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
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—F. Scott Fitzgerald (1896–1940)