In this thesis we study the theory of CW-complexes in a synthetic way. We first introduced some important concepts of homotopy type theory and used them to define cell complexes as a homotopical pushout. After that, we have proven some topological properties of CW-complexes and calculated the homotopy groups of some examples with methods in type theory.

We study the slice filtrations of motivic ring spectra and spectral sequences associated to them. Using décalage, we could relate Adams-Novikov spectral sequences with slice spectral sequences after suitable Betti realization and étale realization, this works in both characteristic zero and $p$ cases.

Slight changes after the submitted version.