Serre fibration Hurewicz fibration Dold fibration shrinkable map
for the set of inclusions of the topological n-disks, into their cylinder objects, along (for definiteness) the left endpoint inclusion.
I.e. is a Serre fibration if for every commuting square of continuous functions of the form
then there exists a continuous function such as to make a commuting diagram of the form
Long exact sequences of homotopy groups
Since Serre fibrations are the abstract fibrations in the Serre-classical model structure on topological spaces, the following statement follows from general model category theory. But it may also be seen by direct inspection, as follows.
First erect a cylinder over all 0-cells
Assume then that the cylinder over all -cells of has been erected using attachment from . Then the union of any -cell of with the cylinder over its boundary is homeomorphic to and is like the cylinder over the cell “with end and interior removed”. Hence via attaching along the cylinder over is erected.
Let be a Serre fibration, def. 2, let be any point and write
for the fiber inclusion over that point. Then for every choice of lift of the point through , the induced sequence of homotopy groups
is exact, in that the kernel of is canonically identified with the image of :
It is clear that the image of is in the kernel of (every sphere in becomes constant on , hence contractible, when sent forward to ).
For the converse, let be represented by some . Assume that is in the kernel of . This means equivalently that fits into a commuting diagram of the form
where is the contracting homotopy witnessing that .
Now since is a lift of , there exists a left homotopy
(for instance: regard as embedded in such that is identified with the basepoint on the boundary of and set ).
The pasting of the top two squares that have appeared this way is equivalent to the following commuting square
Because is a Serre fibration and by lemma 1 and prop. 1, this has a lift
Notice that is a basepoint preserving left homotopy from to some . Being homotopic, they represent the same element of :
But the new representative has the special property that its image in is not just trivializable, but trivialized: combining with the previous diagram shows that it sits in the following commuting diagram
The commutativity of the outer square says that is constant, hence that is entirely contained in the fiber . Said more abstractly, the universal property of fibers gives that factors through , hence that is in the image of .
long exact sequence of homotopy groups
- Every locally trivial topological fiber bundle is in particular a Serre fibration.