# Algebraic Topology – Vector Bundles, Fibrations, and Classifying Spaces

Originally I was thinking of doing a series of posts to develop some of the basic ideas of topological ${K}$-theory (which is still likely going to happen), but it occurred to me that there were many interesting aspects of the theory of vector bundles (or, more generally, ${G}$-bundles or, even more generally, fibrations) that wouldn’t relate to that.  This post aims to introduce the concept of vector bundles and see how they fit in to a broader context of algebraic topology.

Definition.vector bundle over a topological space ${B}$ is a (continuous) map ${p: E \mapsto B}$ satisfying the following properties:

(1) For any ${b \in B}$, the pre-image ${p^{-1}(B)}$ is homeomorphic to ${\mathbb{R}^n}$.  (The pre-image of a point is called a fiber)

(2) (Local Triviality) For any ${b \in B}$, there is an open neighborhood ${U_b}$ around that point such that ${p^{-1}(U_b)}$ is homeomorphic to ${U_b \times \mathbb{R}^n}$

${B}$ is called the base space and ${E}$ is called the total space.

Examples of vector bundles are fairly easy to come by.  We have, for example, the tangent bundle of a manifold, which can easily shown to satisfy the two conditions above.  We will re-visit tangent bundles several times in future blog posts.  Another important example is the Möbius bundle over ${S^1}$.  Intuitively, this assigns a line to each point on the unit circle in such a way that it “twists” once before it comes around the circle (geometrically, this would look like a Möbius strip, hence the name).  This example is important because it is one of the easiest example of a non-trivial line bundle (that is, it doesn’t look globally like ${S^1 \times \mathbb{R}}$).

A mild technical note: in general, we will be assuming that our base spaces are compact and Hausdorff, though a weaker assumption (such as paracompactness) is generally sufficient to get all of the properties that we want.

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While there is a lot more to be said about the theory of vector bundles (which will be the case in future blog posts), instead I will focus on two related notions.  The first is the theory of fibrations.  This generalizes the idea of a vector bundle in a way that yields useful results in homotopy theory.

Definition. fiber bundle is defined in the same way as a vector bundle, but instead of requiring that the fibers be homeomorphic to ${\mathbb{R}^n}$, we allow them to be any topological space ${F}$.  (Of course this also has to satisfy the local triviality condition)

While fiber bundles are useful, we want a slightly more specific homotopy theoretic property:

Definition. A map ${p: E \mapsto B}$ is said to satisfy the homotopy lifting property if, for any space ${X}$, a homotopy ${H: X \times I \mapsto B}$ lifts (not necessarily uniquely) to a homotopy ${\tilde{H}:X \times I \mapsto E}$ satisfying ${p \circ \tilde{H} = H}$.  Such a map is called a fibration.

A reader familiar with some of the basic ideas of homotopy theory might recognize something familiar in these definitions: the idea of a covering space satisfies these properties exactly.  An ${n}$-sheeted covering space if precisely a ${\mathbb{Z}/n\mathbb{Z}}$-fibration.  The most important property of fibrations (from the viewpoint of homotopy theory) is the following:

Claim. Given a fibration ${F \hookrightarrow E \mapsto B}$, there is a long exact sequence of homotopy groups:

${\ldots \rightarrow \pi_n(F) \rightarrow \pi_n(E) \rightarrow \pi_n(B) \rightarrow \pi_{n-1}(F) \rightarrow \ldots}$

The first two homomorphisms are the obvious ones induced by the fibration maps.  The third one can be obtained through some diagram chasing (in a way analogous to the Snake Lemma).

Example. The Hopf fibration is historically an important example of a fibration.  We can think of ${S^3}$ as ${\{(z_1,z_2) \in \mathbb{C}^2 : |z_1|^2 + |z_2|^2 = 1\}}$.  In an analogous way we can define ${S^2 = \mathbb{C}P^1}$ as equivalence classes of pairs of complex numbers with ${(z_1,z_2) \sim (z_3,z_4)}$ if and only if ${z_1z_4 = z_2z_3}$.  We can think of this equivalence relation as sending a pair of complex numbers to their quotient with an additional point at infinity – the one point compactification of ${\mathbb{C}}$.  This, then, naturally induces a map ${p: S^3 \mapsto S^2}$ sending a pair of complex numbers to their equivalence class.  The pre-image of a given point is the set of complex numbers with norm 1, which is precisely ${S^1}$.  This is the Hopf fibration, and it leads to the following result on homotopy groups:

Claim. For ${n \geq 3}$, we have ${\pi_n(S^2) \approx \pi_n(S^3)}$.  In particular, ${\pi_3(S^2) \approx \mathbb{Z}}$.

Proof. We can directly apply our long exact sequence, along with the fact that al higher homotopy groups of ${S^1}$ are trivial, to get the following short exact sequence:

${0 \rightarrow \pi_n(S^3) \rightarrow \pi_n(S^2) \rightarrow 0}$

Because this is exact, the groups must be isomorphic.  Using the fact that ${\pi_n(S^n) \approx \mathbb{Z}}$ for all ${n}$ yields the desired result.

This is just one example of the usefulness of fibrations.  (Ironically, from the perspective of homotopy theory, vector bundles are one of the least interesting example of fibrations because the homotopy groups of ${\mathbb{R}^n}$ are not of much interest.)

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The final area I would like to discuss is the theory of classifying spaces.  First, we will explore a bit more about the theory of vector bundles:

Definition. Define the infinite Grassmanian ${G_n^{\infty}}$ to be the set of all ${n}$-dimensional subspaces of ${\mathbb{R}^{\infty}}$.  We can also define it as a limit of ${G_n^m}$ with the weak limit topology.  We will denote ${G_1^{\infty} = \mathbb{R}P^{\infty}}$ as ${BO}$.  There is an analogous construction in the complex case, which we will denote ${BU}$.

For the sake of simplicity, we will be working with complex vector bundles.  The reason that this case is simpler is because every manifold has a unique complex orientation, so we don’t need to worry about issues of orientability.

Theorem. There is a bijection between complex line bundles ${p: E \mapsto B}$ and maps ${f: B \mapsto BU}$.

The reasoning behind this statement should be apparent: we are assigning to each point in ${B}$ a point in ${BU}$, which is precisely an ${n}$-dimensional vector space.  We would like to generalize this construction to fiber bundles where the fiber is any topological group. (This is a generalization because we can think of a vector bundle as having fibers in the infinite unitary group ${U}$).  Unfortunately, solving this problem for any topological group is extremely difficult.  However, if we assume that we are working with a discrete topological group, then there is a solution.  As it turns out, ${BU}$ is just an example of what is known as a classifying space:

Definition. Let ${G}$ be a discrete (topological) group.  Then define a space ${BG}$ called the classifying space of ${G}$ to be a topological space such that ${\pi_1(BG) \approx G}$ and all higher homotopy groups are trivial.

Of course, the task of constructing such a topological space is non-trivial.  One solution is to resort to the Eilenberg-Maclane space ${K(G,1)}$.  this certainly satisfies the definition.  However, an observant algebraic topologist will notice that this definition only defines a classifying space up to homotopy equivalence, so there are many different models of classifying spaces.  Thus, we will give another construction that is in some ways a “better” model – this will have the property ${B(G \times H) = BG \times BH}$.  As the functor ${K(G,1)}$ doesn’t preserve products, this new model can be seen as “better”.

Definition. Given a small category ${\mathscr{C}}$, define the nerve of the category to be a simplicial set ${N(\mathscr{C})}$ based on morphisms in the category.  We will give the construction here:

Our 0-simplices will simply be the objects in the category.  Our 1-simplices will be morphisms between the objects.  2-simplices will be the diagrams with edges given by two composable morphisms along with their composition (that is, given two composable morphisms ${f,g}$, we take the commutative triangle with edges ${f,g,gf}$).  We can continue this construction to get a simplicial set, which we will call the nerve of the category.  We still need to specify the face and degeneracy maps in order to have the entire structure of a simplicial set.  Define the face map ${d_i: N(\mathscr{C})_k \mapsto N(\mathscr{C})_{k-1}}$ by taking the simplex ${A_1 \mapsto A_2 \mapsto \ldots A_{i-1} \mapsto A_i \mapsto A_{i+1} \mapsto \ldots }$ to the simplex ${A_1 \mapsto A_2 \mapsto \ldots A_{i-1} \mapsto A_{i+1} \mapsto \ldots }$ by composing the morphisms ${A_{i-1} \mapsto A_i \mapsto A_{i+1}}$ into one morphism.  Define the degeneracy map ${s_i}$ by inserting an identity morphism at the object ${A_i}$.

We will use this construction to very easily construct a classifying space.  Given a discrete group ${G}$, we can think of it as a single-element groupoid (that is, a category with one object in which all of the morphisms are isomorphisms).  Then we take the nerve of this category, and finally take the geometric realization of the nerve.  This is a topological space, which will be precisely ${BG}$.  Note that the nerve functor is right adjoint, so it commutes with limits, including products.  Additionally, it is a well-known fact that geometric realization commutes with products, even though it is a left adjoint functor (see, for example, this paper for a motivation of why this is true).

That the result of this construction is a classifying space is not hard to see.  The fundamental groups of a CW complex (and, thus, a simplicial complex) is obtained from the 2-skeleton by taking formal products of the 1-cells and using the 2-cells to apply relations.  In this case, the 1-cells are precisely the elements of the group and the 2-cells will tell us that ${g * h = gh}$, so this will give us the correct fundamental group.  Verifying that the higher homotopy groups are trivial is much more difficult and beyond the scope of this post.

Example. Consider the group with two elements.  Then, in the nerve of the category, there will be exactly one non-degenerate simplex in each dimension.  This will give rise to a simplicial complex with one simplex in each dimension.  This is precisely analogous to infinite real projective space, which can be realized as a CW complex with one cell in each dimension. By making the proper identifications we can see that this the same space.  Because our construction of classifying spaces commutes with products, we have that ${\mathbb{R}P^{\infty}}$ is the classifying space for any product of these spaces as well.