Notes on K-theory: Part I

November 15, 2025

Broadly, $K$-theory of a space $X$ studies vector bundles over $X$. The two main flavors of $K$-theory are $KO$-theory and $KU$-theory, corresponding to real and complex vector bundles, respectively. We shall denote both theories as $K$, and make the distinction whenever necessary for exposition. Throughout the post, all spaces will be ‘nice’ (for instance, CW-complexes) and all pairs of spaces will be ‘nice’ (for instance, a CW-pair).

Let $X$ be a space. Let $\mathrm{Vect}_{\mathbf{k}}(X)$ denote the commutative monoid of isomorphism classes of finite rank vector bundles over $X$ with vector spaces over the field $\mathbf{k}$ as fibers, where $\mathbf{k} = \mathbf{R}, \mathbf{C}$. Addition is given by direct sum of vector bundles, and the zero element is the trivial vector bundle of rank zero. Then one defines $\mathrm{K}(X)$ as the Grothendieck completion of $\mathrm{Vect}_{\mathbf{k}}(X)$, given by formally throwing in the inverses to make $\mathrm{Vect}_{\mathbf{k}}(X)$ into an abelian group. Thus, elements of $\mathrm{K}(X)$ are of the form $[E] - [F]$ where $E, F \to X$ are vector bundles over $X$. We shall call an element of the form $[E] - [F]$ a virtual bundle over $X$. If $X$ is connected, let us define the virtual rank of the virtual vector bundle $[E] - [F]$ to be $\mathrm{rk}(E) - \mathrm{rk}(F)$.

Note that $[E] = [F]$ in $\mathrm{K}(X)$ if and only if there exists a vector bundle $V \to X$ such that $E \oplus V \cong F \oplus V$. Embed $V$ in a trivial bundle $\varepsilon^n := X \times k^n \to X$ for a sufficiently large $n$, and $V^\perp$ be the orthogonal complement of $V$, so that $V \oplus V^\perp \cong \varepsilon^n$. Therefore,

\[\displaystyle E \oplus \varepsilon^n \cong E \oplus V \oplus V^\perp \cong F \oplus V \oplus V^\perp \cong F \oplus \varepsilon^n\]

Thus, geometrically, $\mathrm{K}(X)$ consists of ‘equistable isomorphism’ classes of (virtual) bundles over $X$, where ‘equistably isomorphic’ here means isomorphic upon stabilization of both by a trivial bundle of the same rank. This is a weaker notion than stable isomorphism, which allows stabilization by trivial bundles of different ranks. For a based space $(X, pt)$, i.e. space with a chosen basepoint, we shall soon define a variant $\widetilde{\mathrm{K}}(X)$ of the $K$-theory groups above so that we can upgrade from ‘equistably isomorphic’ to ‘stably isomorphic’ in this geometric interpretation. Roughly, this will be done by restricting attention to virtual bundles $[E] - [F] \in \mathrm{K}(X)$ which have virtual rank over $pt$ zero, i.e.

\[\displaystyle \mathrm{rk}(E_{pt}) - \mathrm{rk}(F_{pt}) = 0\]

The choice of the basepoint is important here because a vector bundle over a disconnected space may have different ranks over different connected components.

First, we begin by noting that for any map $f : X \to Y$, there is an induced homomorphism $f^* : \mathrm{K}(Y) \to \mathrm{K}(X)$ given by pullback, i.e. $f^*[E] = [f^* E]$. Hence, $\mathrm{K}$ is functorial. As an aside, note that pulled back vector bundles under homotopic maps are isomorphism. Therefore, if $f \simeq g$, we have $f^* = g^*$.

Note that $\mathrm{K}(pt) \cong \mathbf{Z}$ since finite-dimensional vector bundles over a point are finite-dimensional vector spaces, classified by dimension. For a based space $(X, pt)$, the inclusion of the basepoint induces a natural surjective homomorphism $\mathrm{K}(X) \to \mathrm{K}(pt)$. We denote the kernel of this homomorphism as $\mathrm{\widetilde{K}}(X)$, which we shall call the reduced $K$-theory of $X$. We stress that this is only defined for a based space $(X, pt)$. The elements of $\widetilde{\mathrm{K}}(X)$ consist of virtual bundles $[E] - [F]$ with virtual rank over $pt$ zero. Further, $[E] - [F] = 0$ in $\widetilde{\mathrm{K}}(X)$ iff $[E] = [F]$ in $\mathrm{K}(X)$ which is true if $E$ and $F$ are equistably isomorphic. However, since rank of $E$ and $F$ agree over $pt$, this notion is completely equivalent to stable isomorphism for these two particular vector bundles $E, F$, as promised. Note that even though for a vector bundle $E$ over $X$, $[E]$ is not a class in $\widetilde{\mathrm{K}}(X)$, we nevertheless always have a distinguished class corresponding to $E$, given by $[E] - [\varepsilon^{\mathrm{rk}(E_0)}] \in \widetilde{\mathrm{K}}(X)$.

For a pair of spaces $(X, A)$, we define $\mathrm{K}(X, A) := \widetilde{\mathrm{K}}(X/A)$. If $A = \emptyset$, we adopt the slightly unnatural convention that the quotient $X/\emptyset$ is the space $X^+ := X \sqcup \{pt\}$ obtained from artificially adjoining a disjoint basepoint to $X$. Note that for an unbased space $X$, there are natural isomorphisms $\mathrm{K}(X) \cong \mathrm{\widetilde{K}}(X^+) \cong \mathrm{K}(X, \emptyset)$. Note that we have an exact sequence

\[\displaystyle \mathrm{K}(X, A) \to \mathrm{K}(X) \to \mathrm{K}(A)\]

Indeed, given a virtual bundle $[E] - [F]$ over $X$ such that $[E] = [F]$ in $\mathrm{K}(A)$, we get that $E$ and $F$ are stably isomorphic over $A$, i.e. $E\vert_A \oplus \varepsilon^n \cong F\vert_A \oplus \varepsilon^n$. Embed $E$ in some large rank trivial bundle $\varepsilon^m$ over $X$, and let $E^\perp$ be its orthogonal complement. Thus, $E^\perp\vert_A \oplus F\vert_A \oplus \varepsilon^n \cong \varepsilon^{m+n}$. Therefore, $E^\perp \oplus F \oplus \varepsilon^n$ descends to a vector bundle over $X/A$, and it has rank $m+n$ over the basepoint. Thus, consider the element

\[[E^\perp \oplus F \oplus \varepsilon^n] - [\varepsilon^{m+n}] \in \mathrm{K}(X, A),\]

whose image is exactly $[E] - [F] \in \mathrm{K}(X)$.

The above discussion is quite suggestive. Following our nose, we turn $K$-theory into a (generalized) cohomology theory, as follows. Recall that for a based space $(X, pt)$, the reduced suspension of $X$ is defined by

\[\displaystyle \Sigma X = \frac{X \times S^1}{(X \times 1) \vee (pt \times S^1)}\]

As such, $\Sigma X$ inherits a natural basepoint given by the image of $(X \times 1) \vee (pt \times S^1)$. Let $i \geq 0$.

If $X$ is an unbased space, set $\mathrm{K}^{-i}(X) := \widetilde{\mathrm{K}}(\Sigma^i(X^+))$.
If $X$ is a based space, set $\widetilde{\mathrm{K}}^{-i}(X) := \widetilde{\mathrm{K}}(\Sigma^i X)$.
If $(X, A)$ is a pair of spaces, set $\mathrm{K}^{-i}(X, A) := \widetilde{\mathrm{K}}^{-i}(X/A) \cong \widetilde{\mathrm{K}}(\Sigma^i(X/A))$.

The negative indexing is perhaps a bit jarring. The point is merely that with this indexing convention, it defines a cohomology theory. Recall that for any pair of spaces $(X, A)$ we have the Barratt-Puppe sequence

\[\displaystyle A \to X \to X/A \to \Sigma A \to \Sigma X \to \Sigma(X/A) \to \Sigma^2A \to \cdots\]

where $\Sigma$ denotes the reduced suspension. The only map in the above chain of arrows that is not immediately clear is $X/A \to \Sigma A$. To define this, observe that $X/A \simeq X \cup_A CA$ where $CA$ denotes the cone over $A$. Then, quotienting by $X$, we obtain a map

\[\displaystyle X/A \simeq X \cup_A CA \to (X \cup_A CA)/X \cong CA/A \simeq \Sigma A\]

Using this, we obtain a complex of groups

\[\displaystyle \cdots \to \widetilde{\mathrm{K}}^{-2}(A) \to \mathrm{K}^{-1}(X, A) \to \mathrm{\widetilde{K}^{-1}}(X) \to \mathrm{\widetilde{K}^{-1}}(A)\to \mathrm{K}^0(X, A) \to \mathrm{\widetilde{K}^0}(X) \to \mathrm{\widetilde{K}^0}(A)\]

This complex is indeed long exact, for details see Atiyah “K-Theory”, Proposition 2.4.4. To illustrate, let us sketch the proof of exactness at $\mathrm{K}^0(X, A)$. The kernel of $\mathrm{K}^0(X, A) \to \widetilde{\mathrm{K}}^0(X)$ consists of rank zero virtual bundles $[E] - [F]$ over $X/A \simeq X \cup_A CA$ which restrict to the zero element over $X$. Thus, $E\vert_X$ and $F\vert_X$ are stably equivalent, i.e., $E\vert_X \oplus \varepsilon^n \cong F\vert_X \oplus \varepsilon^n$. Thus, $[E \oplus \varepsilon^n] - [F \oplus \varepsilon^n]$ descends to a rank zero virtual bundle over $(X \cup_A CA)/X \cong \Sigma A$.

The excision axiom and additivity axiom are more or less tautological in this theory. Indeed, for based spaces $X_1, X_2$, any virtual bundle over $X_1 \vee X_2$ having virtual rank zero over the basepoint is the same as a pair of virtual bundles over $X_1$ and $X_2$, respectively, having virtual rank zero over the respective basepoint. Further for any triple of spaces $(X, A, B)$ with $\overline{B} \subset A^\circ$,

\[\displaystyle \mathrm{K}^{-i}(X \setminus B, A \setminus B) = \widetilde{\mathrm{K}} \left (\Sigma^i \left ( \frac{X \setminus B}{A \setminus B} \right ) \right ) \cong \widetilde{\mathrm{K}}\left (\Sigma^i \left ( \frac{X}{A} \right )\right ) = {\mathrm{K}}^{-i}(X, A)\]

Hence, we obtain as a corollary that $\{\widetilde{\mathrm{K}}^{-n} : n \geq 0\}$ is a generalized (reduced) cohomology theory.

Definition. An $\Omega$-spectrum $E$ is a collection of based spaces $\{E_n : n \in \mathbf{Z}\}$ along with a collection of homotopy equivalences $\varepsilon_n : E_n \to \Omega E_{n+1}$ for each $n \in \mathbf{Z}$.

For any $\Omega$-spectrum $E$, the collection of functors on based spaces

\[\displaystyle E^n : \mathsf{Top}_* \to \mathsf{AbGrp}\]

given by $\widetilde{E}^n(X) := [X, E_n]_*$, defines a generalized reduced cohomology theory. Here, for based spaces $A, B$, we are using $[A, B]_*$ to denote the set of homotopy classes of based maps. Note that since $E_n = \Omega E_{n+1}$ is a loopspace, hence in particular an $H$-space, $\widetilde{E}^n(X)$ is an abelian group due to the Eckmann-Hilton duality argument. Indeed, one defines ${E}^n(X, A) := \widetilde{E}^n(X/A)$ and every other axiom except the exactness axiom is trivial. The exactness axiom follows once again from Barratt-Puppe sequence, while observing that

\[\displaystyle \widetilde{E}^n(\Sigma X) = [\Sigma X, E_n]_* \cong [X, \Omega E_n]_* \simeq [X, E_{n+1}]_* = \widetilde{E}^{n+1}(X),\]

from the suspension-loopspace adjunction and definition of $\Omega$-spectrum. Incidentally, one can also define for any unbased space $X$, the unreduced flavor $E^n(X) := \widetilde{E}^n(X^+)$. Note that the unreduced cohomology theory of a point recovers the negative stable homotopy groups of the spectrum, i.e.

\[\displaystyle E^n(pt) = [pt^+, E_n]_* = \pi_0(E_n) = \pi_{(-n)+k}(E_k),\]

where $k \gg n$ is very large. Note that the right hand side is in fact independent of the choice of a large enough $k$, because $E_k \simeq \Omega E_{k+1}$. Consequently,

\[\displaystyle E^n(pt) = \varinjlim_k ~ \pi_{-n+k}(E_k) = \pi_{-n}(E)\]

It is an easy corollary of the Brown representability theorem (see Chapter 7.7. in Spanier’s “Algebraic Topology”) that any generalized (reduced) cohomology theory is representable by an $\Omega$-spectrum. Therefore, $K$-theory must also correspond to an $\Omega$-spectrum. Indeed, it is not difficult to derive what it will be explicitly in this case. We proceed to do so now.

As is used countless times throughout so far, any vector bundle $E \to X$ of rank $k$ over a field $\mathbf{k}$ ($= \mathbf{R}$ or $\mathbf{C}$) can be embedded in a trivial bundle $\varepsilon^n \to X$ of rank $n \gg k$. Corresponding to this embedding, there is a “fiber Gauss map” $\varphi : X \to \mathrm{Gr}(k, \mathbf{k}^n)$ to the Grassmannian of $k$-dimensional planes in $\mathbf{k}^n$. Namely, for any $x \in X$, the fiber $E_x$ sits inside the fiber $\mathbf{k}^n$ of $\varepsilon^n$ over $x$, as a linear subspace. We define $\varphi(x)$ to be the point in the Grassmannian corresponding to $E_x \subset \mathbf{k}^n$. Let

\[\displaystyle \mathrm{Gr}(k, \mathbf{k}^\infty) := \varinjlim_n~ \mathrm{Gr}(k, \mathbf{k}^n),\]

where the maps $\mathrm{Gr}(k, \mathbf{k}^n) \to \mathrm{Gr}(k, \mathbf{k}^{n+1})$ are induced by the coordinate inclusion $\mathbf{k}^n \to \mathbf{k}^{n+1}$. Then, the homotopy class of the map $\varphi : X \to \mathrm{Gr}(k, \mathbf{k}^\infty)$ is independent of the choice of any embedding of $E$ in a trivial bundle. In fact, the isomorphism class of $E$ is completely determined by the homotopy class of the map $\varphi$. Indeed, there is a tautological vector bundle of rank $k$ over $\mathrm{Gr}(k, \mathbf{k}^\infty)$, given by

\[\displaystyle \gamma_k := \{([V], x) : x \in V \} \subset \mathrm{Gr}(k, \mathbf{k}^\infty) \times \mathbf{k}^\infty \to \mathrm{Gr}(k, \mathbf{k}^\infty)\]

Then, it follows from the definition that $E \cong \varphi^*(\gamma_k)$. We make a side-remark before proceeding. Let $V_k(\mathbf{k}^n)$ denote the space of orthonormal $k$-frames in $\mathbf{k}^n$, with respect to the (Hermitian) inner product on $\mathbf{k}^n$ for $\mathbf{k} = \mathbf{R}, \mathbf{C}$, respectively. Let $V_k(\mathbf{k}^\infty)$ denote the direct limit/increasing union of $V_k(\mathbf{k}^n)$, as before. Then, by the orbit-stabilizer theorem, we have

\[\displaystyle \begin{align*} \mathrm{Gr}(k, \mathbf{R}^\infty) &\cong \frac{V_k(\mathbf{R}^\infty)}{O(k)}, & \mathrm{Gr}(k, \mathbf{C}^\infty) \cong \frac{V_k(\mathbf{C}^\infty)}{U(k)} \end{align*}\]

Since $V_k(\mathbf{k}^\infty)$ is contractible, and the quotient of contractible space with a free action of a topological space $G$ is the classifying space $BG$, we get $\mathrm{Gr}(k, \mathbf{R}^\infty) \cong BO(k)$ and $\mathrm{Gr}(k, \mathbf{C}^\infty) \cong BU(k)$. In conclusion, we get that isomorphism classes of rank real (resp. complex) $k$ vector bundles over $X$ are in 1–1 correspondence with homotopy classes of maps from $X$ to $BO(k)$ (resp. $BU(k)$). To get rid of the dependence on $k$, let us define

\[\displaystyle \begin{align*}BO &:= \varinjlim BO(k),& BU := \varinjlim BU(k)\end{align*}\]

where the maps in the directed system are induced by the maps $O(k) \to O(k+1)$ and $U(k) \to U(k+1)$ given by sending an orthogonal (resp. unitary) matrix $A$ to the block-diagonal matrix $\mathrm{diag}(A, 1)$. Now, to gain some, one must lose some. While we dropped the dependence on $k$, this creates two problems:

Let $X$ be connected, and $E$ be a vector bundle over $X$. The ‘classifying map’ (or the ‘fiber Gauss map’, as defined earlier) of the bundles $E$ and $E \oplus \varepsilon^n$ are homotopic as maps to $BO$ or $BU$. So, the homotopy class of the classifying map to $BO$ or $BU$ is a stable isomorphism invariant. However, $\mathrm{K}(X)$ entails equistable isomorphism classes of vector bundles. To get to stable, one would have to fix a basepoint, but $X$ need not have a canonical choice of a basepoint and such a choice was not made in the discussion above while defining the classifying map.
Not only that, but if $X$ is disconnected, we can stabilize $E$ by trivial vector bundles of different rank over different components of $X$, and the resulting vector bundle will also have its classifying map homotopic to that of $E$. This is a weaker notion than stably isomorphic!

We fix this by considering the homomorphism

\[\mathrm{rk} : \mathrm{K}(X) \to [X, \mathbf{Z}]\]

given by $\mathrm{rk}(E)_x = \mathrm{rk}(E_x)$, and extending in the obvious way to virtual bundles. Note that this is a surjective homomorphism, and there is an obvious splitting given by considering trivial bundle of specified rank over each connected components of $X$. Hence, $\mathrm{K}(X) \cong [X, \mathbf{Z}] \oplus \ker(\mathrm{rk})$. I claim that the discussion above implies that $\ker(\mathrm{rk})$ is isomorphic to $[X, BO]$ in the real case, and $[X, BU]$ in the complex case. To this end, note that any map from $X$ to $BO$ (resp. $BU$) determines a real (resp. complex) vector bundle $E$ over $X$. We consider the union of trivial bundles over each connected component $X_\alpha \subset X$ with rank $\mathrm{rk}(x_\alpha)$ for some $x_\alpha \in X_\alpha$. Let us denote the resulting bundle as $\varepsilon^{\mathrm{rk}(E)}$. Then $[E] - [\varepsilon^{\mathrm{rk}(E)}]$ determines an element of $\ker(\mathrm{rk})$. It is easily checked that this establishes a 1–1 correspondence. Putting it all together, we obtain

\[\displaystyle \begin{align*} \mathrm{KO}(X) &\cong [X, BO] \times [X, \mathbf{Z}] \cong [X, BO \times \mathbf{Z}],\\ \mathrm{KU}(X) &\cong [X, BU] \times [X, \mathbf{Z}] \cong [X, BU \times \mathbf{Z}] \end{align*}\]

Choose a basepoint $o$ on $BO$ (resp. $BU$) and fix them forever. Then $\{0\} \times \{o\}$ is a basepoint for $\mathbf{Z} \times BO$ (resp. $\mathbf{Z} \times BU$). If $(X, pt)$ is a based space, then the reduced theories are given by the based homotopy classes of maps:

\[\displaystyle \begin{align*} \mathrm{\widetilde{KO}}(X) & \cong [X, BO \times \mathbf{Z}]_*, & \mathrm{\widetilde{KU}}(X) \cong [X, BU \times \mathbf{Z}]_* \end{align*}\]

By the loopspace suspension adjunction, we get:

\[\displaystyle \begin{align*} \mathrm{\widetilde{KO}}^{-n}(X) & \cong [X, \Omega^n(BO \times \mathbf{Z})]_*, & \mathrm{\widetilde{KU}}^{-n}(X) \cong [X, \Omega^n(BU \times \mathbf{Z})]_* \end{align*}\]

Recall that the unreduced groups are defined as $\mathrm{KO}^{-n}(X) = \widetilde{\mathrm{KO}}^{-n}(X^+)$ and $\mathrm{KU}^{-n}(X) = \widetilde{\mathrm{KU}}^{-n}(X^+)$, for $n \geq 0$ as usual. This lets us identify the $\Omega$-spectrum of our cohomology theory, but this is not the end of the story. A central theorem in homotopy theory, called the Bott periodicity theorem, shows that there are homotopy equivalences

\[\displaystyle \begin{align*}\Omega^8(BO \times \mathbf{Z}) &\simeq BO \times \mathbf{Z}, & \Omega^2(BU \times \mathbf{Z}) \simeq BU \times \mathbf{Z} \end{align*}\]

Several proofs of these two results are known to exist. Bott’s original proof used Morse theory, details of which can be found in Milnor’s red book. This result allows us to periodically extend our spectrum to negative degrees. Indeed, let us set $\mathrm{KO}_{k+8n} = \Omega^k(BO \times \mathbf{Z})$ for $k = 0, 1, \cdots, 7$ and $\mathrm{KU}_{k+2n} = \Omega^k(BU \times \mathbf{Z})$ for $k = 0, 1$. Then $\mathrm{KO} = \{\mathrm{KO}_n : n \in \mathbf{Z}\}$ and $\mathrm{KU} = \{\mathrm{KU}_n : n \in \mathbf{Z}\}$ defines the real and complex $K$-theory $\Omega$-spectra, and the corresponding generalized reduced cohomology theories are the real and complex $K$-theory, which exist in all integral degrees and are $8$-periodic and $2$-periodic, respectively. For non-negative degrees, the theories agree with the cohomology theories $\widetilde{\mathrm{KO}}^{-n}$ and $\widetilde{\mathrm{KU}}^{-n}$ that we defined concretely.

Bott periodicity theorem gives an easy way to compute the homotopy groups of the $K$-theory spectra, because we need only compute the first eight homotopy groups of $O(\infty)$ and the first two homotopy groups of $U(\infty)$. We reproduce the tables here:

\[\displaystyle \begin{align*} & {\begin{array}{ c | c | c | c | c | c | c | c } \pi_0(\mathrm{KO}) & \pi_1(\mathrm{KO}) & \pi_2(\mathrm{KO}) & \pi_3(\mathrm{KO}) & \pi_4(\mathrm{KO}) & \pi_5(\mathrm{KO}) & \pi_6(\mathrm{KO}) & \pi_7(\mathrm{KO}) \\ \hline \mathbf{Z} & \mathbf{Z}_2 & \mathbf{Z}_2 & 0 & \mathbf{Z} & 0 & 0 & 0\\ \end{array} } \\ \\ & {\begin{array}{c|c} \pi_0(\mathrm{KU}) & \pi_1(\mathrm{KU}) \\ \hline \mathbf{Z} & 0 \end{array} } \end{align*}\]

By our earlier discussion, this gives a complete description of both the real and complex unreduced $K$-theory of a point, as a graded group. Now, let us proceed to study the graded ring structure on $K$-theory. For any two spaces $X, Y$, we have a homomorphism

\[\displaystyle \mu : \mathrm{K}(X) \otimes \mathrm{K}(Y) \to \mathrm{K}(X \times Y),\]

given by $\mu([E], [F]) = [E \boxtimes F]$ and bi-linearly extending. Here, $E \boxtimes F = \pi_X^* E \otimes \pi_Y^* F$ is the external tensor product of the two bundles $E$ and $F$. Note that if $X = Y$, restricting $E \boxtimes Y$ to the diagonal subspace $X \subset X \times X$ gives the tensor product $E \otimes F$. This defines a graded ring structure

\[\mathrm{K}(X) \otimes \mathrm{K}(X) \to \mathrm{K}(X)\]

Note that for any two based spaces $X, Y$, there is a homotopy equivalence (Proposition 4I.1, Hatcher, “Algebraic Topology”)

\[\displaystyle \Sigma (X \times Y) \simeq \Sigma X \vee \Sigma Y \vee \Sigma(X \wedge Y)\]

This shows $\Sigma(X) \vee \Sigma(Y)$ is a retract of $\Sigma(X \times Y)$. Consequently, the long exact sequence in $K$-theory for the pair $(\Sigma(X \times Y), \Sigma(X) \vee \Sigma(Y))$ splits. We get a short exact sequence:

\[\displaystyle 0 \to \widetilde{\mathrm{K}}^{-n}(X \wedge Y) \to \widetilde{\mathrm{K}}^{-n}(X \times Y) \stackrel{r}{\to} \widetilde{\mathrm{K}}^{-n}(X) \oplus \widetilde{\mathrm{K}}^{-n}(Y) \to 0\]

For virtual bundles $E$ over $\Sigma^i X$ and $F$ over $\Sigma^j Y$ with virtual rank zero over the basepoint, $\mu(E, F)$ will restrict to the zero element on $\Sigma^i X \times \{pt_Y\}$ and $\{pt_X\} \times \Sigma^j Y$, as tensor product of the zero vector space with anything is zero. Therefore, $\mu(E, F)$ is in the kernel of the map $r$ in the short exact sequence above. Thus, $\mu$ factors through the following map:

\[\displaystyle \mu : \widetilde{\mathrm{K}}^{-i}(X) \otimes \widetilde{\mathrm{K}}^{-j}(Y) \to \widetilde{\mathrm{K}}^0(\Sigma^i X \wedge \Sigma^j Y) \cong \widetilde{\mathrm{K}}^{-(i+j)}(X \wedge Y)\]

Indeed, $\Sigma^i X \wedge \Sigma^j Y \cong S^i \wedge X \wedge S^j \wedge Y \cong S^{i+j} \wedge (X \wedge Y) \cong \Sigma^{i+j}(X \wedge Y)$. In the unbased case, replacing $X$ and $Y$ with $X^+$ and $Y^+$ respectively, one obtains the product in unreduced $K$-theory:

\[\mu : \mathrm{K}^{-i}(X) \otimes \mathrm{K}^{-j}(Y) \to \mathrm{K}^{-(i+j)}(X \times Y),\]

since $X^+ \wedge Y^+ \cong X \times Y$. In particular, this implies $\mathrm{K}^*(pt)$ is a graded ring and for any unbased space $X$, $\mathrm{K}^*(X)$ is a graded $\mathrm{K}^*(pt)$-module.

We have seen that $\mathrm{K}^*(pt)$ are homotopy groups of the $K$-theory spectrum. The ring structure can be explicitly described as follows. Let us specialize to real $K$-theory. Note that $\mathrm{KO}_0 = BO \times \mathbf{Z}$. There is a map $\psi: \displaystyle O(m) \times O(n) \to O(mn)$ given by $\psi(A, B) = A \otimes B$, where $A \otimes B$ is the $mn \times mn$ matrix given by taking tensor product of the two matrices $A$ and $B$. Taking a colimit as $m, n \to \infty$, one obtains a map

\[\displaystyle \psi : O \times O \to O\]

Recall that for any group $G$, there is an explicit model of $BG$ known as the bar construction, given as follows. Consider the category $\mathbf{B} G$ with a single object $G$ and arrows given by self-loops indexed by elements of the group $g \in G$, composition given by multiplication. We may then form the simplicial nerve $N(\mathbf{B} G)$ of the category, whose $k$-simplices are $k$-tuples of composable morphisms. In this case, that just means that a $k$-simplex is a tuple $(g_1, \cdots, g_k)$. The face maps $d_i : N(\mathbf{B} G)_k \to N(\mathbf{B} G)_{k-1}$ and degeneracy maps $s_i : N(\mathbf{B} G)_k \to N(\mathbf{B} G)_{k+1}$ are given by composing at the $i$-th place and inserting the identity morphism (element) at the $i$-th place, respectively:

\[\displaystyle \begin{align*} d_i (g_1, \cdots, g_k) &= (g_1, \cdots, g_{i-1}, g_i g_{i+1}, g_{i+2}, \cdots, g_k) \\ s_i (g_1, \cdots, g_k) &= (g_1, \cdots, g_{i-1}, 1, g_{i+1}, \cdots, g_k) \end{align*}\]

The classifying space of $G$ can then be modelled by the geometric realization of the nerve, i.e. $BG \simeq \vert N(\mathbf{B} G) \vert$. Note that $BG$ has a rogue $0$-simplex $e^0$, corresponding to the empty tuple of composable morphisms. This seems unnecessary, but is in fact quite important for $BG$ to be a based space.

Clearly, this construction is (bi)functorial and hence applying it to $\psi$ gives rise to a map $B\psi : BO \times BO \to BO$. Note that $B\psi$ sends $(e^0 \times BO) \cup (BO \times e^0)$ to $e^0$. Therefore, the map descends to the smash product. We now include the factor $\mathbf{Z}$:

\[\displaystyle \Phi : (BO \times \mathbf{Z}) \times (BO \times \mathbf{Z}) \to (BO \times \mathbf{Z}) \wedge (BO \times \mathbf{Z}) \to BO \times \mathbf{Z}\]

Given by $\Phi((x, m), (y, b)) = (B\psi(x, y), mn)$. The extra factor of $\mathbf{Z}$ does not cause issues with the smashing, because $0$ is an absorbing element in integers under multiplication, so that indeed $(BO \times \mathbf{Z}) \times \{e^0, 0\}$ and $\{e^0, 0\} \times (BO \times \mathbf{Z})$ are sent to $\{e^0, 0\}$ and the map factors through the smash product. We may now define

\[\displaystyle \mu : \mathrm{KO}_n \wedge \mathrm{KO}_m = \Omega^n(\mathrm{KO}_0) \wedge \Omega^m(\mathrm{KO}_0) \to \Omega^{m+n}(\mathrm{KO}_0 \wedge \mathrm{KO}_0) \xrightarrow{\Omega^{m+n}\Phi} \Omega^{m+n}(\mathrm{KO}_0) = \mathrm{KO}_{m+n}\]

Here, we use that if $X, Y$ are two based spaces and $S^n \to X, S^m \to Y$ are two based maps corresponding to points in $\Omega^n X$ and $\Omega^m X$, we may smash them together to obtain a map $S^{m+n} = S^m \wedge S^n \to X \wedge Y$, giving an element of $\Omega^{m+n}(X \wedge Y)$. This says $\mathrm{KO}$ (and, by a similar argument, $\mathrm{KU}$) admits the structure of a ring spectrum (see Chapter III.4 in Adams’ blue book). This implies $\pi_*(\mathrm{KO})$ and $\pi_*(\mathrm{KU})$ admits the structure of a graded ring.

A more general version of the Bott periodicity theorem pins down what these graded rings are explicitly:

\[\displaystyle \begin{align*}\pi_*(\mathrm{KO}) &\cong \frac{\mathbf{Z}[\eta, x, y]}{\langle 2\eta, \eta^3, \eta y, y^2 - 4x \rangle}, &\vert \eta \vert &= 1, \vert y \vert = 4, \vert x \vert = 8 \\ \pi_*(\mathrm{KU}) &\cong \mathbf{Z}[\xi], &\vert \xi \vert &= 2 \end{align*}\]

Recall that we concluded $\mathrm{KO}^*(X)$ (resp. $\mathrm{KU}^*(X)$) is a module over $\pi_*(\mathrm{KO})$ (resp. $\pi_*\mathrm{KU}$). In fact, the Bott periodicity isomorphisms are given by the module multiplication:

\[\displaystyle \begin{align*} \mathrm{KO}^{-n}(X) &\to \mathrm{KO}^{-n-8}(X), &z&\mapsto z \otimes x\\ \mathrm{KU}^{-n}(X) &\to \mathrm{KU}^{-n-2}(X), &z&\mapsto z \otimes \xi \end{align*}\]

We move on to discuss orientability and Thom isomorphism. We do this in some generality. Let $E = \{E_n : n \in \mathbf{Z}\}$ be an $\Omega$-spectrum with a ring structure. In other words, there is a multiplication $\mu : E \wedge E \to E$ and a map $1 : S^0 \to E$ where $S^0 = \{S^n : n \in \mathbf{Z}_{\geq 0}\}$ is the sphere spectrum (which is not an $\Omega$-spectrum), such that $\mu$ is associative and $1$ is the multiplicative unit “upto homotopy”. The data of the multiplicative unit is the same as choice of a distinguished element $1 \in E^0(pt)$. The details, especially regarding an appropriate definition for smash product of spectra, is tricky: a smash product of two spectra is naturally a bi-spectrum, not a spectrum. One needs to reindex by choosing an appropriate totally ordered subset of $\mathbf{N} \times \mathbf{N}$ and demonstrate independence “upto homotopy”. For details, see Chapter III.4 in Adams’ blue book.

In any case, $E$ gives rise to a multiplicative generalized cohomology theory $E^*$. Let $V$ be a real vector space of rank $k$, and $0 \in V$ be the zero element. Then,

\[\displaystyle E^k(V, V \setminus 0) \cong \widetilde{E}^k(S^k) \cong \widetilde{E}^0(S^0) \cong E^0(pt)\]

We say an $E$-orientation on the vector space $V$ is choice of a generator of $u \in E^k(V, V \setminus 0)$, by which we mean $u$ is the iterated suspension of an element of the form $\varepsilon \cdot 1$ where $\varepsilon$ is a unit (i.e. invertible) in $E^0(pt)$ and $1 \in E^0(pt)$ is the distinguished element mentioned before.

Suppose $\pi : V \to X$ is a real vector bundle of rank $k$ over a space $X$. We shall say the vector bundle $(V, X, \pi)$ has a fiberwise $E$-orientation if there exists a choice of a generator $u_x \in E^k(V_x, V_x \setminus 0_x)$ for every $x \in X$ and fiber $V_x$ over $x$, such that for any two points $x, y \in X$ and any trivializing open set $U \subset X$ containing $x$ and $y$, the composition of the excision isomorphisms

\[\displaystyle E^k(V_x, V_x \setminus 0_x) \xleftarrow{\cong} E^k(V_U, V_U \setminus 0_U) \xrightarrow{\cong} E^k(V_y, V_y \setminus 0_y),\]

takes $u_x$ to $u_y$. In that case, the classes $\{u_x\}$ patch together to a well-defined class $u \in E^k(V, V \setminus 0_X)$ that we call the Thom class of the vector bundle $V$. Note that $(V, V \setminus 0_X)$ deformation retracts to the good pair $(\mathbb{D}(V), \mathbb{S}(V))$ given by the unit disk and sphere bundles of $V$, respectively. Thus, $u$ defines an element of

\[\displaystyle E^k(V, V \setminus 0_X) \cong E^k(\mathbb{D}(V), \mathbb{S}(V)) = \widetilde{E}^k(Th(V)),\]

where $Th(V) = \mathbb{D}(V)/\mathbb{S}(V)$ is the Thom space of $V$. Consider the homomorphism

\[\displaystyle \Phi : E^n(X) \to E^{n+k}(V, V \setminus 0_X) \cong \widetilde{E}^{n+k}(Th(V))\]

given by $\Phi(z) = \pi^*z \smile u$, where $\smile$ denotes the (relative) product structure on the cohomology theory $E^*$. We claim that $\Phi$ is an isomorphism for all $n$. We effectively repeat the same proof as the usual Thom isomorphism theorem. Note that as $E^*$ is a generalized cohomology theory, it satisfies the Mayer-Vietoris sequence (it’s a nice exercise to derive Mayer-Vietoris from the axioms). For any subset $U \subset X$, let the restriction of $V$ to $U$ be denoted $V_U := \pi^{-1}(U)$. Let the zero section of $V$ over $U$ be denoted as $0_U \subset V_U$. For convenience let us denote $\dot{V}_U := V \setminus 0_U$. Let

\[\Phi_U : E^n(U) \to E^{n+k}(V_U, V_U \setminus 0_U)\]

denote the Thom homomorphism restricted to the bundle $V_U$ over $U$. Suppose the Thom isomorphism theorem holds for two open sets $U_1, U_2 \subset X$ and for the intersection $U_1 \cap U_2$. We show it holds for $U_1 \cup U_2$. Let us denote $V_1 := V_{U_1}$, $V_2 := V_{U_2}$, $V_\cap := V_{U_1 \cap U_2}$ and $V_\cup := V_{U_1 \cup U_2}$. By naturality of the Mayer-Vietoris sequence and multiplicativity, we have a commutative diagram:

\[\displaystyle \begin{CD} E^{n+k}(V_\cup, \dot{V}_\cup) @>>> E^{n+k}(V_1, \dot{V}_1) \oplus E^{n+k}(V_2, \dot{V}_2) @>>> E^{n+k}(V_\cap, \dot{V}_\cap)\\ @A{\Phi_{\cup}}AA @A{\Phi_1\oplus \Phi_2}AA @A{\Phi_\cap}AA \\ E^n(U_\cup) @>>> E^n(U_1) \oplus E^n(U_2) @>>> E^n(U_\cap) \end{CD}\]

By hypothesis, the second and third vertical arrows are isomorphisms. Extending both rows of Mayer-Vietoris sequences two steps to the left and using the five-lemma shows the first arrow is also an isomorphism, as desired. The result then boils down to a local computation where $U$ is a contractible open subset of $X$. Upto homotopy, we may take $U = pt$. Then,

\[\displaystyle \Phi_{pt} : \widetilde{E}^n(S^0) \cong E^n(pt) \to E^{n+k}(V_x, V_x \setminus 0_x) \cong \widetilde{E}^{n+k}(S^k)\]

given by $\Phi_{pt}(z) = z \smile u_x$, is simply the suspension isomorphism. This proves the result.

Therefore, the Thom isomorphism theorem in $E^*$-cohomology is applicable whenever the bundle is $E$-orientable. Therefore, it is natural to ask the question: when does a vector bundle $V$ over $X$ admit a $\mathrm{KO}$- or $\mathrm{KU}$-orientation?

Proposition. A real vector bundle $\pi : V \to X$ of rank $k$ admits a $\mathrm{KO}$-orientation if and only if $V$ is spin, and a $\mathrm{KU}$-orientation if and only if $V$ is spinc.

We only prove the “if” direction. Let us deal with the real case; the complex case is completely analogous. Suppose $k \equiv 0 \pmod 8$. Since $V$ is spin, there is a principal $\mathrm{Spin}(k)$-bundle $P \to X$ and an isomorphism $V \cong P \times_{\mathrm{Spin}(k)} \mathbf{R}^k$. Let $S = S^+ \oplus S^-$ be the unique irreducible graded $\mathrm{Cliff}(\mathbf{R}^k)$-module. Set $\mathbb{S}^{\pm} = P \times_{\mathrm{Spin}(k)} S^{\pm}$, so that $\mathbb{S} := \mathbb{S}^+ \oplus \mathbb{S}^-$ is a bundle of Clifford modules over $V$. Consider the two-term complex:

\[0 \to \pi^* \mathbb{S}^+ \stackrel{\sigma}{\to} \pi^* \mathbb{S}^- \to 0\]

where $\sigma_{(x, v)}(e) = v \cdot e$ on the fiber of $\mathbb{S}^+$ over the point $(x, v) \in V$. Here, by $\cdot$ we mean the Clifford multiplication. Note that the complex is exact away from the zero section of $V$, since $v^2 = -|v| \neq 0$ if $v \neq 0$. Consequently, the virtual bundle

\[\chi_V(\mathbb{S}) := [\pi^* \mathbb{S}^+] - [\pi^* \mathbb{S}^-]\]

defines a class in $\mathrm{KO}^0(V, V \setminus 0)$. By Bott $8$-periodicity, we obtain a class $\mu \in \mathrm{KO}^{-k}(V, V \setminus 0)$. This restricts to a generator on every fiber by construction, so $\mu$ defines a Thom class for $V$. If the rank $k \neq 0 \pmod{8}$, we may stabilize $V$ by a trivial real bundle of appropriate rank so that the new rank is $0\pmod{8}$. Note that for any ringed spectrum $E$, the vector bundle $\pi : V \to X$ is $E$-orientable iff $V \oplus \varepsilon^i$ is $E$-orientable; this can be easily seen by the suspension isomorphism in the cohomology theory represented by $E$. This proves spin bundles are $\mathrm{KO}$-orientable.