Comparing two approaches to the K-theory classification of D-branes

We consider the two main classification methods of D-brane charges via K-theory, in type II superstring theory with vanishing BB-field: the Gysin map approach and the one based on the Atiyah–Hirzebruch spectral sequence. Then, we find out an explicit link between these two approaches: the Gysin map provides a representative element of the equivalence class obtained via the spectral sequence. We also briefly discuss the case of rational coefficients, characterized by a complete equivalence between the two classification methods.