nLab
suspension

Context

Topology

Homotopy theory

Contents

Idea

The suspension SXS X of a topological space XX is a space of one higher dimension, which is (for inhabited XX) a quotient space of X×[0,1]X \times [0,1]. The difference between SXS X and X×[0,1]X \times [0,1] is that the copy of XX at each endpoint (00 or 11) is replaced by a single point.

Compare the reduced suspension ΣX\Sigma X, where you start with a pointed space and identify all copies of the base point as well. This is the special case in Top of a general operation in (∞,1)-categories: see suspension object. For CW-complexes the reduced suspension is weakly homotopy equivalent to the ordinary suspension: ΣXSX\Sigma X \simeq S X.

This picture

from Wikimedia shows the suspension of the blue circle as XX; the green dots correspond to 22 in the first definition below.

Note that we replace XX by a single point at each endpoint; we don't merely identify all of the points in XX there. This only makes a difference for the empty space; we should have S={,}S \empty = \{\bot,\top\}, not S=S \empty = \empty. (This is related to the issues in the definition of connected space.)

Definitions

Let XX be a space (such as a topological space, or something more interesting like a generalized smooth space). Let II be the unit interval [0,1][0,1] in the real line; let 22 be the 22-point discrete space {,}\{\bot,\top\}. Let X×I×2X \times I \times 2 be the cartesian product of XX, 22, and II; let X+(X×2×I)+2X + (X \times 2 \times I) + 2 be the disjoint union of XX, X×I×2X \times I \times 2 and 22. We will suppress reference to the inclusion maps into X+(X×I)+2X + (X \times I) + 2; it will be clear from context how to parse an element of the latter.

Generalisable definition

The suspension SXS X of XX is the quotient space of X+(X×I×2)+2X + (X \times I \times 2) + 2 by the equivalence relation \sim generated by:

  • a(a,0,)a \sim (a,0,\bot) for aa in XX;
  • a(a,0,)a \sim (a,0,\top) for aa in XX;
  • (a,1,)(a,1,\bot) \sim \bot for aa in XX;
  • (a,1,)(a,1,\top) \sim \top for aa in XX.

Generalisation: the join

This generalises immediately to an operation called the join XYX \star Y of two spaces XX and YY; this is the quotient space of X+(X×I×Y)+YX + (X \times I \times Y) + Y by the equivalence relation generated by:

  • a(a,0,b)a \sim (a,0,b) for aa in XX and bb in YY;
  • (a,1,b)b(a,1,b) \sim b for aa in XX and bb in YY.

(Compare join of simplicial sets, the same operation in another guise.) Then we have SX=X2S X = X \star 2.

Simplified definition

It is somewhat simpler to define SXS X as the quotient space of (X×I)+2(X \times I) + 2 by the equivalence relation generated by:

  • (a,0)(a,0) \sim \bot for aa in XX;
  • (a,1)(a,1) \sim \top for aa in XX.

This works to define a topological space, but it does not directly give the smooth structure that matches the picture above.

Further simplified for pointed spaces

If XX has a point pp, hence if it is inhabited, then we can define SXS X as the quotient space of X×IX \times I by the equivalence relation generated by:

  • (a,0)(b,0)(a,0) \sim (b,0) for a,ba, b in XX;
  • (a,1)(b,1)(a,1) \sim (b,1) for a,ba, b in XX.

This is probably the most common definition seen, but it only works for XX an inhabited space (and even then gives only the topological structure).

Reduced suspension

To make the suspension of a pointed space (X,p)(X,p) again a pointed space one may further collapse in SXS X the set {p}×I\{p\} \times I to the point. The result then is called the reduced suspension of (X,p)(X,p) and is denoted

ΣXSX/{p}×IX×I/(X×{0,1}{p}×I). \Sigma X \coloneqq S X / \{p \} \times I \simeq X \times I / \left( X \times \{0,1\} \cup \{p\} \times I \right) \,.

Properties

As a functor

It's easy to extend the suspension operation SS to a functor from Top to itself.

Relation between suspension and reduced suspension

For CW-complexes suspension and reduced suspension agree, up to weak homotopy equivalence.

Examples

Suspension of cubes

The suspension of the nn-cube is the (n+1)(n+1)-cube, probably best visualised as a diamond. This gives a recursive definition of cube, starting with the 00-cube as the point, which is not the suspension of anything. Note that this not only gives us the topological structure of the cube, but also (by working in an appropriate category of smooth spaces throughout) the correct smooth structure on the cube as a manifold with corners. You can probably even get the correct metric on the cube (normalised to have diagonals of length 11) automatically by using a more complicated quotienting process.

Suspension of spheres

Up to homeomorphism, the suspension of the nn-sphere is the (n+1)(n+1)-sphere, and the reduced suspension is

ΣS nS n+1. \Sigma S^n \simeq S^{n+1} \,.

See at one-point compactification – Examples – Spheres for details.

Notice that the nn-sphere is (topologically) the boundary of the (n+1)(n+1)-cube. The coincidence that ‘sphere’ and ‘suspension’ both begin with ‘s’ has not been ignored; we can write S nS n(2)S^n \cong S^n(2), where S nS^n on the left is the nn-sphere and S nS^n on the right is the nn-fold composite of the suspension functor. (Actually, you should start with the (1)(-1)-sphere as the empty space, which is not the suspension of anything; then the 00-sphere is S=2S \empty = 2.) However, this does not give the correct smooth structure on the sphere, unless perhaps there is some more sophisticated definition that fixes this (but then that would break the cube). It might be more appropriate to say that the suspension of the nn-globe is the (n+1)(n+1)-globe.

Suspension of simplices

Up to topological structure, the suspension of the nn-simplex is the (n+1)(n+1)-simplex, but now this is not very useful. To study simplices, you should use the cone functor instead, which is ΛX=X1\Lambda X = X \star 1, where 11 is the point.

References

Everybody knows about the suspension, but Wikipedia knows about the join. See also the textbook by Hatcher and Postnikov, Homotopy of CW-complexes.

The question on what is the Eckman-Hilton dual to XYX\star Y find in

  • D. B. Fuks, Eckmann–Hilton duality and the theory of functors in the category of topological spaces, 1966 Russ. Math. Surv. 21 1–33 doi, free Russian original pdf

Here is Chapter 1 (pdf) of a textbook that knows that S=2S \empty = 2, although even it regards this as an exception.

  • George Whitehead, Some aspects of stable homotopy theory (pdf)

  • Ralph Cohen, A model for the free loop space of a suspension Lecture Notes in Mathematics, 1987, Volume 1286/1987, 193-207 ([])

Revised on November 3, 2013 21:32:53 by Urs Schreiber (89.204.139.125)