**1. Introduction**

One of the things I find most fascinating is the interplay between mathematics and physics. One direction of this interplay is well-known i.e., how mathematics forms the basis of all physical concepts. In fact, this is now so ingrained in our minds that we naturally expect any new physical concept to have an appropriate mathematical description. Wigner calls this the “unreasonable effectiveness of mathematics” in his famous article. What is surprising, perhaps, is the other direction i.e., how physical intuition leads to new developments in mathematics. This is especially true with quantum physics and geometry.

Quantum field theories such as Yang-Mills and Chern-Simons theories provide a deep understanding of certain aspects of topology and geometry. Starting around in the 80s, these developments have led to the creation of the field of quantum topology and quantum algebra. Some of these results, which took mathematicians by surprise, were obtained from deep physical intuition. These developments are beautifully explained in a 2010 article by Atiyah, Dijkgraaf and Hitchin. For instance, understanding the space of the so called self-dual solutions of Yang-Mills theories leads to powerful invariants of four manifolds. Similarly, studying the holonomy of Wilson loop operators and their vacuum expectation values gives a new description of the Jones polynomial.

A basic and extremely important structure in mathematical physics and in the results mentioned above is that of a vector bundle. Several physical theories such as Yang-Mills theory and Chern-Simons theory can be cast in terms of vector bundles. Moreover, natural bundles on manifolds such as the tangent and cotangent bundles are useful in descriptions of general relativity and the phase space of classical mechanical systems (the latter, in fact, leads to the very useful concept of geometric quantization, which produces a quantum mechanical Hilbert space from the classical mechanical phase space). The concept of holonomy in a vector bundle, which measures the deviation produced in a standard vector when it is parallel transported around a loop, turns out to be the underlying mathematical description of the Aharonov-Bohm effect and the quantum field theoretic formulation of the Jones polynomial. These are just some examples of the use of vector bundles.

In this post, I want to discuss some aspects of vector bundles that are useful in the above applications. Aside from the definition and examples, I will talk about how finite dimensional vector bundles can be classified using homotopy.

**2. Definitions**

**2.1. Vector bundles**

A dimensional vector bundle over a -dimensional manifold can be specified as a triple , where is a projection map such that for every point on , the inverse image of is a finite dimensional vector space over some field. In addition, we need two other conditions.

**Local triviality.**This condition states that there exists an open cover of the manifold such that when the bundle is restricted to an open set from the cover, it is isomorphic to the trivial bundle i.e., there is a map (when is a dimensional real vector space).**Continuity of transition functions.**We also require that the maps defined above be homeomorphisms i.e., they are continuous maps of topological spaces. This means that on intersections of open sets of the cover, if we define the functions , we have that are homeomorphisms. The (with ) are called*transition*or*clutching*functions since they tell us how to identify the trivial bundles on different open sets of the cover. The group they generate is called the*structure*group.

It is easy to see that for a local trivialization , the transition functions also satisfy the cocycle condition i.e., is the identity. Since the transition functions are essentially invertible matrices, we can define the equivalence of two sets of transition functions and as the equivalence of the matrices representing them. In other words, and are equivalent if, on each open set , there are functions such that . The *structure group* of the vector bundle is the group in which the transition functions lie. Usually, this is . However, sometimes the structure group can be made into a subgroup of . Using the equivalence of transition functions, we can define the reduction of the structure group of a vector bundle. Suppose for a set of transition functions, we can find equivalent ones that lie in a subgroup of , then we say that the structure group is reduced (to that subgroup).

**2.2. Sections**

An important concept pertaining to a vector bundle is the notion of a section of a bundle. This notion and its properties naturally lead one to consider generalizations of vector bundles called *sheaves*. A section of a vector bundle is a map such that i.e., applied to any point takes it to a vector in the space over . Sections are important because they contain important information about the vector bundle. An example of a section is a vector field on a manifold, where the vector bundle is the tangent bundle of the manifold. Sections are useful to figure out if two vector bundles are isomorphic or not. For example, over the circle , one can construct two line bundles–the trivial bundle or the Möbius bundle defined below. Using the notion of the zero section (or rather, its complement), we can show that these two bundles are not isomorphic. The zero section is a map from to that simply assigns the zero vector to every point on the manifold (i.e., the zero vector in the vector space above that point). This map need not always be smooth. However, for the two vector bundles over , it is smooth. The complement of the zero section for the trivial bundle over is clearly not connected, where as in the Möbius bundle, it is connected. This means that they are not isomorphic as vector bundles. In fact, using the classification of vector bundles discussed later, we will see that these are the only possible non-isomorphic line bundles over .

In the discussion above, we have been talking about real vector bundles, but all of the definitions and concepts can be extended to complex vector bundles (or more specifically holomorphic vector bundles). To obtain complex vector bundles, we need to replace (in the definition of vector bundles above) the base manifold and the fibers by their complex counterparts. If we make the maps and holomorphic (i.e., functions that satisfy the Cauchy-Riemann conditions), then we get holomorphic vector bundles.

**3. Examples**

- The trivial bundle , where the map is simply the restriction to the first factor in the direct product. It is easy to verify that it satisfies the two conditions in the definition of a vector bundle.
- The tangent (and co-tangent) bundles on a manifold are, in a sense, the “first” vector bundles. Arbitrary vector bundles are generalizations of these.
- Möbius line bundle is a quotient of the trivial bundle by the equivalence relation which identifies with , where . This is one of the vector bundles over discussed above.
- The Grassmannian is the space of dimensional subspaces of . In other words, any point on the Grassmannian manifold is a dimensional vector space . One can define a bundle called the tautological bundle by taking the fiber over any point in the Grassmannian as the vector space itself.
- Another important class of bundles are
*pullback*or*induced*bundles. Suppose there is a map between two manifolds , then one can pullback any vector bundle on to as follows. The fiber over any point is defined as , where is the projection map of the bundle over . - Using the standard constructions over vector spaces such as direct sums, tensor products, complements etc., one can define equivalent constructions of vector bundles over a base manifold by defining them fiber-wise.
- The direct sum or
*Whitney*sum of vector bundles can also be thought of as the pullback bundle of the diagonal embedding map taking to .

**4. Classification of bundles**

In order to understand the different types of vector bundles one can construct over a manifold, we need to know what it means to say that two bundles are the same i.e., we need the notion of isomorphism of vector bundles. A morphism of two vector bundles and is a morphism of the base spaces and a fiber-wise linear map between the total spaces. In other words, it is a pair of maps such that the following diagram commutes.

Here the vertical arrows are the projection maps of the respective vector bundles. The map is defined fiber-wise. When restricted to any fiber over a point in , it is a *linear* map to the space over the point in . Using this, we can define isomorphism of vector bundles as a morphism such that the composition of it and its inverse is the identity. Once we have a notion of isomorphism of bundles, we would like to know how to classify them up to equivalence.

It turns out that the Grassmannian and its tautological bundle defined above play an important role in the classification of finite dimensional vector bundles. First, notice that we can extend the definition of the Grassmannian above to subspaces of an infinite dimensional vector space i.e., define is the set of -dimensional subspaces of . Similarly, we have the complex Grassmannian .

The first step is to show that homotopic maps between two manifolds lead to isomorphic induced bundles. To be more concrete, suppose that and are two manifolds and let be a vector bundle over with a projection map . Let and be two maps from to that are homotopic. Each of these maps induces a pullback bundle (described above) from to . Let us denote these two bundles on as and . One can show that these two vector bundles over are isomorphic.

Using this, it can be shown that the equivalence classes of -dimensional vector bundles over any compact (or paracompact) topological space are in one-to-one correspondence with equivalence classes of maps (under homotopy) from to . Put another way, this means that every equivalence class of -dimensional vector bundles over a manifold can be obtained as a pullback bundle from the tautological bundle over via some equivalence class of homotopic maps. The space of equivalence classes of maps from to is denoted . So for any manifold , if we have a good understanding of the space , then we would also have an understanding of all possible vector bundles over .

One can apply this machinery to line bundles over and prove that there are only two isomorphism classes of complex line bundles. Using the classification theorem, we need to know the dimension of the space . First, note that is the infinite projective space ). So we need to find out the dimension of the space . This can be done using Eilenberg-McLane spaces and their properties.

Denoted , an Eilenberg-McLane space is one that has the homotopy group isomorphic to and others trivial. These spaces can be used to construct more complicated spaces with non-trivial homotopy groups of all orders. An important property of Eilenberg-McLane spaces is that i.e., the equivalence classes of maps from a manifold to an Eilenberg-McLane space is isomorphic to the singular cohomology group. It turns out that the infinite projective space is an Eilenberg-McLane space, . This means that . For the manifold , can be calculated to be equal to . Thus one can deduce that there are only two isomprphism classes of complex line bundles. In terms of characteristic classes, this just turns out to be the first Chern class and these line bundles are completely characterized by the first Chern class.