Cone of an application – Wikipedia

In mathematics and more precisely in theory of homotopia, the cone of an application is a topological space built from the cone, based on the starting space of the application, by identifying the points of this base with those of the arrival space by means of the application.

Either X And AND two topological spaces and f : X → AND a continuous application. The application cone f or cofibre homotopique of f , note C _f ^{[ first ]} , is the topological space “Obtained by endearing C ( X ) (the cone of X ) To AND along f ^{[ 2 ]} » , that is to say as a quotant the disjoint meeting CX ⊔ AND by identifying each element x of X ⊂ CX image with his image f ( x ) In AND . More explicitly, this is the quotient of the disjoint meeting X × [0, 1] ⊔ AND by the equivalence relationship: ( x , 0) ∼ ( x’ , 0) and ( x , 1) ∼ f ( x ) ^{[ 3 ]} .

For a morphism of pointed spaces f : ( X , x ₀ ) → ( AND , and ₀ ), by quotant more by ( x ₀ , t ) ∼ and ₀ (for everything t ∈ [0, 1] and not only for t = 1), we obtain the “reduced cone” C _✻ f of f . This amounts to replacing, in the above definition, the cone CX space by the reduced cone C _✻ ( X , x ₀ ) Potted space.

• And X is the sphere S ⁿ , CX is (homeomorphic to) the ( n +1)-Closed boule B ^{n +1} . C _f is then the quotient of the disjoint union of this ball with AND , by identifying each point x you table ∂ B ^{n +1} = S ⁿ of this ball with its image f ( x ) In AND .

• And Y = CX you if f is the canonical inclusion of X In his cone, Cf is the quotient of X × [0, 1] for: ( x , 0) ∼ ( x’ , 0) and ( x , 1) ∼ ( x’ , 1). This is the suspension Produce from space X .

• At the intersection of the two previous examples, the cone of the canonical inclusion of S ⁿ In B ^{n +1} East S ^{n +1} .

• For f : X → AND , space AND is, naturally, a subspace of C _f and the inclusion of AND In C _f is a co -firing.
• And f is injective and relatively open, that is to say if it induces a homeomorphism of X on f ( X ), SO CX is also included in C _f (SO X Also).
• The cone of the identity application of X is naturally homeomorphic to the cone of X .

All of these properties are transposed to pointed areas, taking the reduced cones of pointed applications and pointed areas.

The cone reduced by a morphism of well -punctuated space is homotopically equivalent to its uncommon cone.

The cones of two homotopes continuous applications are homotopically equivalent.

The cone of an application f is the double cylinder of applications of the constant application of X On a point and the application f .

CW-complexes [ modifier | Modifier and code ]

For a CW-Complex X , the ( n + 1) -Skeleton X _{n + 1} is homeomorphic at the cone of the application

${displaystyle coprod _{iin I_{n}}S_{i}^{n}to X_{n}}$

of recolling of ( n + 1) -Cellules, along their edge, at n -skeleton.

Effect on the fundamental group [ modifier | Modifier and code ]

For any pointed space ( X , x ₀ ) and all α lace: ( S ^first , 1) → ( X , x ₀ ), representing an element of the fundamental group of ( X , x ₀ ), we can form the cone C _✻ α. In this cone, the α lace becomes contractile therefore its equivalence class in the fundamental group of ( C _✻ a, x ₀ ) is the neutral element.

This allows, for any group G defined by generators and relations, to build a 2-complex whose fundamental group is G .

Relative homology [ modifier | Modifier and code ]

The application cone allows to interpret the relative homology (in) a pair of spaces ( X , A ) like reduced homology (in) du whenever:

and H _✻ is a homological theory and i : A → X a co -branch, then

${Displaystyle h _ {*} (x, a) = H _ {*} (x/a,*) = {tILDE {H}} _ {*} (x/a),}$

by applying excision to the cone of i ^{[ 4 ]} .

Equivalences of homotopia and homology [ modifier | Modifier and code ]

A morphism between two simply related CW-complexes is an equivalence of homotopia if and only if its cone is contractile.

Either H _✻ A homological theory. The application f : X → AND induces isomorphism in H _✻ If and only if the point application in C _f induces isomorphism in H _✻ , that is to say if H _✻ ( C _f , ∙) = 0.

Role in theory of homotopia [ modifier | Modifier and code ]

And A is a closed of X and if inclusion i of A In X is a co -bonding, then the cone of i is homotopically equivalent to X / A . Like the co -fastening of AND In C _f is closed, its cone is homotopically equivalent to C _f / AND therefore to the suspension Produce of X . By continuing, the cone of the inclusion of C _f In Produce gives the suspension of AND , etc.

And h : AND → WITH is another continuous application, the compound h ∘ f is homotopically zero if and only if h is extendable in a continuous application of C _f In WITH .

The pointed version of this equivalence proves the accuracy of the Puppe suite:

${displaystyle ldots to [Sigma Y,Z]to [Sigma X,Z]to [C_{*}f,Z]to [Y,Z]to [X,Z].}$

Mapping CONE (Homological Algebra) (in)