- Categories of manifolds
In
mathematics , specificallygeometry and topology , there are many different notions ofmanifold , with more or less structure, and corresponding notions of"map between manifolds", each of which yields a different category and its own classification question.One can relate these categories in a
partial order viaforgetful functor s: "forgetting additional structure". For instance, aRiemannian manifold has an underlyingdifferentiable manifold . For some purposes, it's useful to compare categories: which manifolds in a given category admit a structure, and how many.In other ways, different categories have completely different theories: compare
symmetric space s withhomology manifold s.This article describes many of the structures on manifolds and their connections, with an emphasis on categories studied in
geometry and topology .Kinds of structures
* Many structures on manifolds are
G-structure s, where containment (or more generally, a map ) yields a forgetful functor between categories.
* Geometric structures often imposeintegrability condition s on aG-structure , and the corresponding structure without the integrability condition is called an almost structure. Examples include complex versus almost complex, symplectic versus almost symplectic, and Kähler versus almost Hermitian.
* Of theseG-structure s, many can be expressed viadifferential form s such as asymplectic form orvolume form , or othertensor field s, such as aRiemannian metric Notable geometric and topological categories
Notable categories of manifolds, in decreasing order of rigidity, include: [The complex (including algebraic and Kähler) and symplectic onlyoccur in even dimension; there are some odd-dimensional analogs.]
* smooth projective algebraic varieties
*Kähler manifold s
*complex manifold s /Riemannian manifold s /symplectic manifold s [This level is suggestive: a Kähler manifold has all of these structures, and any two compatible such structures (withintegrability condition s) yields anKähler manifold .]
* Diff:differentiable manifold s (also known as "smooth manifolds")
* PL:PL manifold s (piecewise-linear)
* Top:topological manifold s
*homology manifold sThese can be divided [detailed distinction between geometry and topology] into geometric and topological categories: Diff and below are topological, while above are geometric.
pecial structures
Certain structures are particularly special:
* special holonomy (includingCalabi–Yau manifold s)The following structures are algebraic and very rigid, and admit elegant algebraic classifications:
*Lie groups
* symmetric spaces (andhomogeneous space s)Relation between categories
These categories are related by
forgetful functor s: for instance, a differentiable manifold is also a topological manifold, and a differentiable map is also continuous, so there is a functor .These functors are in general neither one-to-one nor onto; these failures are generally referred to in terms of "structure", as follows. A topological manifold that is in the image of is said to "admit a differentiable structure", and the fiber over a given topological manifold is "the different differentiable structures on the given topological manifold".
Thus given two categories, the two natural questions are:
* Which manifolds of a given type admit an additional structure?
* If it admits an additional structure, how many does it admit?:More precisely, what is the structure of the set of additional structures?In more general categories, this "structure set" has more structure: in Diff it is simply a set, but in Top it is a group, and functorially so.In the case of
G-structure s, this is exactlyreduction of the structure group ,of which the most familiar example isorientability : not every manifold is orientable, and those that are admit exactly two orientations (which form a -torsor ).In general the picture is more complicated; for differentiable (and PL, and Top) structures, this is
surgery theory , and reduction of the structure group (here called the "normal invariant") is the first step, and the second (and last) step is thesurgery obstruction . For geometric structures like a complex structure or symplectic structure, it is in general much more difficult.Important examples where the forgetful functor is...
* ...not one-to-one:exotic sphere s
* ...not onto: non-smoothable manifolds, like the "E8" manifoldExpanded category yields more elegant theory
Expanding a category (weakening the axioms) often yields a more elegant theory.
For instance, the
surgery exact sequence classifieshomology manifold s.
* In Diff, the structure set has no group structure, and is not functorial
* In PL, the structure set is almost a group and functorial, but there's a error (theKirby-Siebenmann invariant ),
* In Top, the structure set has a group structure and is functorial, but there is a factor of error.
* In homology manifolds, it deals with the factor.Similarly, in the
Enriques-Kodaira classification of complex surfaces,complex surfaces have complicated constrains on theirChern number s (the question of which Chern numbers can be realized by complex surfaces is the geography of Chern numbers, and is still an open question),while almost complex surfaces can have any Chern numbers such that .Conversely, the very constrained categories, such as symmetric spaces, also have elegant theories; the intermediate theories are most complicated. This parallels how the
classification of manifolds proceeds by dimensions: low dimensions are constrained and explicitly classified, high dimensions are flexible and algebraic, and intermediate dimension (4 dimensions) is most complicated.Other categories of manifolds
Point-set generalizations
Relaxing the point-set conditions in the definition of manifold yield broader classes of manifolds, which are studied in
general topology :* non-
second countable manifolds, such as the long line
*non-Hausdorff manifold s, such as "the line with two origins"Analytic categories: infinite-dimensional
Modeling a manifold on a possibly infinite-dimensional
topological vector space over the reals yields the following classes of manifolds, which are studied infunctional analysis :*
Hilbert manifold
*Banach manifold
*Fréchet manifold References
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