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 ∗ : π i ( E , p − 1 ( x ) , y ) → π i ( B , x ) {\displaystyle 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
⋯ → π i + 1 ( B , x ) → π i ( p − 1 ( x ) , y ) → π i ( E , y ) → π i ( B , x ) → … → π 0 ( B , x ) → 0 {\displaystyle {\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 f0 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 Fb of p over b is a weak equivalence for all b ∈ B. To see this, recall that Fb is the fibre of q under b where q: Ep → B is the usual path fibration construction. Thus, one has
E p = { ( e , γ ) ∈ E × B I : γ ( 0 ) = p ( e ) } {\displaystyle 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 → Ep, given by φ(e) = (e, p(e)), where p(e) denotes the corresponding constant path. By definition, p factors through Ep such that one gets a commutative diagram
Applying πn 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 Mf → 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 πi(Mf, p−1(b)) = 0 = πi(I, b) holds for all i ∈ I and b ∈ B. But by the long exact sequence of the pair (Mf, 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 Mf. 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:
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