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Contents

Idea

A manifold is a topological space which looks locally like a Cartesian space, commonly a finite-dimensional Cartesian space $\mathbb{R}^n$, in which case one speaks of a manifold of dimension $n$ or $n$-fold, but possibly an infinite-dimensional topological vector space, in which case one has an infinite-dimensional manifold.

What “locally looks like” means depends on what sort of structure we are considering a Cartesian space to embody. At one extreme, we can think of $\mathbb{R}^n$ as merely a topological space. Or, $\mathbb{R}^n$ may be considered as carrying more rigid types of structure, such as $C^k$-differential structure, smooth structure, piecewise-linear (PL) structure, real analytic structure, affine structure, hyperbolic structure, foliated structure, etc., etc. Accordingly we have notions of topological manifold, differentiable manifold, smooth manifold, analytic manifold etc. By default these are modeled on finite dimensional spaces, but most notions have generalizations to a corresponding notion of infinite dimensional manifold.

graphics grabbed from Frankel

In any case, the type of geometry embodied in a particular flavor of manifold is controlled by a particular groupoid or, more generally, category of transformations which preserves whatever geometric features one is interested in; cf. Felix Klein’s Erlanger Programm.

Definitions

Here we will focus on the general notion of a manifold. More concrete examples can be found in individual pages such as topological manifold and smooth manifold.

We will present two possible definitions. The first, via pseudogroups, has a simpler definition, but has two (rather serious) drawbacks:

1. There is in general no notion of morphisms between manifolds. At best, we can only talk about isomorphisms of manifolds.
2. We can only define smooth $n$-manifolds for each $n$ separately, but not smooth manifolds in general (and the same applies to complex manifolds etc.)

The second definition via cartologies was proposed by Todd Trimble to solve the above two problems.

Definition via pseudogroups

The setting is a topological space $X$ together with a pseudogroup $G$ on $X$. For the sake of concreteness, the reader may as well focus on the case $X = \mathbb{R}^n$ and $G$ is the groupoid of diffeomorphisms between open subsets of $X$.

Definitions

Examples

If the term “manifold” appears without further qualification, what is usually meant is a smooth $n$-manifold of some natural number dimension $n$: a $G$-manifold where $G$ is the pseudogroup of invertible $C^{\infty}$ maps between open sets of $\mathbb{R}^n$. Replacing $\mathbb{R}^n$ here by a half-space $\{x \in \mathbb{R}^n: x_1 \geq 0\}$, one obtains the notion of smooth manifold with boundary. Or, replacing $\mathbb{R}^n$ here by the $n$-cube $I^n$, one obtains the notion of (smooth) $n$-manifold with (cubical) corners. Morphisms of manifolds are here called smooth maps, and isomorphisms are called diffeomorphisms. (In manifold theory, one usually reserves the term smooth function for smooth maps to $\mathbb{R}$.)

A topological $n$-manifold is a manifold with respect to the pseudogroup of homeomorphisms between open sets of $\mathbb{R}^n$. Any continuous function between topological manifolds is a morphisms, and any homeomorphism is an isomorphism. A piecewise-linear (PL) $n$-manifold is where the pseudogroup consists of piecewise-linear homeomorphisms between such open sets; morphisms are called piecewise-linear (PL) maps.

One can go on to define, in a straightforward way, real analytic manifolds, complex analytic manifolds, elliptic manifolds, hyperbolic manifolds, and so on, using the general notion of pseudogroup.

Any space $X$ can always be turned into a manifold modelled on itself, using any pseudogroup $G$. Simply take the inclusions of open sets as charts.

Isomorphisms of manifolds

Alternatively, atlases are ordered by inclusion, and two atlases define the same manifold structure on $M$ if they have a common upper bound. Equivalently, two atlases define the same manifold structure if each chart of one is compatible with each chart of the other. Or, one could extend any atlas to the (unique) maximal atlas containing it, which consists of all charts compatible with each of the charts in the original atlas, and simply identify a manifold structure with a maximal atlas.

Definition via cartologies

We begin by defining the 2-poset (i.e., locally preordered bicategory) of regions, denoted $Reg$. The objects are topological spaces (or locales if you prefer); the morphisms are partial functions with open domain, that is spans

where $f$ is continuous and $i$ is an open embedding. The spans are locally (that is, for fixed $X$ and $Y$) ordered by inclusion.

These local posets are not cocomplete, but they admit certain obvious joins: given a family of regional maps

the join $\bigvee_\alpha (U_\alpha, f_\alpha)$ exists iff we have local compatibility:

for all $\alpha, \beta$. Notice that composition on either side with a $1$-cell preserves any local joins which exist.

Every coreflexive morphism $r \leq 1_X$ in $Reg$ splits: there is a map in $Reg$,

whose opposite $i^op: X \to Ext(r)$ also belongs to $Reg$ (that is, $i$ is an open embedding), and the equations

hold. The object $Ext(r)$ may be called the extension of $r$. This splitting is a kind of comprehension principle? familiar from the theory of allegories, among other things.

A cartology is a (locally full) subbicategory $i: C \hookrightarrow Reg$ such that

• (Closure under open subspaces) If $X \in Ob(C)$ and $r \leq 1_X$ in $Reg$, then $i: Ext(r) \to X$ and its opposite $i^{op}$ are morphisms of $C$.
• (“Sheaf condition”) The inclusion $i: C \to Reg$ reflects and preserves local joins.

Intended examples include the case where the objects of $C$ are Euclidean spaces $\mathbb{R}^n$, and morphisms are spans

where $f$ is smooth.

Given a cartology $C$, a morphism $r = (U, f): X \to Y$ in $C$ is pseudo-invertible if there exists $s = (V, g): Y \to X$ such that $s \circ r = 1_U$ and $r \circ s = 1_V$.

The notion of a $C$-manifold modeled on an object $X$ of $C$ is defined just as before, using the pseudogroup on $X$ implied by the previous lemma. In particular, we have $C$-charts of an atlas structure on $M$, which are morphisms in $Reg$

satisfying the expected properties. We can thus speak of $C$-manifolds (or $(C, X)$-manifolds if we want to make explicit the modeling space $X$).

Now, given a cartology $C$, we define the category of $C$-manifolds. Let $M$ be a $(C, X)$-manifold and $N$ a $(C, Y)$-manifold. Then, a $C$-morphism from $M$ to $N$ is a continuous map $f: M \to N$ such that the $Reg$-composite

belongs to $C$, for every pair of charts $(U, \phi): X \to M$ and $(V, \psi): Y \to N$.

These definitions need to be carefully checked against known examples (e.g., the categories $Top$, $PL$, and $Smooth$, among others).

Generalizations

• generalized smooth space

• differentiable stack

• derived smooth manifold

• supermanifold

  • Euclidean supermanifold (notice that the definition of that is very much along the lines of the Klein-program-style definition above).

• semialgebraic manifold

Related concepts

• dimension of a manifold

• low-dimensional topology

• projective manifold

• manifold with boundary

• closed manifold

• product manifold

References

For more see the references at differentiable manifold and smooth manifold.

The vision of the modern notion of manifolds is attributed to:

• Bernhard Riemann, Über die Hypothesen, welche der Geometrie zu Grunde liegen, Göttingen (1845) [[doi:10.1007/978-3-642-35121-1](https://doi.org/10.1007/978-3-642-35121-1)]

  Engl. transl: William Clifford: On the hypotheses which underlie geometry, Nature VIII (1873) 183-184 [[doi:10.1007/978-3-319-26042-6](https://doi.org/10.1007/978-3-319-26042-6)]

See also:

• John Loftin, The real definition of a smooth manifold (pdf)