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| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Classification of manifolds | 3/3 | https://en.wikipedia.org/wiki/Classification_of_manifolds | reference | science, encyclopedia | 2026-05-05T09:08:22.311632+00:00 | kb-cron |
Four-dimensional manifolds are the most unusual: they are not geometrizable (as in lower dimensions), and surgery works topologically, but not differentiably. Since topologically, 4-manifolds are classified by surgery, the differentiable classification question is phrased in terms of "differentiable structures": "which (topological) 4-manifolds admit a differentiable structure, and on those that do, how many differentiable structures are there?" Four-manifolds often admit many unusual differentiable structures, most strikingly the uncountably infinitely many exotic differentiable structures on R4. Similarly, differentiable 4-manifolds is the only remaining open case of the generalized Poincaré conjecture.
== Dimension 5 and more: surgery ==
In dimension 5 and above (and 4 dimensions topologically), manifolds are classified by surgery theory.
The reason for dimension 5 is that the Whitney trick works in the middle dimension in dimension 5 and more: two Whitney disks generically don't intersect in dimension 5 and above, by general position (
2
+
2
<
5
{\displaystyle 2+2<5}
). In dimension 4, one can resolve intersections of two Whitney disks via Casson handles, which works topologically but not differentiably; see Geometric topology: Dimension for details on dimension. More subtly, dimension 5 is the cut-off because the middle dimension has codimension more than 2: when the codimension is 2, one encounters knot theory, but when the codimension is more than 2, embedding theory is tractable, via the calculus of functors. This is discussed further below.
== 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".
=== Low-dimensional self-maps === 1-dimensional: homeomorphisms of the circle 2-dimensional: mapping class group and Torelli group
=== Low codimension === Analogously to the classification of manifolds, in high codimension (meaning more than 2), embeddings are classified by surgery, while in low codimension or in relative dimension, they are rigid and geometric, and in the middle (codimension 2), one has a difficult exotic theory (knot theory).
In codimension greater than 2, embeddings are classified by surgery theory. In codimension 2, particularly embeddings of 1-dimensional manifolds in 3-dimensional ones, one has knot theory. In codimension 1, a codimension 1 embedding separates a manifold, and these are tractable. In codimension 0, a codimension 0 (proper) immersion is a covering space, which are classified algebraically, and these are more naturally thought of as submersions. In relative dimension, a submersion with compact domain is a fiber bundle (just as in codimension 0 = relative dimension 0), which are classified algebraically.
=== High dimensions === Particularly topologically interesting classes of maps include embeddings, immersions, and submersions. Geometrically interesting are isometries and isometric immersions. Fundamental results in embeddings and immersions include:
Whitney embedding theorem Whitney immersion theorem Nash embedding theorem Smale-Hirsch theorem Key tools in studying these maps are:
Gromov's h-principles Calculus of functors One may classify maps up to various equivalences:
homotopy cobordism concordance isotopy Diffeomorphisms up to cobordism have been classified by Matthias Kreck
== See also == The Berger classification of holonomy groups.
== References ==
== Further reading == Kreck, Matthias (2000). "A guide to the classification of manifolds". Princeton University Press eBook Package 2014. Surveys on Surgery Theory (AM-145). Vol. 1. Princeton University Press. pp. 121–134. doi:10.1515/9781400865192-009. ISBN 978-1-4008-6519-2.