# 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 (relative) homotopy groups of E, B and p$^{−1}$(x). 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
    $p_{*}\colon \pi _{i}(E,p^{-1}(x),y)\to \pi _{i}(B,x)$
for all x ∈ B, y ∈ p$^{−1}$(x) 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
    ${\begin{aligned}\dots \to \pi _{i+1}(B,x)\to \pi _{i}(p^{-1}(x),y)\to \pi _{i}(E,y)&\to \pi _{i}(B,x)\to \dots \\&\to \pi _{0}(B,x)\to 0\end{aligned}}$
as follows directly from the long exact sequence for the pair (E, p$^{−1}$(x)).
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 (E, p$^{−1}$(x)) and (E′, p′$^{−1}$(x)) and therefore a morphism between the exact sequences of a quasifibration. Hence, the diagram
commutes with f$_0$ being the restriction of f to p$^{−1}$(x) and x′ being an element of the form p′(f(e)) for an e ∈ p$^{−1}$(x).
An equivalent definition is saying that a surjective map p: E → B is a quasifibration if the inclusion of the fibre p$^{−1}$(b) 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
    $E_{p}=\{(e,\gamma )\in E\times B^{I}:\gamma (0)=p(e)\}$
and q is given by q(e, γ) = γ(1). Now consider the natural homotopy equivalence φ : E → E$_p$, given by φ(e) = (e, p(e)), where p(e) 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$_i$(M$_f$, p$^{−1}$(b)) = 0 = π$_i$(I, b) holds for all i ∈ I and b ∈ B. But by the long exact sequence of the pair (M$_f$, p$^{−1}$(b)) 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(p) : SP(X) → SP(X/A) induced by the projection p: X → X/A is a quasifibration for a CW pair (X, A) 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:
    Theorem. Every quasifibration p: E → B factors through a fibration whose fibres are weakly homotopy equivalent to the ones of p.
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(E) is called distinguished (with respect to p) if p: p$^{−1}$(U) → U is a quasifibration.
    Theorem. If the open subsets U,V and U ∩ V are distinguished with respect to the continuous map p: E → B, then so is U ∪ V.
    Theorem. Let p: E → B be a continuous map where B is the inductive limit of a sequence B$_1$ ⊂ B$_2$ ⊂ ... All B$_n$ are moreover assumed to satisfy the first separation axiom. If all the B$_n$ are distinguished, then p 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.