Quasi-fibration

 ( Algebraic Topology ) 

In algebraic topology, a quasifibration is a generalisation of 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.

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Definition A continuous surjective map of topological spaces p: E → B is called a quasifibration if it induces isomorphisms

$p\_*\backslash colon\; \backslash pi\_i(E,p^\{-1\}(x),y)\; \backslash to\; \backslash pi\_i(B,x)$

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

$\backslash begin\{align\}$

\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{align}

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 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

$E\_p=\backslash \{(e,\backslash gamma)\backslash in\; E\backslash times\; B^I:\backslash gamma(0)=p(e)\backslash \}$

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.

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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 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.

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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.Dold and Thom (1958), Satz 2.2

Theorem. Let p: E → B be a continuous map where B is the inductive limit of a sequence B₁ ⊂ B₂ ⊂ ... 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 space 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.

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Notes

• (2026). 9780387224893, Springer Science & Business Media. ISBN 9780387224893
• Allen Hatcher (2026). 9780521795401, Cambridge University Press. . ISBN 9780521795401
• Renzo A. Piccinini (1992). 9780080872827, Elsevier. ISBN 9780080872827

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Further reading

• Quasifibrations and homotopy pullbacks on MathOverflow
• Quasifibrations from the Lehigh University
• http://pantodon.jp/index.rb?body=quasifibration in Japanese

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