Topology/Manifolds/Categories of Manifolds

There are many different notions of manifold, 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 via forgetful functors: “forgetting additional structure”. For instance, a Riemannian manifold has an underlying differentiable 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 spaces with homology manifolds.

This article describes many of the structures on manifolds and their connections, with an emphasis on categories studied in geometry and topology; in some cases the formal categorical point of view is an important part of the subject, while in other cases one less formally simply discusses various types of manifolds and maps, without the apparatus of category theory.

Kinds of structures

 * Many structures on manifolds are G-structures, where containment (or more generally, a map $$H \to G$$) yields a forgetful functor between categories.
 * Geometric structures often impose integrability conditions on a G-structure, and the corresponding structure without the integrability condition is called an almost structure. Examples include complex versus almost complex, symplectic versus almost symplectic, Hermitian versus almost Hermitian, and Kähler versus almost Kähler.
 * Of these G-structures, many can be expressed via differential forms such as a symplectic form or volume form, or other tensor fields, such as a Riemannian metric.

Notable geometric and topological structures
Notable structures on manifolds, in decreasing order of rigidity, include:
 * smooth projective algebraic varieties
 * Kähler manifolds
 * complex manifolds / Riemannian manifolds / symplectic manifolds
 * Note that for symplectic manifolds, there are various notions of what maps are of interest; there is no generally agreed “category of symplectic manifolds”.
 * Diff: differentiable manifolds (also known as smooth manifolds)
 * PL: PL manifolds (piecewise-linear)
 * Formally, PL and Diff are not directly comparable, so one must introduce the category PDIFF (piecewise-differentiable), which is however equivalent to PL, and thus not distinguished except in careful usage.
 * Top: topological manifolds
 * homology manifolds

These can be divided into geometric and topological categories: Diff and below are topological, while above are geometric. The topological structures have agreed category structures (such as differentiable maps), while the geometric structures have various notions of maps, and no single categorical structure – when defined by a G-structure one can take “maps respecting the G-structure”, such as isometric immersions of Riemannian manifolds, but these are not necessarily the maps of most interest.

Special structures
Certain structures are particularly special:
 * special holonomy (including Calabi–Yau manifolds)

The following structures are algebraic and very rigid, and admit elegant algebraic classifications:
 * Lie groups
 * symmetric spaces (and homogeneous spaces)

Relation between categories
These categories are related by forgetful functors: for instance, a differentiable manifold is also a topological manifold, and a differentiable map is also continuous, so there is a functor $$\operatorname{Diff} \to \operatorname{Top}$$.

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 $$\operatorname{Diff} \to \operatorname{Top}$$ 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-structures, this is exactly reduction of the structure group, of which the most familiar example is orientability: not every manifold is orientable, and those that are admit exactly two orientations (which form a $$\mathbf{Z}/2$$-torsor).

In general the picture is more complicated; for the forgetful functor from differentiable (and PL, and Top) manifolds to the category of spaces with Poincaré duality, 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 the surgery 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 spheres
 * ...not onto: non-smoothable manifolds, like the E8 manifold

Examples
Expanding a category (weakening the axioms) often yields a more flexible theory from the categorical viewpoint. On the other hand, the very constrained categories, such as Lie groups or symmetric spaces, often have very 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.

Geometric Topology
For instance, the surgery exact sequence classifies homology manifolds.
 * 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 $$\mathbf{Z}/2$$ error (the Kirby–Siebenmann invariant),
 * In Top, the structure set has a group structure and is functorial, but there is a factor of $$\mathbf{Z}$$ error.
 * In homology manifolds, it deals with the $$\mathbf{Z}$$ factor.

Algebraic Geometry
Similarly, in the Enriques–Kodaira classification of complex surfaces, complex surfaces have complicated constrains on their Chern numbers (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 $$c_1^2+c_2 \equiv 0 \pmod{12}$$.

Other categories of manifolds
See also: Categories of manifolds Manifolds are studied outside of geometric topology; the approaches and theories vary significantly between fields.

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 manifolds, 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 in functional analysis:


 * Hilbert manifold
 * Banach manifold
 * Fréchet manifold