Complex K-theory of an even-cell complex

If a finite CW-complex $Y$ has only even-dimensional cells, thenand $K^0(Y)$ is free abelian, with one generator for each cell. This follows by induction from the six-term exact sequence for adjoining a wedge of even-dimensional cells.

If $Y$ is a finite even-cell complex, the exterior product is an isomorphismCellular induction proves this because $K^0(Y)$ is free and $K^{-1}(Y)=0$.