# Introduction and Basic Definitions

We introduce ${\displaystyle C^{\infty }}$ vector bundles, sections and other basic definitions before building up to the Chern classes of line bundles through a sequence of examples.

## What are Chern classes?

It is usually a difficult problem to classify all non-isomorphic vector bundles over a given fixed base space. The idea of characteristic classes is to provide a topological invariant which allows us to distinguish some classes of vector bundles. Characteristic classes are elements of the cohomology groups of the base space. Chern classes are particular characteristic classes which we associate to complex vector bundles. They are a sequence of functions ${\displaystyle c_{1},c_{2},\ldots }$ assigning to each complex vector bundle ${\displaystyle E\rightarrow B}$ a class ${\displaystyle c_{i}(E)\in H^{2i}(B;\mathbb {Z} )}$ depending only on the isomorphism type of ${\displaystyle E}$ (and are the unique such sequence satisfying a list of properties which we'll list towards the end). The Chern classes are formally akin to the Stiefel-Whitney classes, which are defined for real vector bundles. The Stiefel-Whitney classes take coefficients in ${\displaystyle \mathbb {Z} _{2}}$.

## Vector Bundles

In the course of this lecture we denote by ${\displaystyle M}$, a smooth manifold of real dimension ${\displaystyle n}$. A rank ${\displaystyle r}$ smooth complex vector bundle is a map ${\displaystyle \pi :E\rightarrow M}$ together with a complex vector space structure on the fibres${\displaystyle \pi ^{-1}(b)}$ for each ${\displaystyle m\in M}$ such that we have the following local triviality condition: for each ${\displaystyle m\in M}$ there exists an open neighbourhood ${\displaystyle U_{m}}$ of ${\displaystyle m}$ such that there is a diffeomorphism ${\displaystyle d_{m}:\pi ^{-1}(U_{m})\rightarrow U_{m}\times \mathbb {C} ^{r}:\pi ^{-1}(b)\mapsto \{b\}\times \mathbb {C} ^{r}}$ which is a vector space isomorphism. We call ${\displaystyle d_{m}}$ the local trivialisation of the vector bundle at ${\displaystyle m}$. We call ${\displaystyle E}$ the total space and call ${\displaystyle M}$ the base space of the vector bundle. For convenience, we refer to the bundle simply as ${\displaystyle E}$ with all of the other data above quietly suppressed.

A smooth section of the vector bundle ${\displaystyle E}$ is a smooth right inverse of ${\displaystyle \pi }$. The trivial bundle over ${\displaystyle M}$ is the vector bundle ${\displaystyle M\times \mathbb {C} ^{r}}$. Any vector bundle ${\displaystyle E}$ is said to be trivial if it is isomorphic to the trivial bundle.

Exercise 4.1

Show how to put the real line bundle structure on the open Mobius band

Exercise 4.2

Show that ${\displaystyle L^{\otimes 2}}$ is isotopic in ${\displaystyle \mathbb {R} ^{4}}$ to the trivial embedding of the cylinder.

Remark

Bundles on contractible manifolds are trivial

Sketch proof: After trivialising at some fibre ${\displaystyle E_{x}}$ lying over ${\displaystyle x}$, we extend the trivialisation along the homotopy that contracts M to ${\displaystyle x}$ using a version of Tietze's Extension theorem. We then lift the homotopy to E so that it covers the homotopy on ${\displaystyle M}$.