geometric topology In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another. History Geometric topology as an area distinct from algebraic topology may be said to have originated ...

, PDIFF, for ''p''iecewise ''diff''erentiable, is the category of

piecewise In mathematics, a piecewise-defined function (also called a piecewise function, a hybrid function, or definition by cases) is a function defined by multiple sub-functions, where each sub-function applies to a different interval in the domain. ...

- smooth

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...

s and piecewise-smooth

maps A map is a symbolic depiction emphasizing relationships between elements of some space, such as objects, regions, or themes. Many maps are static, fixed to paper or some other durable medium, while others are dynamic or interactive. Althoug ...

between them. It properly contains DIFF (the category of

smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...

s and

smooth functions In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if ...

between them) and PL (the category of

piecewise linear manifold In mathematics, a piecewise linear (PL) manifold is a topological manifold together with a piecewise linear structure on it. Such a structure can be defined by means of an atlas, such that one can pass from chart to chart in it by piecewise linear ...

s and piecewise

linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...

s between them), and the reason it is defined is to allow one to relate these two categories. Further, piecewise functions such as splines and

polygonal chain In geometry, a polygonal chain is a connected series of line segments. More formally, a polygonal chain is a curve specified by a sequence of points (A_1, A_2, \dots, A_n) called its vertices. The curve itself consists of the line segments co ...

s are common in mathematics, and PDIFF provides a category for discussing them.

Motivation

PDIFF is mostly a technical point: smooth maps are not piecewise linear (unless linear), and piecewise linear maps are not smooth (unless globally linear) – the intersection is

s, or more precisely affine maps (because not based) – so they cannot directly be related: they are separate generalizations of the notion of an affine map. However, while a smooth manifold is not a PL manifold, it carries a canonical PL structure – it is uniquely triangularizable; conversely, not every PL manifold is smoothable. For a particular smooth manifold or smooth map between smooth manifolds, this can be shown by breaking up the manifold into small enough pieces, and then linearizing the manifold or map on each piece: for example, a circle in the plane can be approximated by a triangle, but not by a 2-gon, since this latter cannot be linearly embedded. This relation between Diff and PL requires choices, however, and is more naturally shown and understood by including both categories in a larger category, and then showing that the inclusion of PL is an equivalence: every smooth manifold and every PL manifold ''is'' a PDiff manifold. Thus, going from Diff to PDiff and PL to PDiff are natural – they are just inclusion. The map PL to PDiff, while not an equality – not every piecewise smooth function is piecewise linear – is an equivalence: one can go backwards by linearize pieces. Thus it can for some purposes be inverted, or considered an isomorphism, which gives a map

\text \to \text \to \text.

These categories all sit inside TOP, the category of topological manifold and continuous maps between them. In summary, PDiff is more general than Diff because it allows pieces (corners), and one cannot in general smooth corners, while PL is no less general that PDiff because one can linearize pieces (more precisely, one may need to break them up into smaller pieces and then linearize, which is allowed in PDiff).

History

That every smooth (indeed, ''C''¹) manifold has a unique PL structure was originally proven in . A detailed expositionary proof is given in . The result is elementary and rather technical to prove in detail, so it is generally only sketched in modern texts, as in the brief proof outline given in . A very brief outline is given in , while a short but detailed proof is given in .

References

* * * * * {{refend Geometric topology