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:



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:


  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.


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