Complex (topological) K-theory of $S \Sigma$ for a surface $\Sigma$. In the following question here:
What's the K-group of a surface?
The $K$-theory of a compact orientable surface is computed. I was curious if it was also possible to compute the "higher" $K$-groups as well. My understanding is that, by Bott periodicity, it would suffice to compute $K(S \Sigma)$, where $S \Sigma$ is the suspension of $\Sigma$. Moreover, I think that one can do this rationally by comparing it to rational cohomology. Is it easy to compute $K(S \Sigma)$ integrally?
 A: Below by $B^n A$ I mean the Eilenberg-MacLane space $K(A, n)$.
The Chern character isomorphism tells you how to compute rational K-theory; in this case it gives
$$K^0(\Sigma) \otimes \mathbb{Q} \cong H^0(\Sigma, \mathbb{Q}) \times H^2(\Sigma, \mathbb{Q}) \cong \mathbb{Q}^2$$
$$K^1(\Sigma) \otimes \mathbb{Q} \cong H^1(\Sigma, \mathbb{Q}) \cong \mathbb{Q}^{2g}.$$
In other words, rational K-theory is periodic rational cohomology. It turns out that in this case integral K-theory is also periodic integral cohomology; that is, the following holds.

Theorem: Let $\Sigma$ be a compact orientable surface of genus $g$. Then
$$K^0(\Sigma) \cong [\Sigma, \mathbb{Z} \times BU] \cong H^0(\Sigma, \mathbb{Z}) \times H^2(\Sigma, \mathbb{Z}) \cong \mathbb{Z}^2$$
$$K^1(\Sigma) \cong [\Sigma, U] \cong H^1(\Sigma, \mathbb{Z}) \cong \mathbb{Z}^{2g}.$$

We will give three proofs of this.
Proof 1: the Atiyah-Hirzebruch spectral sequence.
This is a spectral sequence
$$H^p(\Sigma, K^q(\text{pt})) \to K^{p+q}(\Sigma)$$
which in this case degenerates at the $E_2$ page. All of the groups appearing on the $E_2$ page are free abelian, so there are no extension problems and we get isomorphisms
$$K^0(\Sigma) \cong H^0(\Sigma, \mathbb{Z}) \times H^2(\Sigma, \mathbb{Z}) \cong \mathbb{Z}^2$$
$$K^1(\Sigma) \cong H^1(\Sigma, \mathbb{Z}) \cong \mathbb{Z}^{2g}.$$
as desired. $\Box$
Proof 2: obstruction theory in the source.
We'll describe more generally a computation of $[\Sigma, Y]$ for a nice space $Y$.

Theorem: Let $Y$ be a path-connected space with $\pi_1(Y)$ abelian and the action of $\pi_1(Y)$ on $\pi_2(Y)$ trivial. Then
$$[\Sigma, Y] \cong \text{Hom}(\pi_1(\Sigma), \pi_1(Y)) \times \pi_2(Y).$$

Proof. Pick a basepoint for $\Sigma$ and for $Y$. We will first work out the answer with a basepoint and then remove the dependence on it later.
As Mariano says in the comments, the starting point is to think of a compact orientable surface $\Sigma$ of genus $g$ as the mapping cone, or homotopy cofiber, of the map $S^1 \to (S^1)^{\vee 2g}$ given on fundamental groups by the inclusion of $[a_1, b_1] ... [a_g, b_g]$. We know how to map the $1$-skeleton $(S^1)^{\vee 2g}$ into an arbitrary pointed space; this is precisely a $2g$-tuple of loops in $Y$ based at the same point. The identification of $\Sigma$ as the homotopy cofiber of a map from $S^1$ tells us that the space of extensions of a pointed map $(S^1)^{\vee 2g} \to Y$ to a pointed map $\Sigma \to Y$ is precisely the space of null-homotopies of the image of the loop $[a_1, b_1] ... [a_g, b_g]$ in $Y$.
Passing to homotopy classes, we want to identify the space of homotopy classes of pairs consisting of $2g$ based loops $a_1, b_1, ..., a_g, b_g$ in $Y$ and a null-homotopy of the loop $[a_1, b_1] ... [a_g, b_g]$. This data gives us in particular an element of $\text{Hom}(\pi_1(\Sigma), \pi_1(Y))$, but there is extra data given by the choice of null-homotopy. In general, the homotopy classes of null-homotopies of a loop in $Y$ are an affine space over $\pi_2(Y)$, and so in particular noncanonically in bijection with $\pi_2(Y)$. We may furthermore replace the loop by a homotopic loop, but this doesn't matter since the action of $\pi_1(Y)$ on $\pi_2(Y)$ is trivial. This gives
$$[\Sigma, Y]_{\ast} \cong \text{Hom}(\pi_1(\Sigma), \pi_1(Y)) \times \pi_2(Y)$$
(noncanonically), where $[-, -]_{\ast}$ denotes based homotopy classes. The above argument works with any choice of basepoint in $Y$, so now consider the effect of freely changing basepoints in $Y$, which is one way to think of what happens when we pass to unbased homotopy classes. Picking distinguished paths from the basepoint we started with in $Y$ to all other basepoints, the result, after translating to the basepoint we started with, is that all of the data above gets acted on by $\pi_1(Y)$ (by conjugation on $\pi_1(Y)$ and by the usual action on $\pi_2(Y)$), but by hypothesis this action is trivial and the result follows. $\Box$
To complete the computation, applying the theorem above gives
$$[\Sigma, \mathbb{Z} \times BU] \cong \mathbb{Z} \times \pi_2(BU) \cong \mathbb{Z}^2$$
$$[\Sigma, U] \cong \text{Hom}(\pi_1(\Sigma), \mathbb{Z}) \cong \mathbb{Z}^{2g}$$
since in the first case $\pi_1$ vanishes and in the second case $\pi_1$ is abelian and $\pi_2$ vanishes. $\Box$
Proof 3: obstruction theory in the target.
Mostly as an exercise for me, I want to make use of the following general result.

Theorem: Let $X$ be a finite CW-complex of dimension at most $n$, let $Y$ be a space with a Postnikov tower of principal fibrations, and let $Y_n$ be the $n$-truncation of $Y$. Then the map $Y \to Y_n$ induces a bijection
$$[X, Y] \cong [X, Y_n].$$

The $n$-truncation $Y_n$ of a space is a space together with a map $Y \to Y_n$ inducing isomorphisms on $\pi_k, 0 \le k \le n$ and such that all higher homotopy of $Y_n$ vanishes. It can be constructed from $Y$ by attaching cells of suitably high dimensions to kill all of the higher homotopy of $Y$. In particular, by definition it satisfies $[S^k, Y] \cong [S^k, Y_n]$ for all $0 \le k \le n$, and the theorem just says that this implies the corresponding fact not only for these spheres but for spaces built out of them.
Proof. The truncation map $Y \to Y_n$ induces a map $[X, Y] \to [X, Y_n]$ and we want to show that it is a bijection. We'll do this by proving a corresponding bijection for the entire Postnikov tower
$$\cdots \to Y_{n+2} \to Y_{n+1} \to Y_n \to \cdots$$
of $Y$. By hypothesis, the spaces in the Postnikov tower fit into principal fibrations giving fibration sequences
$$B^{n+1} \pi_{n+1}(Y) \to Y_{n+1} \to Y_n \xrightarrow{k_n} B^{n+2} \pi_{n+1}(Y).$$
Any map $X \to Y_n$ induces a map $X \to B^{n+2} \pi_{n+1}(Y)$ giving a cohomology class in $H^{n+2}(X, \pi_{n+1}(Y))$, and $X$ lifts to a map $X \to Y_{n+1}$ if and only if this cohomology class vanishes. But in this case we know that $H^{n+2}(X, -)$ vanishes with any coefficients, so a lift always exists; hence $[X, Y_{n+1}] \to [X, Y_n]$ is surjective.
Knowing that a lift exists, the set of (weak) homotopy classes of lifts is an affine space over $[X, B^{n+1} \pi_{n+1}(Y)] \cong H^{n+1}(X, \pi_{n+1}(Y))$, which also vanishes; hence lifts are unique up to homotopy, and $[X, Y_{n+1}] \to [X, Y_n]$ is a bijection. By induction $[X, Y_{n+k}] \to [X, Y_n]$ is a bijection for all $k \ge 0$, and by reconstructing $Y$ as the homotopy limit over the $Y_{n+k}, k \ge 0$ we get a bijection $[X, Y] \cong [X, Y_{n+k}], k \ge 0$. $\Box$
To complete the computation, $U$ and $BU$ both have abelian fundamental groups which act trivially on their higher homotopy (since they are both infinite loop spaces); these are the spaces which have Postnikov towers of principal fibrations, so the above theorem applies to them. Applying the theorem gives
$$[\Sigma, \mathbb{Z} \times BU] \cong \mathbb{Z} \times H^2(\Sigma, \mathbb{Z}) \cong \mathbb{Z}^2$$
$$[\Sigma, U] \cong H^1(\Sigma, \mathbb{Z}) \cong \mathbb{Z}^{2g}$$
since the $2$-truncations of $BU$ and $U$ both have a single nonzero homotopy group and hence must be Eilenberg-MacLane spaces, namely $B^2 \mathbb{Z}$ and $B \mathbb{Z}$ respectively. $\Box$
All three proofs show more generally that integral K-theory agrees with periodic integral cohomology for any finite CW-complex of dimension at most $2$. For finite CW-complexes of dimension $3$, the third proof tells you that this continues to hold for $K^0$, but for $K^1$ the $3$-truncation of $U$ may be an interesting homotopy $3$-type.
