Quasi-fibration

In algebraic topology, a quasifibration is a generalisation of fibre bundles and fibrations introduced by Albrecht Dold and René Thom. Roughly speaking, it is a continuous map p: E → B having the same behaviour as a fibration regarding the homotopy groups of E, B and p⁻¹. Equivalently, one can define a quasifibration to be a continuous map such that the inclusion of each fibre into its homotopy fibre is a weak equivalence. One of the main applications of quasifibrations lies in proving the Dold-Thom theorem.

Definition

A continuous surjective map of topological spaces p: E → B is called a quasifibration if it induces isomorphisms
for all x ∈ B, y ∈ p⁻¹ and i ≥ 0. For i = 0,1 one can only speak of bijections between the two sets.
By definition, quasifibrations share a key property of fibrations, namely that a quasifibration p: E → B induces a long exact sequence of homotopy groups
as follows directly from the long exact sequence for the pair.
This long exact sequence is also functorial in the following sense: Any fibrewise map f: E → E′ induces a morphism between the exact sequences of the pairs and and therefore a morphism between the exact sequences of a quasifibration. Hence, the diagram
commutes with f₀ being the restriction of f to p⁻¹ and x′ being an element of the form p′ for an e ∈ p⁻¹.
An equivalent definition is saying that a surjective map p: E → B is a quasifibration if the inclusion of the fibre p⁻¹ into the homotopy fibre F_b of p over b is a weak equivalence for all b ∈ B. To see this, recall that F_b is the fibre of q under b where q: E_p → B is the usual path fibration construction. Thus, one has
and q is given by q = γ. Now consider the natural homotopy equivalence φ : E → E_p, given by φ =, where p denotes the corresponding constant path. By definition, p factors through E_p such that one gets a commutative diagram
Applying π_n yields the alternative definition.

Examples

Every Serre fibration is a quasifibration. This follows from the Homotopy lifting property.
The projection of the letter L onto its base interval is a quasifibration, but not a fibration. More generally, the projection M_f → I of the mapping cylinder of a map f: X → Y between connected CW complexes onto the unit interval is a quasifibration if and only if π_i = 0 = π_i holds for all i ∈ I and b ∈ B. But by the long exact sequence of the pair and by Whitehead's theorem, this is equivalent to f being a homotopy equivalence. For topological spaces X and Y in general, it is equivalent to f being a weak homotopy equivalence. Furthermore, if f is not surjective, non-constant paths in I starting at 0 cannot be lifted to paths starting at a point of Y outside the image of f in M_f. This means that the projection is not a fibration in this case.
The map SP : SP → SP induced by the projection p: X → X/''A'' is a quasifibration for a CW pair consisting of two connected spaces. This is one of the main statements used in the proof of the Dold-Thom theorem. In general, this map also fails to be a fibration.

Properties

The following is a direct consequence of the alternative definition of a fibration using the homotopy fibre:
A corollary of this theorem is that all fibres of a quasifibration are weakly homotopy equivalent if the base space is path-connected, as this is the case for fibrations.
Checking whether a given map is a quasifibration tends to be quite tedious. The following two theorems are designed to make this problem easier. They will make use of the following notion: Let p: E → B be a continuous map. A subset U ⊂ p is called distinguished if p: p⁻¹ → U is a quasifibration.
To see that the latter statement holds, one only needs to bear in mind that continuous images of compact sets in B already lie in some B_n. That way, one can reduce it to the case where the assertion is known.
These two theorems mean that it suffices to show that a given map is a quasifibration on certain subsets. Then one can patch these together in order to see that it holds on bigger subsets and finally, using a limiting argument, one sees that the map is a quasifibration on the whole space. This procedure has e.g. been used in the proof of the Dold-Thom theorem.