Mapping cone

Mapping cone

In mathematics, especially homotopy theory, the mapping cone is a construction Cf of topology, analogous to a quotient space. It is also called the homotopy cofiber, and also notated Cf.

Contents

Definition

Given a map f\colon X \to Y, the mapping cone Cf is defined to be the quotient topological space of (X \times I) \sqcup Y with respect to the equivalence relation (x, 0) \sim (x',0)\,, (x,1) \sim f(x)\, on X. Here I denotes the unit interval [0,1] with its standard topology. Note that some (like May) use the opposite convention, switching 0 and 1.

Visually, one takes the cone on X (the cylinder X \times I with one end (the 0 end) identified to a point), and glues the other end onto Y via the map f (the identification of the 1 end).

Coarsely, one is taking the quotient space by the image of X, so Cf "=" Y/f(X); this is not precisely correct because of point-set issues, but is the philosophy, and is made precise by such results as the homology of a pair and the notion of an n-connected map.

The above is the definition for a map of unpointed spaces; for a map of pointed spaces f\colon (X,x_0) \to (Y,y_0), (so f\colon x_0 \mapsto y_0), one also identifies all of {x_0}\times I; formally, (x_0,t) \sim (x_0,t')\,. Thus one end and the "seam" are all identified with y0.

Example of circle

If X is the circle S1, Cf can be considered as the quotient space of the disjoint union of Y with the disk D2 formed by identifying a point x on the boundary of D2 to the point f(x) in Y.

Consider, for example, the case where Y is the disc D2, and

f: S1Y = D2

is the standard inclusion of the circle S1 as the boundary of D2. Then the mapping cone Cf is homeomorphic to two disks joined on their boundary, which is topologically the sphere S2.

Double mapping cylinder

The mapping cone is a special case of the double mapping cylinder. This is basically a cylinder joined on one end to a space X1 via the map

f1: S1X1

and joined on the other end to a space X2 via the map

f2: S1X2.

The mapping cone is the degenerate case of the double mapping cylinder (also known as the homotopy pushout), in which one space is a single point.

Applications

CW-complexes

Attaching a cell

Effect on fundamental group

Given a space X and a loop

\alpha\colon S^1 \to X

representing an element of the fundamental group of X, we can form the mapping cone Cα. The effect of this is to make the loop α contractible in Cα, and therefore the equivalence class of α in the fundamental group of Cα will be simply the identity element.

Given a group presentation by generators and relations, one gets a 2-complex with that fundamental group.

Homology of a pair

The mapping cone lets one interpret the homology of a pair as the reduced homology of the quotient:

If E is a homology theory, and i\colon A \to X is a cofibration, then E_*(X,A) = E_*(X/A,*) = \tilde E_*(X/A), which follows by applying excision to the mapping cone.[1]

Relation to homotopy (homology) equivalences

A map  f\colon X\rightarrow Y between simply-connected CW complexes is a homotopy equivalence if its mapping cone is contractible.

More generally, a map is called n-connected (as a map) if its mapping cone is n-connected (as a space), plus a little more. See A. Hatcher Algebraic Topology.

Let  \mathbb{}H_* be a fixed homology theory. The map  f:X\rightarrow Y induces isomorphisms on \mathbb{}H_*, if and only if the map  \{\cdot\}\hookrightarrow C_f induces an isomorphism on  \mathbb{}H_*, i.e.  \mathbb{}H_*(C_f,pt)=0.

See also

References

  1. ^ Peter May "A Concise Course in Algebraic Topology", section 14.2

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