I'm reading the paper Loop groups and twisted K-theory I by Freed, Hopkins, and Teleman. They give some examples of computing (twisted) K groups using the Mayer-Vietoris sequence.
I'm a bit confused with some of their computations, for instance $S^3$ (their example 1.4 in the first section). They take subsets $U_+ = S^3 \backslash(0,0,0,-1)$ and $U_- = S^3 \backslash(0,0,0,1)$ and then they say that $K^0(U_\pm) \simeq \mathbb Z$. I don't understand where this comes from since $U_\pm$ are non-compact so I believe $K^0(U_\pm)$ should be the reduced $K^0$ of a 1-point compactification. The compactifications of these spaces are $S^3$ so shouldn't $K^0(U_\pm) = \tilde {K^0}(S^3) = 0$? However, it seems like if you replace $U_\pm$ by shrinking it a bit to make it closed, these computations work out.
So I'm wondering what the exact statement of Mayer-Vietoris is for $K$-theory (specifically, what type of covers you can take) or if Freed, Hopkins, and Teleman are using a different definition of $K^0$ for which $K^0(U_\pm)$ is indeed $\mathbb Z$. Any references would also be appreciated since I couldn't find much in the literature about a Mayer-Vietoris sequence for $K$-theory.
Best Answer
Freed, Hopkins and Teleman will be using the homotopical definition $K^0(X)=[X,\mathbb{Z}\times BU]$. For many spaces $X$ this is the same as the Grothendieck group of vector bundles on $X$; in particular this holds if $X$ is compact Hausdorff, or if it is a finite-dimensional CW complex. This definition is visibly homotopy invariant, and the spaces $U_{\pm}$ are contractible, so $K^0(U_{\pm})=K^0(\text{point})=\mathbb{Z}$.
For a noncompact manifold $M$ we can also consider $\widetilde{K}^0(M\cup\{\infty\})$. This is the Grothendieck group of vector bundles on $M$ with a specified trivialisation outside of a compact set. This is also interesting, but different.
With the homotopical version of $K$-theory, you get a Mayer-Vietoris sequence for $K^0(A\cup B)$ whenever the map from the homotopy pushout of $A\xleftarrow{}A\cap B\to B$ to $A\cup B$ is a homotopy equivalence. Provided that all the relevant spaces have the homotopy type of CW complexes, this essentially means that you get a Mayer-Vietoris sequence in $K$-theory iff you get one in ordinary singular cohomology. In particular, this will certainly work if $A$ and $B$ are open subsets of a manifold.