j)$ has a convergent subsequence in $M$ . It is also equivalent to $f$ being a closed map with compact fibers. These equivalences use the Hausdorff, locally compact, second-countable properties built into the manifold convention; they need not hold in arbitrary topological spaces. See Lee, Appendix A and Chapter 4.

Stability and geometric consequences

Composites of proper maps are proper, and the base change of a proper smooth map along any smooth map is proper whenever the fiber product is formed in the usual manifold setting. A proper injective immersion is a smooth embedding. Properness also prevents points from escaping to infinity while their images remain bounded, which is why it appears in global inverse results, degree theory, and compactness arguments.

Examples and non-examples

The constant map $M\to\{\ast\}$ is proper exactly when $M$ is compact. If $K$ is a compact manifold, the projection $M\times K\to M$ from the product manifold is proper. By contrast, the projection $\mathbb R^2\to\mathbb R$ , $(x,y)\mapsto x$ , is not proper because the inverse image of $\{0\}$ is a noncompact line. A proper smooth map need not be a submersion, immersion, or injective.

References

John M. Lee, Introduction to Smooth Manifolds, 2nd ed., Graduate Texts in Mathematics 218, Springer, 2012. Publisher record. Relevant: Chapter 4 and Appendix A, proper maps and embeddings.
Morris W. Hirsch, Differential Topology, Graduate Texts in Mathematics 33, Springer, 1976. Publisher record. Relevant: Chapter 1, proper maps and differential-topological conventions.