Topology, Manifolds, and Smooth Structure

This document collects the point-set topology and differential-geometry definitions that physics texts (including this repository) generally take for granted: topological space, continuity, manifold, smooth structure, smooth map, tangent space, and friends. It is intended as a reference, not a course; for a real treatment see Lee, Introduction to Smooth Manifolds; Munkres, Topology; Nakahara, Geometry, Topology and Physics.

1. Topological Spaces

1.1 Definition

A topological space is a pair $(X,\tau )$ where $X$ is a set and $\tau \subseteq \mathcal{P}(X)$ is a collection of subsets called open sets, satisfying:

• (T1) $\varnothing \in \tau$ and $X\in \tau$.
• (T2) Arbitrary unions of open sets are open: $\{U_{\alpha }{\}}_{\alpha \in A}\subseteq \tau \Rightarrow {\bigcup }_{\alpha }{U}_{\alpha }\in \tau$.
• (T3) Finite intersections of open sets are open: $U_1,\dots ,U_n\in \tau \Rightarrow {\bigcap }_{i=1}^n{U}_i\in \tau$.

The collection $\tau$ is called the topology on $X$. A subset $C\subseteq X$ is closed if its complement $X\setminus C$ is open.

1.2 Continuity

A map $f:X\to Y$ between topological spaces is continuous if the preimage of every open set is open:

$$U\in {\tau }_Y⟹f^{-1}(U)\in {\tau }_X.$$

For $X=Y=\mathbb{R}$ with the standard topology this is equivalent to the $ϵ$-$\delta$ definition. The advantage of the topological formulation is that it generalizes to arbitrary spaces with no notion of distance.

A homeomorphism is a bijective continuous map with continuous inverse. Two spaces are homeomorphic ($X\cong Y$) if there exists a homeomorphism between them — they are then "the same" topologically.

1.3 Examples

• Discrete topology: $\tau =\mathcal{P}(X)$ — every set is open. All maps from a discrete space are continuous.
• Indiscrete topology: $\tau =\{\varnothing ,X\}$ — only $\varnothing$ and $X$ are open. Few maps to an indiscrete space are continuous.
• Standard topology on ${\mathbb{R}}^n$: open sets are arbitrary unions of open balls $B_r(x)=\{y:\Vert y-x\Vert <r\}$.
• Subspace topology: for $Y\subseteq X$, ${\tau }_Y=\{U\cap Y:U\in {\tau }_X\}$.
• Product topology: for $X\times Y$, generated by sets of the form $U\times V$ with $U\in {\tau }_X,V\in {\tau }_Y$.
• Quotient topology: for an equivalence relation $\sim$ on $X$, declare $U\subseteq X/\sim$ open iff its preimage in $X$ is open.

1.4 Useful properties

• Hausdorff (or $T_2$): for any two distinct points $x,y\in X$, there exist disjoint open sets $U\ni x$ and $V\ni y$. This separates points by neighbourhoods. (Most physically reasonable spaces are Hausdorff.)
• Second countable: there exists a countable collection of open sets $\{U_n{\}}_{n\in \mathbb{N}}$ such that every open set is a union of $U_n$'s. Roughly, "the space is not too big" — necessary to avoid pathologies in manifold theory.
• Connected: $X$ cannot be written as a disjoint union of two non-empty open sets. The Lorentz group has four connected components (see QFT/preliminaries.md).
• Compact: every open cover $\{U_{\alpha }\}$ has a finite subcover. For subsets of ${\mathbb{R}}^n$, compact = closed and bounded (Heine–Borel).
• Path-connected: any two points can be joined by a continuous path $\gamma :[0,1]\to X$.
• Simply connected: path-connected, and every loop $\gamma :S^1\to X$ contracts to a point. The universal cover of a connected space is the simply connected version (see Universal cover in QFT/preliminaries.md).

2. Manifolds

2.1 Topological manifold

A topological manifold of dimension $n$ is a topological space $M$ that is:

1. Hausdorff,
2. Second countable,
3. Locally Euclidean of dimension $n$: every point $p\in M$ has an open neighbourhood $U\subseteq M$ together with a homeomorphism $\varphi :U\to \varphi (U)\subseteq {\mathbb{R}}^n$ to an open subset of ${\mathbb{R}}^n$.

The pair $(U,\varphi )$ is called a chart (or local coordinate system); the components of $\varphi (p)=(x^1(p),\dots ,x^n(p))$ are local coordinates.

The intuition: $M$ "looks like ${\mathbb{R}}^n$ near every point", but globally may have non-trivial topology (a sphere $S^2$ is a 2-manifold, but no single chart covers all of it).

2.2 Atlas and smooth structure

A smooth atlas on $M$ is a collection of charts $\{(U_{\alpha },{\varphi }_{\alpha }){\}}_{\alpha \in A}$ covering $M$ (i.e. ${\bigcup }_{\alpha }{U}_{\alpha }=M$), such that any two charts are smoothly compatible: whenever $U_{\alpha }\cap U_{\beta }\ne \varnothing$, the transition map

$${\varphi }_{\beta }\circ {\varphi }_{\alpha }^{-1}:{\varphi }_{\alpha }(U_{\alpha }\cap U_{\beta })⟶{\varphi }_{\beta }(U_{\alpha }\cap U_{\beta })$$

is a smooth ($C^{\infty }$) map between open subsets of ${\mathbb{R}}^n$.

What "smooth" means. A map $f:V\to W$ between open subsets of ${\mathbb{R}}^n$ and ${\mathbb{R}}^m$ is smooth (or $C^{\infty }$) if all partial derivatives of all orders exist and are continuous. In coordinates, $f$ has components $f^i(x^1,\dots ,x^n)$, and smoothness means ${\partial }^k{f}^i/\partial x^{j_1}\cdots \partial x^{j_k}$ exists and is continuous for all $k\ge 0$. This is the standard multivariable-calculus notion; the manifold framework just lets us say "smooth" on more general spaces by reducing to coordinate patches.

A smooth structure on $M$ is a maximal smooth atlas (one that contains every chart smoothly compatible with all charts in the atlas). A smooth manifold is a topological manifold equipped with a smooth structure.

2.3 Smooth maps between manifolds

Given smooth manifolds $(M,{\mathcal{A}}_M)$ and $(N,{\mathcal{A}}_N)$, a map $f:M\to N$ is smooth if for every chart $(U,\varphi )\in {\mathcal{A}}_M$ and $(V,\psi )\in {\mathcal{A}}_N$ with $f(U)\subseteq V$, the coordinate representation

$$\psi \circ f\circ {\varphi }^{-1}:\varphi (U)\subseteq {\mathbb{R}}^m⟶\psi (V)\subseteq {\mathbb{R}}^n$$

is smooth in the standard ${\mathbb{R}}^m\to {\mathbb{R}}^n$ sense.

A diffeomorphism is a smooth bijective map with smooth inverse. Two smooth manifolds are diffeomorphic if there is a diffeomorphism between them.

2.4 Examples

• ${\mathbb{R}}^n$ itself is a smooth $n$-manifold, with a single global chart $\varphi =\mathrm{id}$.
• The $n$-sphere $S^n=\{x\in {\mathbb{R}}^{n+1}:\Vert x\Vert =1\}$ is a smooth $n$-manifold; it requires (at minimum) two charts (e.g. stereographic projection from north and south poles).
• The torus $T^n=(S^1{)}^n$.
• The real projective space ${\mathbb{RP}}^n$ = lines through the origin in ${\mathbb{R}}^{n+1}$.
• Lie groups: $GL(n,\mathbb{R})$ is an open subset of ${\mathbb{R}}^{n^2}$ and inherits a smooth structure; $SU(n)$, $SO(n)$, etc. are smooth submanifolds carved out by polynomial constraints.
• The graph of any smooth function $f:{\mathbb{R}}^n\to {\mathbb{R}}^m$ is a smooth $n$-manifold sitting inside ${\mathbb{R}}^{n+m}$.

2.5 Manifolds with boundary

Replace "locally Euclidean of dimension $n$" with "locally homeomorphic to either ${\mathbb{R}}^n$ or the half-space ${\mathbb{H}}^n=\{x\in {\mathbb{R}}^n:x^1\ge 0\}$". Points mapped to $\partial {\mathbb{H}}^n$ form the boundary $\partial M$, itself a manifold of dimension $n-1$. Examples: the closed disk ${\bar{D}}^2$, the closed interval $[0,1]$.

3. Tangent Spaces

At each point $p\in M$ of a smooth $n$-manifold there is a vector space $T_{p}M$ of dimension $n$, the tangent space at $p$. Three equivalent definitions, each useful in different contexts:

3.1 Velocities of curves

A smooth curve through $p$ is a smooth map $\gamma :(-ϵ,ϵ)\to M$ with $\gamma (0)=p$. Two such curves are equivalent (${\gamma }_1\sim {\gamma }_2$) if for any chart $(U,\varphi )$ around $p$,

$$\frac{d}{dt}{|}_{t=0}\varphi ({\gamma }_1(t))=\frac{d}{dt}{|}_{t=0}\varphi ({\gamma }_2(t))\text{in }{\mathbb{R}}^n.$$

The tangent space $T_{p}M$ is the set of equivalence classes $[\gamma ]$.

3.2 Derivations

A derivation at $p$ is a linear map $X:C^{\infty }(M)\to \mathbb{R}$ satisfying the Leibniz rule

$$X(fg)=X(f)g(p)+f(p)X(g).$$

The space of all derivations at $p$ is $T_{p}M$. In a chart with coordinates $x^i$, every derivation is uniquely a linear combination

$$X=X^i\frac{\partial }{\partial x^i}{|}_p,X^i\in \mathbb{R},$$

so the partial derivatives $\partial /\partial x^i{|}_p$ form a basis.

3.3 Coordinate vectors

In a chosen chart, a tangent vector is just an $n$-tuple $(X^1,\dots ,X^n)\in {\mathbb{R}}^n$, with the transformation rule under change of chart $x^i=x^i(x^j)$:

$$X^i=\frac{\partial x^i}{\partial x^j}{X}^j.$$

This is the "physicist's definition" — a tangent vector is "something with an upper index that transforms as a contravariant vector".

3.4 The tangent bundle

The disjoint union $TM={⨆}_{p\in M}{T}_{p}M$ is itself a smooth manifold of dimension $2n$, called the tangent bundle. A vector field is a smooth map $X:M\to TM$ with $X(p)\in T_{p}M$ — equivalently, a smooth section of the projection $TM\to M$.

4. Lie Groups (where physics meets topology)

A Lie group is a smooth manifold $G$ that is also a group, such that the group operations

$$\mu :G\times G\to G,\mu (g,h)=gh,\iota :G\to G,\iota (g)=g^{-1}$$

are smooth maps. Equivalently, multiplication and inversion are differentiable when expressed in local coordinates.

For matrix Lie groups (subgroups of $GL(n,\mathbb{F})$ defined by smooth equations on matrix entries — see group-theory.md § Standard matrix groups), the definition reduces to: the group elements depend smoothly on a finite number of real parameters, and so do products and inverses. Concretely:

• $SO(n)$, $SU(n)$, $SL(n,\mathbb{F})$ are all smooth submanifolds of the appropriate $GL$, defined by smooth polynomial equations ($\det =1$, $M^{T}M=1$, etc.).
• The Lorentz group $O(1,3)$ is a smooth submanifold of $GL(4,\mathbb{R})$ defined by ${\Lambda }^T\eta \Lambda =\eta$.
• The Poincaré group is the semidirect product of ${\mathbb{R}}^{1,3}$ (a manifold) and ${\mathcal{L}}_{+}^{\uparrow }$ (a manifold) — itself a manifold.

Lie algebra as the tangent space at the identity

The Lie algebra $\mathfrak{g}$ of $G$ is by definition

$$\mathfrak{g}\equiv T_{e}G,$$

the tangent space at the identity element $e\in G$. It is a vector space of dimension $\dim G$, equipped with a Lie bracket $[\cdot ,\cdot ]:\mathfrak{g}\times \mathfrak{g}\to \mathfrak{g}$ inherited from the group structure (for matrix Lie groups, this is the matrix commutator $[X,Y]=XY-YX$).

The exponential map $\exp :\mathfrak{g}\to G$ takes a Lie algebra element to a one-parameter subgroup. See group-theory.md § Lie groups and Lie algebras for the physics-oriented treatment using these definitions.

5. Why physics gets away with skipping all this

Almost every Lie group used in physics is a matrix group — a closed subgroup of $GL(n,\mathbb{F})$ for some $n,\mathbb{F}$. For matrix groups:

• "Smooth manifold structure" reduces to "matrix entries depend smoothly on parameters", which is intuitively obvious from calculus.
• The Lie algebra is a subspace of the matrices $M_n(\mathbb{F})$, with bracket = commutator.
• The exponential map is the matrix exponential $\exp (X)={\sum }_k{X}^k/k!$.

So most QFT texts (Peskin–Schroeder, Schwartz, Srednicki, Weinberg, Mandl–Shaw, Georgi) skip the manifold formalism entirely and proceed by example, defining each Lie group concretely as a matrix group. This document exists for the reader who wants the formal definitions filled in.

When the manifold structure does become essential:

• General relativity: spacetime is a non-trivial 4-manifold, and the entire formalism of GR (covariant derivatives, curvature, geodesics) is differential geometry on this manifold. See e.g. Carroll, Spacetime and Geometry; Wald, General Relativity.
• Gauge theory / fibre bundles: the global structure of gauge fields (instantons, monopoles, anomalies) requires viewing gauge fields as connections on principal bundles, which needs full manifold theory. See Nakahara, Geometry, Topology and Physics.
• Topological QFT: the entire framework lives on manifolds, where the topology of the manifold is the central object of study.

6. Pointers

• This repository: math/group-theory.md, QFT/preliminaries.md § Lorentz and Poincaré Groups.
• Lee, Introduction to Topological Manifolds / Introduction to Smooth Manifolds: standard mathematics references.
• Munkres, Topology: standard reference for point-set topology (Chs. 2–4).
• Nakahara, Geometry, Topology and Physics: physics-friendly comprehensive treatment.
• Bredon, Topology and Geometry: a more advanced unified reference.