A long exact sequence in topological K-theory

Let $X$ be a fixed base compact and Hausdorff space. Denote by $K(X)$ the Grothendieck group of $\mathrm{Vect}_\mathbf{C}(X)$ of isomorphism classes of finite-dimensional complex vector bundles over $X$ with addition given by $\oplus$ and multiplication given by $\otimes$. For example, if $X=\ast$, then $\mathrm{Vect}_\mathbf{C}(\ast)=\mathbf{N}$ and thus $K(\ast)=\mathbf{Z}$. Denote by $\widetilde{K}(X)$ the kernel of the map $K(X)\to K(\ast)$, induced by the inclusion $\ast\to X$. This is called reduced K-theory. K-theory has been used, for example, to answer the Hopf invariant one problem, which we won’t discuss here. (There are many more applications of topological K-theory, but we won’t go into those in this post, as some of them require more algebro-topological stuff.) One of the many reasons why K-theory is so interesting is because of the following beautiful result by Bott:

Theorem (Bott periodicity): There’s an isomorphism $\widetilde{K}(X)\cong\widetilde{K}(\Sigma^2 X)$.

In future posts, we’ll prove the Bott periodicity theorem. In this post, however, what we want to prove in this post is the following result, which hints to us that K-theory might be some kind of cohomology theory:

Theorem (LES): Let $A$ be a closed subset of a compact Hausdorff space $X$. There’s a long exact sequence: $\widetilde{K}(A)\leftarrow \widetilde{K}(X)\leftarrow \widetilde{K}(X/A)\leftarrow\widetilde{K}(\Sigma A)\leftarrow\widetilde{K}(\Sigma X)\leftarrow\cdots$

The sequence $A\to X\to X/A\to \Sigma A\to \Sigma X\to\Sigma(X/A)\to\cdots$ is called the Puppe sequence. Consider the sequence $X\to X\cup CA\to (X\cup CA)/X$ where $CA$ is the cone of $A$. First, since $CA$ is contractible, $X\cup CA$ is homotopy equivalent to $X/A$, and second, $(X\cup CA)/X$ is simply $\Sigma A$. Thus sticking $\Sigma A$ to the end of $A\to X\to X/A$, etc., gives us the Puppe sequence. Now, a very important property of the Puppe sequence is that if $X$ and $Y$ are path-connected CW-complexes, then there’s a long exact sequence $[X,Z]\leftarrow [Y,Z]\leftarrow [X/A,Z]\leftarrow[\Sigma A,Z]\leftarrow[\Sigma X,Z]\leftarrow\cdots$. But, well, this looks somewhat like the required long exact sequence for reduced K-theory. Hmm…

Let’s go back to ordinary vector bundles. Let $p:E\to X$ be a vector bundle over $X$, and let $f:Y\to X$ be a continuous map. There’s an induced bundle over $Y$, given by $f^\ast(E) = \{(x,e)\in Y\times E|p(e)=f(x)\}$ (what’s the projection map?). How does this help us? Suppose $p:E\to X$ is a vector bundle; if this is classified by a map from $X$ to some space, then, well, we could use the long exact sequence from the Puppe sequence to prove the required result.

If $p:E\to X$ is a vector bundle of rank $n$, there indeed is such a space! Let $G^\mathbf{C}_{n,k}$ denote the space of $n$-dimensional complex linear subspaces of $\mathbf{C}^{n+k}$. The open subsets of $G^\mathbf{C}_{n,k}$ are sets of subspaces which intersect an open subset of $\mathbf{C}^{n+k}$. How is this helping us? We can define a vector bundle $\xi^\mathbf{C}_{n,k}$ over $G^\mathbf{C}_{n,k}$ via $\xi^\mathbf{C}_{n,k} = \{(x,e)\in G^\mathbf{C}_{n,k}\times\mathbf{C}^{n+k}|e\in x\}$. If $i:G^\mathbf{C}_{n,k}\to G^\mathbf{C}_{n,k+1}$ is induced by the inclusion $\mathbf{C}^{n+k}\to\mathbf{C}^{n+k+1}$, then $i^\ast(\xi^\mathbf{C}_{n,k+1})=\xi^\mathbf{C}_{n,k}$. So, what we can do is consider the union of $G^\mathbf{C}_{n,k}$ under the inclusion maps in the previous sentence. This gives the classifying space $BU(n)$ of the unitary group $U(n)$, which is also the Eilenberg-Maclane space $K(U(n),1)$. We have the following important result:

Important result: There’s a universal bundle, denoted $\xi^\mathbf{C}_n$, over $BU(n)$, which induces $\xi^\mathbf{C}_{n,k}$, such that any rank $n$ vector bundle over a paracompact space $X$ is induced by a map $X\to BU(n)$, and two such bundles are isomorphic if they’re induced by homotopic maps.

That’s a really cool result, but it doesn’t seem to help us much in the case of a general vector bundle over $X$, which is what K-theory is about. Let’s consider the inclusions $U(1)\to U(2)\to\cdots U(n)\to U(n+1)\to\cdots$, and denote by $U$ its direct limit. The classifying space of $U$ is denoted $BU$. This is also the direct limit of the inclusions $BU(1)\to BU(2)\to\cdots BU(n)\to BU(n+1)\to\cdots$. It’s easy to see that a general vector bundle over $X$ corresponds via the above important result to a map $X\to BU$, and this vector bundle is of rank $n$ if $n$ is the minimal integer such that the map $X\to BU$ factors as $X\to BU(n)\hookrightarrow BU$.

Well, by the definition of $\widetilde{K}(X)$, we see that $\widetilde{K}(X)\simeq[X,BU]$! Note also that $K(X)\simeq [X,BU\times\mathbf{Z}]$; to see this, note that $K(X)\simeq \mathrm{H}^0(X;\mathbf{Z})\oplus\widetilde{K}(X)$. Because $\mathrm{H}^0(X;\mathbf{Z})\simeq[X,\mathbf{Z}]$, the required result follows. By the long exact sequence induced by the Puppe sequence, we’ve finished the proof of the Theorem (LES). The above discussion also implies, via Bott periodicity, that $\Omega^2 BU\simeq BU\times\mathbf{Z}$. As mentioned in the beginning of the post, K-theory might be a cohomology theory; it indeed is! K-theory is an example of something called an extraordinary cohomology theory, in the sense that it satisfies all of the Eilenberg-Steenrod axioms except for the dimension axiom. (I’ll note here that there’s also an analogue of all of this stuff for real K-theory, where we consider real vector bundles.)