# Hopf invariant one, III: K-theory

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?). Suppose $p:E\to X$ is a vector bundle; the following result states that this is classified by a map from $X$ to some space.

Lemma: There’s a universal bundle, denoted $\xi^\mathbf{C}_n$, over a space $BU(n)$, 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.

Proof. 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 $\mathbf{C}^{n+k}$. 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 $i^\ast(\xi^\mathbf{C}_{n,k+1})=\xi^\mathbf{C}_{n,k}$. Thus we can consider the union of $G^\mathbf{C}_{n,k}$ under the inclusion maps to define the classifying space $BU(n)$ of the unitary group $U(n)$. QED.

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 $X\to BU(n)\hookrightarrow BU$.

Definition: Let $X$ be a fixed base space that is compact and Hausdorff. 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 $\otimes$. 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. There is an analogous definition of $KO$ using real vector bundles.

Example: If $X=\ast$, then $\mathrm{Vect}_\mathbf{C}(\ast)=\mathbf{N}$ and thus $K(\ast)=\mathbf{Z}$.

Before going to the next example, we will prove the following lemma from linear algebra.

Lemma: $\mathrm{GL}_n(\mathbf{C})$ deformation retracts onto $U(n)$.

Proof. Let $A\in\mathrm{GL}_n(\mathbf{C})$, and consider the QR decomposition $A=UP$ where $U$ is unitary and $P$ is a upper triangular matrix. Define $h:\mathrm{GL}_n(\mathbf{C})\times I\to U(n)$ as:

QED.

What follows is a more nontrivial computation, namely that of $\widetilde{K}(S^2)$. I learnt this from Professor Miller’s lecture notes (http://www.math.harvard.edu/~ecp/latex/misc/haynes-notes/haynes-notes.pdf). We can write $S^2$ as two disks glued at the equator (which is $S^1$). The disks are contractible and hence any vector bundle is trivial, i.e., is of the form $D^2\times\mathbf{C}^n$. Any vector bundle on $S^2$ can be trivialized on the hemispheres, and thus are completely determined by choosing a linear isomorphism $\mathbf{C}^n\to\mathbf{C}^n$ in a continuous manner. In other words, $\mathrm{Vect}_n(S^2)=[S^1,\mathrm{GL}_n(\mathbf{C})]=\pi_1\mathrm{GL}_n(\mathbf{C})$. The map $S^1\to\mathrm{GL}_n(\mathbf{C})$ associated to a complex $n$-dimensional vector bundle $E\to S^2$ is called the clutching function. This statement is true even if we place $S^2$ with $S^k$ for some $k$ (and replace $S^1$ with $S^{k-1}$).

By the lemma, $\mathrm{GL}_n(\mathbf{C})$ deformation retracts onto $U(n)$. Now, $U(n+1)$ acts on $S^{2n+1}$, and the stabilizer of this action is $U(n)$. The map $U(n+1)\to S^{2n+1}$ given by $A\mapsto Ax$ for some fixed basepoint $x\in S^{2n+1}$ is a fibration with fibers $U(n)$. Consider the long exact sequence in homotopy groups $\cdots\to \pi_{k+1}(S^{2n+1})\to \pi_k(U(n))\to \pi_k(U(n+1))\to \pi_k(S^{2n+1})\to\pi_{k-1}(U(n))\to\cdots$.

This shows that $\pi_1(U(n))\cong\pi_1(U(n+1))$ for $n\geq 1$. Now, $U(1)=S^1$, so $\pi_1(U(1))\cong\mathbf{Z}$. An element $m\in \pi_1(U(1))$ therefore corresponds to the clutching function $z\mapsto z^m$ for a complex line bundle ${}^mE_1\to S^2\simeq\mathbf{CP}^1$. In particular, the universal line bundle over $\mathbf{CP}^1$ corresponds to the clutching function $z\mapsto z$. The inclusion $U(n)\hookrightarrow U(n+1)$ shows that the element $m\in \mathbf{Z}=\pi_1(U(n))$ corresponds to the clutching function $z\mapsto\begin{pmatrix}z^m & & & \\ & 1 & & \\ & & \ddots & \\ & & & 1\end{pmatrix}$ of the bundle ${}^mE_n$. Thus ${}^mE_k\oplus{}^\ell E_j={}^{m+\ell}E_{k+j}$. We will now prove an intermediary result.

Lemma: Let $\xi\to \mathbf{CP}^1$ denote the universal line bundle over $\mathbf{CP}^1=BU(1)$. Then $(\xi\otimes\xi)\oplus \mathbf{C}\simeq\xi\oplus\xi$.

Proof. By our above analysis, it suffices to show that the respective clutching functions $f:z\mapsto\begin{pmatrix}z^2 & \\ & 1\end{pmatrix}$ and $g:z\mapsto\begin{pmatrix}z & \\ & z\end{pmatrix}$ are homotopic. We know that $\mathrm{GL}_2(\mathbf{C})$ is path-connected. There is therefore a path, say $h_t$, from $I$ to the matrix that flips the factors in $\mathbf{C}^2\times\mathbf{C}^2$. The composition $(f\oplus 1)h_t(1\oplus g)h_t$ is a homotopy from $f\oplus g$ to $b\oplus 1$ where $b(z)=f(z)g(z)$ for $z\in S^1$. (When $t=0$, this is $(f\oplus 1)(1\oplus g)=f\oplus g$, and when $t=1$, this is $(f\oplus 1)(1\oplus g)$.) QED.

Finally, this means that:

Result: $K(S^2)=\mathbf{Z}[\xi]/(\xi-[\mathbf{C}])^2$, and $\widetilde{K}(S^2)=\mathbf{Z}\langle[\xi]-[\mathbf{C}]\rangle$.

Another example: recall the statement $\mathrm{Vect}_n(S^k)=[S^{k-1},\mathrm{GL}_n(\mathbf{C})]=\pi_{k-1}\mathrm{GL}_n(\mathbf{C})$. Suppose $k=1$. Then $\mathrm{Vect}_n(S^1)=\pi_0\mathrm{GL}_n(\mathbf{C})$, but $\mathrm{GL}_n(\mathbf{C})$ is connected, so $\widetilde{K}(S^1)=0$.

Recall the following result from https://erdosninth.wordpress.com/2016/04/17/a-long-exact-sequence-in-topological-k-theory/, which hints to us that K-theory might be some kind of cohomology theory.

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

Another important result:

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

Proof. Omitted. Maybe I should include this? It’s similar to the computation of $\widetilde{K}(S^2)$.

Definition: Define $\widetilde{K}^{-n}(X)=\widetilde{K}(\Sigma^n X)$ for $n\geq 0$, and $\widetilde{K}^n(X)=\widetilde{K}(\Sigma^{2k-n}X)$ for some $k$ such that $2k>n$.

The $\widetilde{K}^n$ form a generalized cohomology theory, called (reduced) complex K-theory. Unreduced complex K-theory can also be defined. In fact, $\widetilde{K}^n(X_+)=:K^n(X)$ where $K(X)=K^0(X)$ is as defined above. Brown representability tells us that complex K-theory is represented by a spectrum $KU$. We have already shown that $KU_{2k}=BU\times\mathbf{Z}$. It’s easy to determine $KU_{2k+1}=\Omega(BU\times\mathbf{Z})=\Omega BU\simeq U$.

Complex K-theory is a multiplicative cohomology theory. This comes from the following H-space structure on $BU\times\mathbf{Z}$. Define maps $p_{m,n}:BU(m)\times BU(n)\to BU(m+n)$ that is characterized up to homotopy as $p_{m,n}^\ast(\xi^\mathbf{C}_{m+n})=\xi^\mathbf{C}_m\times\xi^\mathbf{C}_n$ where the $\xi^\mathbf{C}_k$ are the universal $k$-bundles over $BU(k)$.

For example, since $X\wedge S^2=\Sigma^2 X$, it follows that:

Recall that $\widetilde{K}^0(S^2)$ is generated by $[\xi]-[\mathbf{C}]=[\xi]-1$ (where we just write $1$ for $[\mathbf{C}]$). This is isomorphic to $\widetilde{K}^{-2}(S^0)$, and the generator of $\widetilde{K}^{-2}(S^0)=K^{-2}(\ast)$ is denoted $\beta$ and called the Bott element. It follows that as a graded ring, $K^\ast=K^\ast(\ast)=\mathbf{Z}[\beta^{\pm 1}]$.

Also, recall that one can define $KO(X)$ via real vector bundles. One can run through the same constructions as above, and find that:

Proposition:

1. There are spaces $BO$ constructed as the direct limit of the $G^\mathbf{R}_{n,k}$ (defined as the space of $n$-dimensional linear subspaces of $\mathbf{R}^{n+k}$) such that $\widetilde{KO}(X)=[X,BO\times\mathbf{Z}]$.
2. The real K-theory of the $8$-sphere is $\widetilde{KO}(S^8)=\mathbf{Z}\langle[\xi\to\mathbf{H}\mathbf{P}^2]-[\mathbf{C}^4]\rangle$.
3. There is an isomorphism $\widetilde{KO}(X)\otimes \widetilde{KO}(S^8)\cong \widetilde{KO}(X\wedge S^8)$. Define $\widetilde{KO}^{-n}(X)=\widetilde{KO}(\Sigma^n X)$ for $n\geq 0$, and $\widetilde{KO}^n(X)=\widetilde{KO}(\Sigma^{8k-n}X)$ for some $k$ such that $8k>n$.
4. Let $KO^n(X)=\widetilde{KO}^n(X_+)$. Then $KO$ is a generalized cohomology theory, represented by a spectrum $KO$ with $KO_n=\Omega^n(BO\times\mathbf{Z})$. Note that Bott periodicity for $KO$ says that $\Omega^8(BO\times\mathbf{Z})\simeq BO\times\mathbf{Z}$.

It turns out that $KO^\ast=KO^\ast(\ast)$ is much more complicated. There is an isomorphism of graded rings $KO^\ast=\mathbf{Z}[\eta,\alpha,\zeta^{\pm 1}]/(2\eta,\eta^3,\eta\alpha,\alpha^2-4\zeta)$ where $|\eta|=-1$, $|\alpha|=-4$, and $|\zeta|=-8$.

One might wonder about the relationship between $KO$ and $KU$ — after all, both are “K-theories” defined using vector bundles! Complexification of vector bundles does indeed give a multiplicative homomorphism $KO^\ast(X)\to KU^\ast(X)$, and realification gives a homomorphism of $KO^\ast$-modules $KU^\ast(X)\to KO^\ast(X)$. On homotopy groups (cf. Rognes’ notes on topological K-theory), complexification gives a map $\pi_\ast KO\to \pi_\ast KU$ sending $\eta\mapsto 0$, $\alpha\mapsto 2\beta^2$, and $\zeta\mapsto \beta^4$. Realification gives a map $\pi_\ast KU\to\pi_\ast KO$ sending $\beta^{4k}\mapsto 2\zeta^k, \beta^{4k+1}\mapsto\eta^2\zeta^k,\beta^{4k+2}\mapsto\alpha\zeta^k,\beta^{4k+3}\mapsto 0$.

Next time we’ll finish the proof of the Hopf invariant one problem. The way we’ll do this is by redefining the Hopf invariant using K-theory, and proving that our new definition is equivalent to the definition we already have. Then we’ll have to do a little more analysis. This part of the series might be a bit delayed since final exams are coming up in $\leq 2$ weeks.