and the vertical mappings are weak homotopy equivalences. Principal fibrations play an important role in Postnikov towers. Long exact sequence of homotopy groups For a Serre fibration there exists a long exact sequence of homotopy groups. For base points and this is given by:The homomorphisms and are the induced homomorphisms of the inclusion and the projection Hopf fibration Hopf fibrations are a family of fiber bundles whose fiber, total space and base space are spheres:The long exact sequence of homotopy groups of the hopf fibration yields: This sequence splits into short exact sequences, as the fiber in is contractible to a point:This short exact sequence splits because of the suspension homomorphism and there are isomorphisms:The homotopy groups are trivial for so there exist isomorphisms between and for Analog the fibers in and in are contractible to a point. Further the short exact sequences split and there are families of isomorphisms: and Spectral sequence Spectral sequences are important tools in algebraic topology for computing (co-)homology groups. The Leray-Serre spectral sequence connects the (co-)homology of the total space and the fiber with the (co-)homology of the base space of a fibration. For a fibration with fiber where the base space is a path connected CW-complex, and an additive homology theory there exists a spectral sequence: Fibrations do not yield long exact sequences in homology, as they do in homotopy. But under certain conditions, fibrations provide exact sequences in homology. For a fibration with fiber where base space and fiber are path connected, the fundamental group acts trivially on and in addition the conditions for and for hold, an exact sequence exists (also known under the name Serre exact sequence):This sequence can be used, for example, to prove Hurewicz`s theorem or to compute the homology of loopspaces of the form For the special case of a fibration where the base space is a -sphere with fiber there exist exact sequences (also called Wang sequences) for homology and cohomology: Orientability For a fibration with fiber and a fixed commuative ring with a unit, there exists a contravariant functor from the fundamental groupoid of to the category of graded -modules, which assigns to the module and to the path class the homomorphism where is a homotopy class in A fibration is called orientable over if for any closed path in the following holds: Euler characteristic For an orientable fibration over the field with fiber and