Relation To Other Categories of Manifolds
PL, or more precisely PDIFF, sits between DIFF (the category of smooth manifolds) and TOP (the category of topological manifolds): it is categorically "better behaved" than DIFF – for example, the Generalized Poincaré conjecture is true in PL (with the possible exception of dimension 4, where it is equivalent to DIFF), but is false generally in DIFF – but is "worse behaved" than TOP, as elaborated in surgery theory.
Read more about this topic: Piecewise Linear Manifold
Famous quotes containing the words relation to, relation and/or categories:
“We must get back into relation, vivid and nourishing relation to the cosmos and the universe. The way is through daily ritual, and is an affair of the individual and the household, a ritual of dawn and noon and sunset, the ritual of the kindling fire and pouring water, the ritual of the first breath, and the last.”
—D.H. (David Herbert)
“When needs and means become abstract in quality, abstraction is also a character of the reciprocal relation of individuals to one another. This abstract character, universality, is the character of being recognized and is the moment which makes concrete, i.e. social, the isolated and abstract needs and their ways and means of satisfaction.”
—Georg Wilhelm Friedrich Hegel (1770–1831)
“Kitsch ... is one of the major categories of the modern object. Knick-knacks, rustic odds-and-ends, souvenirs, lampshades, and African masks: the kitsch-object is collectively this whole plethora of “trashy,” sham or faked objects, this whole museum of junk which proliferates everywhere.... Kitsch is the equivalent to the “cliché” in discourse.”
—Jean Baudrillard (b. 1929)