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Path space fibration

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In algebraic topology, the path space fibration over a based space (X, *) is a fibration of the form

Contents

Ω X ↪ P X → χ ↦ χ ( 1 ) X

where

  • P X = Map ⁡ ( I , X ) = { f : I → X | f ( 0 ) = ∗ } is the space called the path space of X.
  • Ω X is the fiber of χ ↦ χ ( 1 ) over the base point of X; thus it is the loop space of X.

    • The space X I consists of all maps from I to X that may not preserve the base points; it is called the free path space of X and the fibration X I → X given by, say, χ ↦ χ ( 1 ) , is called the free path space fibration.

    The path space fibration can be understood to be dual to the mapping cone. The reduced fibration is called the mapping fiber or, equivalently, the homotopy fiber.

    Mapping path space

    If ƒ:X→Y is any map, then the mapping path space Pƒ of ƒ is the pullback of Y I → Y , χ ↦ χ ( 1 ) along ƒ. Since a fibration pullbacks to a fibration, if Y is based, one has the fibration

    F f ↪ P f → p Y

    where p ( x , χ ) = χ ( 0 ) and F f is the homotopy fiber, the pullback of P Y → χ ↦ χ ( 1 ) Y along ƒ.

    Note also ƒ is the composition

    X → ϕ P f → p Y

    where the first map φ sends x to ( x , c f ( x ) ) , c f ( x ) the constant path with value ƒ(x). Clearly, φ is a homotopy equivalence; thus, the above decomposition says that any map is a fibration up to homotopy equivalence.

    If ƒ is a fibration to begin with, then ϕ : X → P f is a fiber-homotopy equivalence and, consequently, the fibers of f over the path-component of the base point are homotopy equivalent to the homotopy fiber F f of ƒ.

    Moore's path space

    By definition, a path in a space X is a map from the unit interval I to X. Again by definition, the product of two paths α, β such that α(1) = β(0) is the path β · α: I → X given by:

    ( β ⋅ α ) ( t ) = { α ( 2 t ) if  0 ≤ t ≤ 1 / 2 β ( 2 t − 1 ) if  1 / 2 ≤ t ≤ 1 .

    This product, in general, fails to be associative on the nose: (γ · β) · α ≠ γ · (β · α), as seen directly. One solution to this failure is to pass to homotopy classes: one has [(γ · β) · α ] = [γ · (β · α)]. Another solution is to work with paths of arbitrary length, leading to the notions of Moore's path space and Moore's path space fibration.

    Given a based space (X, *), we let

    P ′ X = { f : [ 0 , r ] → X | r ≥ 0 , f ( 0 ) = ∗ } .

    An element f of this set has the unique extension f ~ to the interval [ 0 , ∞ ) such that f ~ ( t ) = f ( r ) , t ≥ r . Thus, the set can be identified as a subspace of Map ⁡ ( [ 0 , ∞ ) , X ) . The resulting space is called Moore's path space of X. Then, just as before, there is a fibration, Moore's path space fibration:

    Ω ′ X ↪ P ′ X → p X

    where p sends each f: [0, r] → X to f(r) and Ω ′ X = p − 1 ( ∗ ) is the fiber. It turns out that Ω X and Ω ′ X are homotopy equivalent.

    Now, we define the product map:

    μ : P ′ X × Ω ′ X → P ′ X

    by: for f : [ 0 , r ] → X and g : [ 0 , s ] → X ,

    μ ( g , f ) ( t ) = { f ( t ) if  0 ≤ t ≤ r g ( t − r ) if  r ≤ t ≤ s + r .

    This product is manifestly associative. In particular, with μ restricted to Ω'X × Ω'X, we have that Ω'X is a topological monoid (in the category of all spaces). Moreover, this monoid Ω'X acts on P'X through the original μ. In fact, p : P ′ X → X is an Ω'X-fibration.

    References

    Path space fibration Wikipedia
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