Contents

Idea

A manifold is a space which looks locally like a Euclidean space, most commonly a finite-dimensional Euclidean space ${ℝ}^n$.

What “locally looks like” means really depends on what sort of structure we are considering Euclidean space to embody. At one extreme, we can think of ${ℝ}^n$ as merely a topological space. Or, ${ℝ}^n$ may be considered as carrying more rigid types of structure ($C^k$, smooth, PL, real analytic, affine, hyperbolic, foliated, etc., etc.). In any case, the type of geometry embodied in a particular flavor of manifold is controlled by a particular groupoid of transformations which preserves whatever geometric features one is interested in; cf. Felix Klein’s Erlanger Programm.

Definitions

To give a reasonably general notion of manifold, we first specify the kinds of concrete geometric groupoids which come into play.

A pseudogroup on a topological space (or locale) $X$ is a groupoid $G$ each of whose objects is an open set of $X$, and whose morphisms are homeomorphisms between such open sets, satisfying the following conditions:

• The objects cover $X$. (Equivalently, in light of the last axiom, every open set of $X$ is an object of $G$.)
• If $g:V\to W$ belongs to $G$ and $U\subseteq V$ is an open set, then the restriction $g{\mid }_U:U\to g(U)$ belongs to $G$. (Equivalently, in light of the other axioms, every inclusion map $\mathrm{id}{\mid }_U:U\to V$ belongs to $G$.)
• (sheaf property) If $g:U\to V$ is a homeomorphism and if there is a covering $U_{\alpha }$ of $U$ such that the restrictions $g{\mid }_{U_{\alpha }}:U_{\alpha }\to g(U_{\alpha })$ are morphisms of $G$, then $g$ is also morphism of $G$.

Commonly used choices for $X$ include ${ℝ}^n$ or ${ℂ}^n$, the half-space

or the $n$-cube $I^n=[0,1{]}^n$. For the sake of concreteness, the reader may as well focus on the case $X={ℝ}^n$.

Let $G$ be a pseudogroup on $X$. A $G$-chart on a topological space $M$ is an open subset $U$ of $M$ together with an embedding

Two charts $\varphi :U\to X$ and $\psi :V\to X$ are compatible if

belongs to $G$. A $G$-atlas on $M$ is a family of compatible charts $({\varphi }_{\alpha }:U_{\alpha }\to X{)}_{\alpha }$ such that $(U_{\alpha }{)}_{\alpha }{)}_{\alpha }$ covers $M$. The (restricted) maps ${\varphi }_{\alpha \beta }={\varphi }_{\beta }\circ {\varphi }_{\alpha }^{-1}$ are called transition functions between the charts of the atlas.

Finally, a $G$-manifold is a topological space equipped with a $G$-atlas.

We can think of a $G$-manifold as a space which is locally modeled on $X$ according to the geometry $G$.

• It is almost invariably the case in classical manifold theory that one makes some technical niceness assumptions on the space $M$: viz. it is Hausdorff and paracompact; often one assumes its topology has a countable basis as well. In the typical cases mentioned above for $X$, this will mean that $M$ is metrizable. In many studies, for example in cobordism theory, one goes even further and assumes the manifolds are compact.

An atlas is not considered an essential part of the structure of a manifold: two different atlases may yield the same manifold structure. Here are the relevant definitions:

An isomorphism of $G$-manifolds $f:M\to N$ (defined by chosen atlas structures) is a homeomorphism $f$ such that

is in $G$ whenever $(U,\varphi )$ is a coordinate chart of $x\in M$, and $(V,\psi )$ is a coordinate chart of $f(x)\in N$. If $M_1$ and $M_2$ are two $G$-manifold structures on the same topological space $M$, then $M_1$ and $M_2$ are considered equal as $G$-manifolds if $\mathrm{id}:M\to M$ is an isomorphism from $M_1$ to $M_2$ (and hence also from $M_2$ to $M_1$).

• 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.

Morphisms of Manifolds

We begin by defining the 2-poset (i.e., locally preordered bicategory) of regions, denoted $\mathrm{Reg}$. 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 $r\le 1_X$ in $\mathrm{Reg}$ splits: there is a map in $\mathrm{Reg}$,

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

hold. The object $\mathrm{Ext}(r)$ 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 $i:C\hookrightarrow \mathrm{Reg}$ such that

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

Intended examples include the case where the objects of $C$ are Euclidean spaces ${ℝ}^n$, 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 $\mathrm{Reg}$

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 $\mathrm{Reg}$-composite

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

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

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 $C^{\mathrm{\infty }}$ maps between open sets of ${ℝ}^n$. Replacing ${ℝ}^n$ here by a half-space $\{x\in {ℝ}^n:x_1\ge 0\}$, one obtains the notion of smooth manifold with boundary. Or, replacing ${ℝ}^n$ 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 $ℝ$.)

A topological $n$-manifold is a manifold with respect to the pseudogroup of homeomorphisms between open sets of ${ℝ}^n$. 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 straighforward 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.

Tangent Bundle

Many species of manifolds (Riemannian, Lorentzian, symplectic, and so on) involve extra structures defined on the tangent bundle of a smooth manifold. This is perhaps the most fundamental construction in manifold theory.

If $M$ is a smooth $n$-manifold defined by an atlas $(U_{\alpha },{\varphi }_{\alpha })$, then we may define its tangent bundle $TM$ by a gluing construction in $\mathrm{Top}$, taking $TM$ to be the quotient of the disjoint sum

obtained by dividing by the equivalence relation

where $p\in U_{\alpha }\cap U_{\beta }$, and $g_{\alpha \beta }(p)\in \mathrm{GL}({ℝ}^n)$ is the result of differentiating the transition function ${\varphi }_{\alpha \beta }$ at the point ${\varphi }_{\alpha }(p)$. We thus obtain a covering $U_{\alpha }\times {ℝ}^n$ of $TM$, and these form coordinate charts of a smooth manifold structure on $TM$ in a more or less evident way. There is an obvious projection map $\pi :TM\to M$, called the tangent bundle; the fiber ${\pi }^{-1}(p)$ over a point $p\in M$ is called the tangent space at $p$, denoted $T_{p}M$. Elements $v\in T_{p}M$ are called tangent vectors at $p$.

• It is not immediately apparent that this construction yields the same manifold (in the sense described earlier) independent of the atlas chosen. To make this manifest, it is preferable to deal with coordinate-free expressions, defining for example tangent vectors with reference to the sheaf of smooth functions on $M$. We discuss this below.

The functions

satisfy Čech 1-cocycle relations

These 1-cocycle data make the tangent bundle an $n$-dimensional vector bundle with structure group $\mathrm{GL}({ℝ}^n)$.

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).