This course develops the basic language of fibre bundles as geometric objects that look locally like products but may have nontrivial global structure. It begins with the notion of local triviality and bundle atlases, then explains how transition functions encode the data needed to glue local pieces into a global bundle. From there, the course studies sections as the natural objects living inside a bundle and uses them to formulate local-to-global questions that recur throughout geometry and topology.

The central themes are classification, functoriality, and the interaction between local descriptions and global invariants. After establishing the elementary bundle formalism, the course moves to operations on vector bundles, bundle metrics, partitions of unity, and bundle maps, showing how these tools make the category of bundles workable and natural under pullback. It then develops classifying maps in simple cases, clutching constructions, and the sheaf of sections viewpoint, before turning to examples beyond vector bundles and the obstructions that prevent trivialization.

The later chapters synthesize these ideas by comparing different ways of building bundles and recognizing when two presentations define the same object. By the end, the student should be able to move fluently between atlases, transition data, sections, and maps, and to understand how global bundle phenomena arise from local geometric information.

# Introduction

This opening chapter sets the direction for the course before the technical development begins. Fibre bundles are a language for spaces that look like products locally but may fail to be products globally. The central theme is that the failure of a global product structure is not mysterious: it is recorded by transition functions, cocycle identities, and the way local sections transform across overlaps.

The course assumes familiarity with smooth manifolds, tangent and cotangent bundles, basic point-set topology, linear algebra, and partitions of unity. Many familiar geometric objects already have the shape of a bundle; the point of the first lectures is to isolate the structure they share and to make the local-to-global mechanism precise.

## Why Bundles Appear

What goes wrong if every geometric family over a space is treated as a product? A product $B \times F$ has a fixed fibre $F$ over every point of $B$, but it also has a distinguished global identification of all fibres with the same model. Geometry often gives the first feature without the second.

The tangent spaces $T_pM$ of a smooth manifold $M$ all have dimension $n$ when $M$ is $n$-dimensional, so each is abstractly isomorphic to $\mathbb R^n$. There is usually no preferred way to identify $T_pM$ with $T_qM$ for distinct points $p,q \in M$. This tension is the basic reason for introducing fibre bundles.

[example: Tangent Spaces as a Local Product] Let $M$ be a smooth $n$-manifold and let $(U,\varphi)$ have coordinates $(x_1,\dots,x_n)$. For $p\in U$, the coordinate vector fields form a basis of $T_pM$, so every $v\in T_pM$ has a unique expression \begin{align*} v=\sum_{j=1}^n a^j\frac{\partial}{\partial x_j}\Big|_p. \end{align*} This gives a local product identification \begin{align*} \Phi_U:\pi^{-1}(U)\to U\times \mathbb R^n,\qquad \Phi_U(v)=\bigl(p,(a^1,\dots,a^n)^\top\bigr). \end{align*} Now let $(V,\psi)$ be another coordinate chart with coordinates $(y_1,\dots,y_n)$, and fix $p\in U\cap V$. On the overlap, each $y_i$ is a smooth function of the $x$-coordinates, so the chain rule gives \begin{align*} \frac{\partial}{\partial x_j}\Big|_p=\sum_{i=1}^n\frac{\partial y_i}{\partial x_j}(p)\frac{\partial}{\partial y_i}\Big|_p. \end{align*} Substituting this basis change into the expression for $v$ gives \begin{align*} v=\sum_{j=1}^n a^j\sum_{i=1}^n\frac{\partial y_i}{\partial x_j}(p)\frac{\partial}{\partial y_i}\Big|_p. \end{align*} Reordering the finite sums, \begin{align*} v=\sum_{i=1}^n\left(\sum_{j=1}^n\frac{\partial y_i}{\partial x_j}(p)a^j\right)\frac{\partial}{\partial y_i}\Big|_p. \end{align*} Thus, if $(a^1,\dots,a^n)^\top$ is the $x$-coordinate column of $v$ and $(b^1,\dots,b^n)^\top$ is the $y$-coordinate column, then \begin{align*} b^i=\sum_{j=1}^n\frac{\partial y_i}{\partial x_j}(p)a^j. \end{align*} Equivalently, \begin{align*} (b^1,\dots,b^n)^\top=\left(\frac{\partial y_i}{\partial x_j}(p)\right)_{i,j}(a^1,\dots,a^n)^\top. \end{align*} So $TM$ is locally identified with $U\times\mathbb R^n$, but two such identifications are glued on overlaps by the Jacobian matrix of the coordinate change, not by a single global choice of coordinates. [/example]

The example also hints at the two layers of the subject. First we need a topological or smooth object whose fibres vary locally like a product. Then we need to understand how changing local product descriptions records the global geometry.

[definition: Informal Fibre Bundle] A fibre bundle consists of a total space $E$, a base space $B$, a projection map $\pi:E\to B$, a model fibre $F$, an open cover $(U_i)_{i\in I}$ of $B$, and local identification maps \begin{align*} \phi_i:\pi^{-1}(U_i)\to U_i\times F \end{align*} such that $\operatorname{pr}_1\circ \phi_i=\pi|_{\pi^{-1}(U_i)}$ for every $i\in I$. [/definition]

This is only an informal definition because the precise version must specify smoothness, compatibility with projections, and the regularity of the changes of local product coordinates. Chapter 1 turns this informal picture into the formal language of smooth fibre bundles, local trivializations, and bundle atlases.

## Local Data and Global Geometry

How can local product charts determine a global object? The answer is that two local product descriptions over overlapping open sets must be compared, and these comparisons satisfy consistency identities on triple overlaps. These comparison maps are the transition functions of the bundle.

[example: The Mobius Band as a Twisted Line Bundle] Realize the Mobius band as \begin{align*} E=([0,2\pi]\times \mathbb R)/\bigl((0,r)\sim(2\pi,-r)\bigr), \end{align*} with projection $\pi([\theta,r])=e^{i\theta}$. Let $U_0=S^1\setminus\{-1\}$ and $U_1=S^1\setminus\{1\}$. Then $U_0\cap U_1$ has two connected components \begin{align*} C_+=\{e^{i\theta}:0<\theta<\pi\} \end{align*} and \begin{align*} C_-=\{e^{i\theta}:\pi<\theta<2\pi\}. \end{align*} Choose the local product description over $U_1$ by \begin{align*} \phi_1([\theta,r])=(e^{i\theta},r),\qquad 0<\theta<2\pi. \end{align*} Over $U_0$, use the angle interval $-\pi<\alpha<\pi$ and set \begin{align*} \phi_0([\alpha,r])=(e^{i\alpha},r). \end{align*} On $C_+$, the same point has $0<\theta<\pi$ and $\alpha=\theta$, so \begin{align*} \phi_0\circ \phi_1^{-1}(e^{i\theta},r) =\phi_0([\theta,r]) =(e^{i\theta},r). \end{align*} On $C_-$, write $\alpha=\theta-2\pi$. Then $-\pi<\alpha<0$, $e^{i\alpha}=e^{i\theta}$, and the Mobius identification gives \begin{align*} [\theta,r]=[\alpha+2\pi,r]=[\alpha,-r]. \end{align*} Therefore \begin{align*} \phi_0\circ \phi_1^{-1}(e^{i\theta},r) =\phi_0([\theta,r]) =\phi_0([\alpha,-r]) =(e^{i\theta},-r). \end{align*} Thus the transition function $g_{01}:U_0\cap U_1\to \mathbb R^\times$ satisfies $g_{01}(p)=1$ for $p\in C_+$ and $g_{01}(p)=-1$ for $p\in C_-$. The fibres are locally copies of $\mathbb R$, but the sign change between the two overlap components records the half-twist that prevents the Mobius band from being the product $S^1\times\mathbb R$. [/example]

The Mobius band is useful because the fibre is as simple as possible, yet the total space is not globally $S^1\times \mathbb R$. It also raises a new structural question: what extra algebraic conditions should be imposed when the fibre carries vector-space operations? This leads from general fibre bundles to vector bundles, where the transition functions must respect linear structure.

[definition: Vector Bundle in Preview] A rank $k$ smooth vector bundle over a smooth manifold $B$ is a smooth fibre bundle $\pi:E\to B$ with model fibre $\mathbb R^k$, local trivialisations \begin{align*} \phi_i:\pi^{-1}(U_i)\to U_i\times \mathbb R^k \end{align*} over an open cover $(U_i)_{i\in I}$ of $B$, and transition maps \begin{align*} g_{ij}:U_i\cap U_j\to GL(k,\mathbb R) \end{align*} for all $i,j\in I$ with $U_i\cap U_j\ne\varnothing$, defined by \begin{align*} \phi_i\circ\phi_j^{-1}(b,v)=(b,g_{ij}(b)v). \end{align*} [/definition]

The linearity condition makes vector bundles suitable for differential geometry. It lets us add sections, multiply them by smooth functions, form dual bundles, and construct tensor bundles.

[example: Tautological Line Bundle Over Real Projective Space] Let $\mathbb{RP}^n$ be the set of lines through $0$ in $\mathbb R^{n+1}$, and let \begin{align*} \gamma^1=\{(\ell,v):\ell\in \mathbb{RP}^n,\ v\in \ell\} \end{align*} with projection $\pi(\ell,v)=\ell$. For each $i$, define \begin{align*} U_i=\{\ell\in \mathbb{RP}^n:\text{some, equivalently every, nonzero }z\in \ell\text{ has }z_i\ne 0\}. \end{align*} On $U_i$, every line $\ell$ has a unique representative $u_i(\ell)\in \ell$ whose $i$th coordinate is $1$: if $z=(z_0,\dots,z_n)\in \ell$ and $z_i\ne 0$, then \begin{align*} u_i(\ell)=\frac{1}{z_i}z. \end{align*} If $z'=cz$ is another nonzero representative of the same line, with $c\ne 0$, then $z_i'=cz_i$ and \begin{align*} \frac{1}{z_i'}z'=\frac{1}{cz_i}(cz)=\frac{1}{z_i}z, \end{align*} so $u_i(\ell)$ is independent of the chosen representative. Since $\ell$ is one-dimensional and $u_i(\ell)\ne 0$, every $v\in \ell$ has a unique form \begin{align*} v=a\,u_i(\ell) \end{align*} for some $a\in \mathbb R$. This gives the local product map \begin{align*} \phi_i:\pi^{-1}(U_i)\to U_i\times \mathbb R,\qquad \phi_i(\ell,v)=(\ell,a) \end{align*} where $v=a\,u_i(\ell)$. Now take $\ell\in U_i\cap U_j$. Write \begin{align*} u_i(\ell)=(u_i^0,\dots,u_i^n). \end{align*} Because $u_i(\ell)$ has $i$th coordinate $1$ and $\ell\in U_j$, its $j$th coordinate $u_i^j$ is nonzero. The representative with $j$th coordinate $1$ is therefore \begin{align*} u_j(\ell)=\frac{1}{u_i^j}u_i(\ell). \end{align*} Multiplying both sides by $u_i^j$ gives \begin{align*} u_i(\ell)=u_i^j\,u_j(\ell). \end{align*} If $\phi_i(\ell,v)=(\ell,a)$, then \begin{align*} v=a\,u_i(\ell). \end{align*} Substituting $u_i(\ell)=u_i^j\,u_j(\ell)$ gives \begin{align*} v=a\,u_i^j\,u_j(\ell). \end{align*} Thus the $j$-coordinate of the same vector is $u_i^j a$, and \begin{align*} \phi_j\circ \phi_i^{-1}(\ell,a)=(\ell,u_i^j a). \end{align*} The transition function on $U_i\cap U_j$ is multiplication by the nonzero scalar $u_i^j$, so the fibre over $\ell$ is the actual line $\ell\subset \mathbb R^{n+1}$, while the local identification with $\mathbb R$ changes by rescaling when the normalized representative of $\ell$ changes. [/example]