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Manifold

A presentation of a topological manifold is a second countable Hausdorff space that is locally homeomorphic to a linear space, by a collection (called an atlas) of homeomorphisms called charts. The composition of one chart with the inverse of another chart is a function called a transition map, and defines a homeomorphism of an open subset of the linear space onto another open subset of the linear space. This formalizes the notion of "patching together pieces of a space to make a manifold" – the manifold produced also contains the data of how it has been patched together. However, different atlases (patchings) may produce "the same" manifold; a manifold does not come with a preferred atlas. And, thus, one defines a topological manifold to be a space as above with an equivalence class of atlases, where one defines equivalence of atlases below.

There are a number of different types of differentiable manifolds, depending on the precise differentiability requirements on the transition functions. Some common examples include the following.

• A differentiable manifold is a topological manifold equipped with an equivalence class of atlases whose transition maps are all differentiable. In broader terms, a C^k-manifold is a topological manifold with an atlas whose transition maps are all k-times continuously differentiable.

• A smooth manifold or C^∞-manifold is a differentiable manifold for which all the transition maps are smooth. That is, derivatives of all orders exist; so it is a C^k-manifold for all k. An equivalence class of such atlases is said to be a smooth structure.

• An analytic manifold, or C^ω-manifold is a smooth manifold with the additional condition that each transition map is analytic: the Taylor expansion is absolutely convergent and equals the function on some open ball.

• A complex manifold is a topological space modeled on a Euclidean space over the complex field and for which all the transition maps are holomorphic.

While there is a meaningful notion of a C^k atlas, there is no distinct notion of a C^k manifold other than C⁰ (continuous maps: a topological manifold) and C^∞ (smooth maps: a smooth manifold), because for every C^k-structure with k > 0, there is a unique C^k-equivalent C^∞-structure (every C^k-structure is uniquely smoothable to a C^∞-structure) – a result of Whitney. In fact, every C^k-structure is uniquely smoothable to a C^ω-structure. Furthermore, two C^k atlases that are equivalent to a single C^∞ atlas are equivalent as C^k atlases, so two distinct C^k atlases do not collide. See Differential structure: Existence and uniqueness theorems for details. Thus one uses the terms "differentiable manifold" and "smooth manifold" interchangeably; this is in stark contrast to C^k maps, where there are meaningful differences for different k. For example, the Nash embedding theorem states that any manifold can be C^k isometrically embedded in Euclidean space R^N – for any 1 ≤ k ≤ ∞ there is a sufficiently large N, but N depends on k.

On the other hand, complex manifolds are significantly more restrictive. As an example, Chow's theorem states that any projective complex manifold is in fact a projective variety – it has an algebraic structure.

Atlases

$X$

$U_{\alpha }$

$U_{\beta }$

$\varphi _{\alpha }$

$\varphi _{\beta }$

$\varphi _{\alpha \beta }$

$\varphi _{\beta \alpha }$

$\mathbf {R} ^{n}$

$\mathbf {R} ^{n}$

Charts on a manifold

An atlas on a topological space X is a collection of pairs {(U_α,φ_α)} called charts, where the U_α are open sets that cover X, and for each index α

$\varphi _{\alpha }\colon U_{\alpha }\to {\mathbf {R} }^{n}$

is a homeomorphism of U_α onto an open subset of n-dimensional real space. The transition maps of the atlas are the functions

$\varphi _{\alpha \beta }=\varphi _{\beta }\circ \varphi _{\alpha }^{-1}|_{\varphi _{\alpha }(U_{\alpha }\cap U_{\beta })}\colon \varphi _{\alpha }(U_{\alpha }\cap U_{\beta })\to \varphi _{\beta }(U_{\alpha }\cap U_{\beta }).$

Every topological manifold has an atlas. A C^k-atlas is an atlas whose transition maps are C^k. A topological manifold has a C⁰-atlas and in general a C^k-manifold has a C^k-atlas. A continuous atlas is a C⁰ atlas, a smooth atlas is a C^∞ atlas and an analytic atlas is a C^ω atlas. If the atlas is at least C¹, it is also called a differential structure or differentiable structure. A holomorphic atlas is an atlas whose underlying Euclidean space is defined on the complex field and whose transition maps are biholomorphic.

Immersed submanifolds

Immersed submanifold open interval with interval ends mapped to arrow marked ends

An immersed submanifold of a manifold M is the image S of an immersion map f: N → M; in general this image will not be a submanifold as a subset, and an immersion map need not even be injective (one-to-one) – it can have self-intersections.

More narrowly, one can require that the map f: N → M be an inclusion (one-to-one), in which we call it an injective immersion, and define an immersed submanifold to be the image subset S together with a topology and differential structure such that S is a manifold and the inclusion f is a diffeomorphism: this is just the topology on N, which in general will not agree with the subset topology: in general the subset S is not a submanifold of M, in the subset topology.

Given any injective immersion f : N → M the image of N in M can be uniquely given the structure of an immersed submanifold so that f : N → f(N) is a diffeomorphism. It follows that immersed submanifolds are precisely the images of injective immersions.

The submanifold topology on an immersed submanifold need not be the relative topology inherited from M. In general, it will be finer than the subspace topology (i.e. have more open sets).

Immersed submanifolds occur in the theory of Lie groups where Lie subgroups are naturally immersed submanifolds.

Embedded submanifolds

An embedded submanifold (also called a regular submanifold), is an immersed submanifold for which the inclusion map is a topological embedding. That is, the submanifold topology on S is the same as the subspace topology.

Given any embedding f : N → M of a manifold N in M the image f(N) naturally has the structure of an embedded submanifold. That is, embedded submanifolds are precisely the images of embeddings.

There is an intrinsic definition of an embedded submanifold which is often useful. Let M be an n-dimensional manifold, and let k be an integer such that 0 ≤ k ≤ n. A k-dimensional embedded submanifold of M is a subset S ⊂ M such that for every point p ∈ S there exists a chart (U ⊂ M, φ : U → Rⁿ) containing p such that φ(S ∩ U) is the intersection of a k-dimensional plane with φ(U). The pairs (S ∩ U, φ|_{S ∩ U}) form an atlas for the differential structure on S.

Alexander's theorem and the Jordan-Schoenflies theorem are good examples of smooth embeddings.

Subspace Topology

Given a topological space $(X,\tau )$ and a subset $S$ of $X$, the subspace topology on $S$ is defined by

$\tau _{S}=\lbrace S\cap U\mid U\in \tau \rbrace .$

That is, a subset of $S$ is open in the subspace topology if and only if it is the intersection of $S$ with an open set in $(X,\tau )$. If $S$ is equipped with the subspace topology then it is a topological space in its own right, and is called a subspace of $(X,\tau )$. Subsets of topological spaces are usually assumed to be equipped with the subspace topology unless otherwise stated.

Alternatively we can define the subspace topology for a subset $S$ of $X$ as the coarsest topology for which the inclusion map

$\iota :S\hookrightarrow X$

is continuous.

More generally, suppose $\iota$ is an injection from a set $S$ to a topological space $X$. Then the subspace topology on $S$ is defined as the coarsest topology for which $\iota$ is continuous. The open sets in this topology are precisely the ones of the form $\iota ^{-1}(U)$ for $U$ open in $X$. $S$ is then homeomorphic to its image in $X$ (also with the subspace topology) and $\iota$ is called a topological embedding.

A subspace $S$ is called an open subspace if the injection $\iota$ is an open map, i.e., if the forward image of an open set of $S$ is open in $X$. Likewise it is called a closed subspace if the injection $\iota$ is a closed map.

Continuous functions between topological spaces

Continuity of a function at a point.

Another, more abstract, notion of continuity is continuity of functions between topological spaces in which there generally is no formal notion of distance, as there is in the case of metric spaces. A topological space is a set X together with a topology on X, which is a set of subsets of X satisfying a few requirements with respect to their unions and intersections that generalize the properties of the open balls in metric spaces while still allowing to talk about the neighbourhoods of a given point. The elements of a topology are called open subsets of X (with respect to the topology).

A function

$f\colon X\rightarrow Y$

between two topological spaces X and Y is continuous if for every open set V ⊆ Y, the inverse image

$f^{-1}(V)=\{x\in X\;|\;f(x)\in V\}$

is an open subset of X. That is, f is a function between the sets X and Y (not on the elements of the topology T_X), but the continuity of f depends on the topologies used on X and Y.

This is equivalent to the condition that the preimages of the closed sets (which are the complements of the open subsets) in Y are closed in X.

Homeomorphisms

A function f: X → Y between two topological spaces (X, T_X) and (Y, T_Y) is called a homeomorphism if it has the following properties:

• f is a bijection (one-to-one and onto),
• f is continuous,
• the inverse function f ⁻¹ is continuous (f is an open mapping).

A function with these three properties is sometimes called bicontinuous. If such a function exists, we say X and Y are homeomorphic. A self-homeomorphism is a homeomorphism of a topological space and itself. The homeomorphisms form an equivalence relation on the class of all topological spaces. The resulting equivalence classes are called homeomorphism classes.

Embedding General topology

In general topology, an embedding is a homeomorphism onto its image. More explicitly, an injective continuous map $f:X\to Y$ between topological spaces $X$ and $Y$ is a topological embedding if $f$ yields a homeomorphism between $X$ and $f(X)$ (where $f(X)$ carries the subspace topology inherited from $Y$). Intuitively then, the embedding $f:X\to Y$ lets us treat $X$ as a subspace of $Y$. Every embedding is injective and continuous. Every map that is injective, continuous and either open or closed is an embedding; however there are also embeddings which are neither open nor closed. The latter happens if the image $f(X)$ is neither an open set nor a closed set in $Y$.

For a given space $X$, the existence of an embedding $X\to Y$ is a topological invariant of $X$. This allows two spaces to be distinguished if one is able to be embedded into a space while the other is not.

Embedding Differential topology

In differential topology: Let $M$ and $N$ be smooth manifolds and $f:M\to N$ be a smooth map. Then $f$ is called an immersion if its derivative is everywhere injective. An embedding, or a smooth embedding, is defined to be an injective immersion which is an embedding in the topological sense mentioned above (i.e. homeomorphism onto its image).

In other words, an embedding is diffeomorphic to its image, and in particular the image of an embedding must be a submanifold. An immersion is a local embedding (i.e. for any point $x\in M$ there is a neighborhood $x\in U\subset M$ such that $f:U\to N$ is an embedding.)

When the domain manifold is compact, the notion of a smooth embedding is equivalent to that of an injective immersion.

An important case is $N=\mathbb {R} ^{n}$. The interest here is in how large $n$ must be, in terms of the dimension $m$ of $M$. The Whitney embedding theorem states that $n=2m$ is enough, and is the best possible linear bound. For example the real projective space of dimension $m$ requires $n=2m$ for an embedding. An immersion of this surface is, however, possible in $\mathbb {R} ^{3}$, and one example is Boy's surface—which has self-intersections. The Roman surface fails to be an immersion as it contains cross-caps.

An embedding is proper if it behaves well w.r.t. boundaries: one requires the map $f:X\rightarrow Y$ to be such that

• $f(\partial X)=f(X)\cap \partial Y$, and
• $f(X)$ is transverse to $\partial Y$ in any point of $f(\partial X)$.

The first condition is equivalent to having $f(\partial X)\subseteq \partial Y$ and $f(X\setminus \partial X)\subseteq Y\setminus \partial Y$. The second condition, roughly speaking, says that f(X) is not tangent to the boundary of Y.

The differential of a smooth map

Let φ : M → N be a smooth map of smooth manifolds. Given some x ∈ M, the differential of φ at x is a linear map

$\mathrm {d} \varphi _{x}:T_{x}M\to T_{\varphi (x)}N\,$

from the tangent space of M at x to the tangent space of N at φ(x). The application of dφ_x to a tangent vector X is sometimes called the pushforward of X by φ. The exact definition of this pushforward depends on the definition one uses for tangent vectors (for the various definitions see tangent space).

If one defines tangent vectors as equivalence classes of curves through x then the differential is given by

$\mathrm {d} \varphi _{x}(\gamma '(0))=(\varphi \circ \gamma )'(0).$

Here γ is a curve in M with γ(0) = x. In other words, the pushforward of the tangent vector to the curve γ at 0 is just the tangent vector to the curve φ∘γ at 0.

Alternatively, if tangent vectors are defined as derivations acting on smooth real-valued functions, then the differential is given by

$\mathrm {d} \varphi _{x}(X)(f)=X(f\circ \varphi ).$

Here X ∈ T_xM, therefore X is a derivation defined on M and f is a smooth real-valued function on N. By definition, the pushforward of X at a given x in M is in T_φ(x)N and therefore itself is a derivation.

After choosing charts around x and φ(x), φ is locally determined by a smooth map

${\widehat {\varphi }}:U\to V$

between open sets of R^m and Rⁿ, and dφ_x has representation (at x)

$\mathrm {d} \varphi _{x}\left({\frac {\partial }{\partial u^{a}}}\right)={\frac {\partial {\widehat {\varphi }}^{b}}{\partial u^{a}}}{\frac {\partial }{\partial v^{b}}},$

in the Einstein summation notation, where the partial derivatives are evaluated at the point in U corresponding to x in the given chart.

Extending by linearity gives the following matrix

$(\mathrm {d} \varphi _{x})_{a}^{\;b}={\frac {\partial {\widehat {\varphi }}^{b}}{\partial u^{a}}}.$

Thus the differential is a linear transformation, between tangent spaces, associated to the smooth map φ at each point. Therefore, in some chosen local coordinates, it is represented by the Jacobian matrix of the corresponding smooth map from R^m to Rⁿ. In general the differential need not be invertible. If φ is a local diffeomorphism, then the pushforward at x is invertible and its inverse gives the pullback of T_φ(x)N.

The differential is frequently expressed using a variety of other notations such as

$D\varphi _{x},\;(\varphi _{*})_{x},\;\varphi '(x),\;T_{x}\varphi .$

It follows from the definition that the differential of a composite is the composite of the differentials (i.e., functorial behaviour). This is the chain rule for smooth maps.

Also, the differential of a local diffeomorphism is a linear isomorphism of tangent spaces.

Covering Space

Let X be a topological space. A covering space of X is a space C together with a continuous surjective map

$p\colon C\to X\,$

such that for every x ∈ X, there exists an open neighborhood U of x, such that p⁻¹(U) (the inverse image of U under p) is a union of disjoint open sets in C, each of which is mapped homeomorphically onto U by p.

The map p is called the covering map, the space X is often called the base space of the covering, and the space C is called the total space of the covering. For any point x in the base the inverse image of x in C is necessarily a discrete space called the fiber over x.

The special open neighborhoods U of x given in the definition are called evenly-covered neighborhoods. The evenly-covered neighborhoods form an open cover of the space X. The homeomorphic copies in C of an evenly-covered neighborhood U are called the sheets over U. One generally pictures C as "hovering above" X, with p mapping "downwards", the sheets over U being horizontally stacked above each other and above U, and the fiber over x consisting of those points of C that lie "vertically above" x. In particular, covering maps are locally trivial. This means that locally, each covering map is 'isomorphic' to a projection in the sense that there is a homeomorphism, h, from the pre-image p⁻¹(U), of an evenly covered neighbourhood U, onto U × F, where F is the fiber, satisfying the local trivialization condition, which is that, if we project U × F onto U, π : U × F → U, so the composition of the projection π with the homeomorphism h will be a map π ∘ h from the pre-image p⁻¹(U) onto U, then the derived composition π ∘ h will equal p locally (within p⁻¹(U)).

Topological Group

A topological group G is a topological space and group such that the group operations of product:

$G\times G\to G:(x,y)\mapsto xy$

and taking inverses:

$G\to G:x\mapsto x^{-1}$

are continuous functions. Here, G × G is viewed as a topological space by using the product topology.

Although not part of this definition, many authors require that the topology on G be Hausdorff; this corresponds to the identity map $*\to G$ being a closed inclusion (hence also a cofibration). The reasons, and some equivalent conditions, are discussed below. In the end, this is not a serious restriction—any topological group can be made Hausdorff in a canonical fashion.

In the language of category theory, topological groups can be defined concisely as group objects in the category of topological spaces, in the same way that ordinary groups are group objects in the category of sets. Note that the axioms are given in terms of the maps (binary product, unary inverse, and nullary identity), hence are categorical definitions. Adding the further requirement of Hausdorff (and cofibration) corresponds to refining to a model category.

Inclusion map

A is a subset of B, and B is a superset of A.

In mathematics, if $A$ is a subset of $B$, then the inclusion map (also inclusion function, insertion, or canonical injection) is the function $\iota$ that sends each element, $x$ of $A$ to $x$, treated as an element of $B$:

$\iota :A\rightarrow B,\qquad \iota (x)=x.$

A "hooked arrow" $\hookrightarrow$ is sometimes used in place of the function arrow above to denote an inclusion map.

This and other analogous injective functions from substructures are sometimes called natural injections.

Given any morphism f between objects X and Y, if there is an inclusion map into the domain $\iota :A\rightarrow X$, then one can form the restriction fi of f. In many instances, one can also construct a canonical inclusion into the codomain R→Y known as the range of f.

Immersion

The Klein bottle, immersed in 3-space.

For a closed immersion in algebraic geometry, see closed immersion.

In mathematics, an immersion is a differentiable function between differentiable manifolds whose derivative is everywhere injective. Explicitly, f : M → N is an immersion if

$D_{p}f:T_{p}M\to T_{f(p)}N\,$

is an injective function at every point p of M (where T_pX denotes the tangent space of a manifold X at a point p in X). Equivalently, f is an immersion if its derivative has constant rank equal to the dimension of M:

$\operatorname {rank} \,D_{p}f=\dim M.$

The function f itself need not be injective, only its derivative.

A related concept is that of an embedding. A smooth embedding is an injective immersion f : M → N which is also a topological embedding, so that M is diffeomorphic to its image in N. An immersion is precisely a local embedding – i.e. for any point x ∈ M there is a neighbourhood, U ⊂ M, of x such that f : U → N is an embedding, and conversely a local embedding is an immersion. For infinite dimensional manifolds, this is sometimes taken to be the definition of an immersion.