Maps Between Manifolds
From the point of view of category theory, the classification of manifolds is one piece of understanding the category: it's classifying the objects. The other question is classifying maps of manifolds up to various equivalences, and there are many results and open questions in this area.
For maps, the appropriate notion of "low dimension" is for some purposes "self maps of low dimensional manifolds", and for other purposes "low codimension".
Read more about this topic: Classification Of Manifolds
Famous quotes containing the word maps:
“And now good morrow to our waking souls,
Which watch not one another out of fear;
For love all love of other sights controls,
And makes one little room an everywhere.
Let sea-discoverers to new worlds have gone,
Let maps to other, worlds on worlds have shown,
Let us possess one world; each hath one, and is one.”
—John Donne (1572–1631)