The notion of a fibration generalizes the notion of a fiber bundle and plays an important role in algebraic topology, a branch of mathematics.

Fibrations are used, for example, in Postnikov systems or obstruction theory.

In this article, all mappings are continuous mappings between topological spaces.

Formal definitionsEdit

Homotopy lifting propertyEdit

A mapping   satisfies the homotopy lifting property for a space   if:

  • for every homotopy   and
  • for every mapping (also called lift)   lifting   (i.e.  )

there exists a (not necessarily unique) homotopy   lifting   (i.e.  ) with  

The following commutative diagram shows the situation: 



A fibration (also called Hurewicz fibration) is a mapping   satisfying the homotopy lifting property for all spaces   The space   is called base space and the space   is called total space. The fiber over   is the subspace   

Serre fibrationEdit

A Serre fibration (also called weak fibration) is a mapping   satisfying the homotopy lifting property for all CW-complexes. 

Every Hurewicz fibration is a Serre fibration.


A mapping   is called quasifibration, if for every     and   holds that the induced mapping   is an isomorphism.

Every Serre fibration is a quasifibration. 


  • The projection onto the first factor   is a fibration. That is, trivial bundles are fibrations.
  • Every covering   satisfies the homotopy lifting property for all spaces. Specifically, for every homotopy   and every lift   there exists a uniquely defined lift   with    
  • Every fiber bundle   satisfies the homotopy lifting property for every CW-complex. 
  • A fiber bundle with a paracompact and Hausdorff base space satisfies the homotopy lifting property for all spaces. 
  • An example for a fibration, which is not a fiber bundle, is given by the mapping   induced by the inclusion   where     a topological space and   is the space of all continuous mappings with the compact-open topology. 
  • The Hopf fibration   is a non trivial fiber bundle and specifically a Serre fibration.

Basic conceptsEdit

Fiber homotopy equivalenceEdit

A mapping   between total spaces of two fibrations   and   with the same base space is a fibration homomorphism if the following diagram commutes:


The mapping   is a fiber homotopy equivalence if in addition a fibration homomorphism   exists, such that the mappings   and   are homotopic, by fibration homomorphisms, to the identities   and   

Pullback fibrationEdit

Given a fibration   and a mapping  , the mapping   is a fibration, where   is the pullback and the projections of   onto   and   yield the following commutative diagram:


The fibration   is called the pullback fibration or induced fibration. 

Pathspace fibrationEdit

With the pathspace construction, any continuous mapping can be extended to a fibration by enlarging its domain to a homotopy equivalent space. This fibration is called pathspace fibration.

The total space   of the pathspace fibration for a continuous mapping   between topological spaces consists of pairs   with   and paths   with starting point   where   is the unit interval. The space   carries the subspace topology of   where   describes the space of all mappings   and carries the compact-open topology.

The pathspace fibration is given by the mapping   with   The fiber   is also called the homotopy fiber of   and consists of the pairs   with   and paths   where   and   holds.

For the special case of the inclusion of the base point  , an important example of the pathspace fibration emerges. The total space   consists of all paths in   which starts at   This space is denoted by   and is called path space. The pathspace fibration   maps each path to its endpoint, hence the fiber   consists of all closed paths. The fiber is denoted by   and is called loop space. 


  • The fibers   over   are homotopy equivalent for each path component of   
  • For a homotopy   the pullback fibrations   and   are fiber homotopy equivalent. 
  • If the base space   is contractible, then the fibration   is fiber homotopy equivalent to the product fibration   
  • The pathspace fibration of a fibration   is very similar to itself. More precisely, the inclusion   is a fiber homotopy equivalence. 
  • For a fibration   with fiber   and contractible total space, there is a weak homotopy equivalence   

Puppe sequenceEdit

For a fibration   with fiber   and base point   the inclusion   of the fiber into the homotopy fiber is a homotopy equivalence. The mapping   with  , where   and   is a path from   to   in the base space, is a fibration. Specifically it is the pullback fibration of the pathspace fibration  . This procedure can now be applied again to the fibration   and so on. This leads to a long sequence:


The fiber of   over a point   consists of the pairs   with closed paths   and starting point  , i.e. the loop space  . The inclusion   is a homotopy equivalence and iteration yields the sequence:


Due to the duality of fibration and cofibration, there also exists a sequence of cofibrations. These two sequences are known as the Puppe sequences or the sequences of fibrations and cofibrations. 

Principal fibrationEdit

A fibration   with fiber   is called principal, if there exists a commutative diagram:


The bottom row is a sequence of fibrations and the vertical mappings are weak homotopy equivalences. Principal fibrations play an important role in Postnikov towers. 

Long exact sequence of homotopy groupsEdit

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 fibrationEdit

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:


Spectral sequenceEdit

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:



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 characteristicEdit

For an orientable fibration   over the field   with fiber   and path connected base space, the Euler characteristic of the total space is given by:


Here the Euler characteristics of the base space and the fiber are defined over the field  . 

See alsoEdit


  1. Hatcher, Allen (2001). Algebraic Topology. NY: Cambridge University Press. ISBN 0-521-79160-X.
  2. Laures, Gerd; Szymik, Markus (2014). Grundkurs Topologie (in German) (2nd ed.). Springer Spektrum. doi:10.1007/978-3-662-45953-9. ISBN 978-3-662-45952-2.
  3. May, J.P. (1999). A Concise Course in Algebraic Topology (PDF). University of Chicago Press. ISBN 0-226-51182-0. OCLC 41266205.
  4. Spanier, Edwin H. (1966). Algebraic Topology. McGraw-Hill Book Company. ISBN 978-0-387-90646-1.
  5. Dold, Albrecht; Thom, René (1958). "Quasifaserungen und Unendliche Symmetrische Produkte". Annals of Mathematics. 67 (2): 239–281. doi:10.2307/1970005. JSTOR 1970005.
  6. Steenrod, Norman (1951). The Topology of Fibre Bundles. Princeton University Press. ISBN 0-691-08055-0.
  7. Davis, James F.; Kirk, Paul (1991). Lecture Notes in Algebraic Topology (PDF). Department of Mathematics, Indiana University.
  8. Cohen, Ralph L. (1998). The Topology of Fiber Bundles Lecture Notes (PDF). Stanford University.