Introduction

Chern classes are part of algebraic topology, as well as other math groups, and are characteristic classes related to complex vector bundles. The first Chern class is the only nontrivial Chern class and is an element of the second cohomology group of \(X\), where \(X\) is a topological space of closure-finite weak CW complex.

Chern Classes Defined

The Chern classes \(c_{i}(\omega)\in H^{2i}(B;\mathbb{Z})\), where \(B\) is a paracompact base space admitting a Hermation metric, are found through induction on the complex dimension \(n\) of the complex real vector bundle \(\omega\), defined with in following conditionals. The top Chern class \(c_{n}(\omega)\) is equivalent to the Euler class \(e(\omega\mathbb{R})\). For any given Chern class less than the top Chern classes, where \(n > i\), it is defined as:

\(c_{i}(\omega)=\pi_{0}^{*-1}c_{i}(\omega_{0})\), s.t.

\(\pi_{0}^{*-1}:H^{2i}(B)\rightarrow H^{2i}(E_{0})\)

where \(E_{0}\) denotes the deleted total space in the real case of \(E_{0}(\omega)\) and \(\pi_{0}^{*-1}\) is an isomorphism for \(n > i\).

Lastly, for all \(c_{i}(\omega)\) where \(i > n\), the Chern classes are defined as zero.

The formal sum of \(c_{\omega}=1+c_{1}(\omega)+c_{2}(\omega)...+c_{n}(\omega)\) for the given ring \(H^{\Pi}(B;\mathbb{Z})\) is termed the total Chern class of the complex \(n\)-plane bundle \(\omega\).

Construction of Chern Classes

For the construction of chern classes, let's give an inductive definition for the characterstics of classes for a complex \(n\)-plane bundle \(\omega\). Note, that there exists an underlying real vector bundle \(\omega_{\mathbb{R}}\) which has a prefered canonical orientation for the complex vector bundle \(\omega\). Where the oriented bundle to \((\omega\oplus w')_{\mathbb{R}}\) is an isomoprhic bundle, but only if \(\omega'\) is a complex \(m\)-plane bundle over the same basis as \(\omega\).

Consider an inductive definition of characterstics classes for a complex \((n-1)\)-plane bundle. To start the construction, first, we need to build a canonical \((n-1)\)-plane bundle \(\omega_{0}\) over \(E_{0}\), which denotes the deleted total space. The deleted total space in the real case \(E_{0}=E_{0}(\omega)\), dentoes the set of all of the non-zero vectors in the total space \(E_{0}=E_{0}(\omega_{\mathbb{R}})\). There is a point on the set of all non-zero vectors \(E_{0}\) is specified with a fiber \(F\) of \(\omega\) and a non-zero vector \(v\) in that fiber \(F\). Given a Hermatian metric defined on \(\omega\), the fiber of \(\omega_{0}\) over the non-zero vector \(v\) in the total space of non-zero vectors, by defintion, is orthogonal complement of the vector \(v\) in the vector space \(F\), by definition.

This new (\(n-1\)) dimensional complex vector space is where all vector spaces can be considered as new vector bundle \(w_{0}\) over \(E_{0}\).

Product Theorem for Chern Classes

The following formula shows that the total Chern class of Whiteny sum of \(\omega\oplus\phi\) is equivalant to the total Chern clesses of \(\omega\) and \(\phi\):

\(c(\omega\oplus\phi)=c(\omega)c(\phi)\)

where \(\omega\) and \(\phi\) are two complex vector bundles over a shared paracompact base space \(B\).

Glossary