Mapping cylinder

In mathematics, specifically algebraic topology, the mapping cylinder of a continuous function f {displaystyle f} between topological spaces X {displaystyle X} and Y {displaystyle Y} is the quotient In mathematics, specifically algebraic topology, the mapping cylinder of a continuous function f {displaystyle f} between topological spaces X {displaystyle X} and Y {displaystyle Y} is the quotient where the ⨿ {displaystyle amalg } denotes the disjoint union, and ∼ is the equivalence relation generated by That is, the mapping cylinder M f {displaystyle M_{f}} is obtained by gluing one end of X × [ 0 , 1 ] {displaystyle X imes } to Y {displaystyle Y} via the map f {displaystyle f} . Notice that the 'top' of the cylinder { 1 } × X {displaystyle {1} imes X} is homeomorphic to X {displaystyle X} , while the 'bottom' is the space f ( X ) ⊂ Y {displaystyle f(X)subset Y} . It is common to write M f {displaystyle Mf} for M f {displaystyle M_{f}} , and to use the notation ⊔ f {displaystyle sqcup _{f}} or ∪ f {displaystyle cup _{f}} for the mapping cylinder construction. That is, one writes with the subscripted cup symbol denoting the equivalence. The mapping cylinder is commonly used to construct the mapping cone C f {displaystyle Cf} , obtained by collapsing one end of the cylinder to a point. Mapping cylinders are central to the definition of cofibrations. The bottom Y is a deformation retract of M f {displaystyle M_{f}} .The projection M f → Y {displaystyle M_{f} o Y} splits (via Y ∋ y ↦ y ∈ Y ⊂ M f {displaystyle Y i ymapsto yin Ysubset M_{f}} ), and the deformation retraction R {displaystyle R} is given by: (where points in Y {displaystyle Y} stay fixed because [ 0 , x ] = [ s ⋅ 0 , x ] {displaystyle =} for all s {displaystyle s} ). The map f : X → Y {displaystyle f:X o Y} is a homotopy equivalence if and only if the 'top' { 1 } × X {displaystyle {1} imes X} is a strong deformation retract of M f {displaystyle M_{f}} . An explicit formula for the strong deformation retraction can be worked out. The mapping cylinder may be viewed as a way to replace an arbitrary map by an equivalent cofibration, in the following sense: Given a map f : X → Y {displaystyle fcolon X o Y} , the mapping cylinder is a space M f {displaystyle M_{f}} , together with a cofibration f ~ : X → M f {displaystyle { ilde {f}}colon X o M_{f}} and a surjective homotopy equivalence M f → Y {displaystyle M_{f} o Y} (indeed, Y is a deformation retract of M f {displaystyle M_{f}} ), such that the composition X → M f → Y {displaystyle X o M_{f} o Y} equals f.