nLab
suspension

Idea

The suspension $SX$ of a topological space $X$ is a space of one higher dimension, which is (for inhabited $X$) a quotient space of $X\times [\mathrm{0,1}]$. The difference between $SX$ and $X\times [\mathrm{0,1}]$ is that the copy of $X$ at each endpoint ($0$ or $1$) is replaced by a single point. The generalisation to arbitrary $(\mathrm{\infty }\mathrm{,1})$-categories is the suspension object.

This picture

from Wikimedia shows the suspension of the blue circle as $X$; the green dots correspond to $2$ in the first definition below.

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

Compare the reduced suspension, where you start with a pointed space and identify all copies of the base point as well.

Definitions

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

The suspension $SX$ of $X$ is the quotient space? of $X+(X\times I\times 2)+2$ by the equivalence relation $\sim$ generated by:

• $a\sim (a\mathrm{,0},\perp )$ for $a$ in $X$;
• $a\sim (a\mathrm{,0},\top )$ for $a$ in $X$;
• $(a\mathrm{,1},\perp )\sim \perp$ for $a$ in $X$;
• $(a\mathrm{,1},\top )\sim \top$ for $a$ in $X$.

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

• $a\sim (a\mathrm{,0},b)$ for $a$ in $X$ and $b$ in $Y$;
• $(a\mathrm{,1},b)\sim b$ for $a$ in $X$ and $b$ in $Y$.

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

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

• $(a\mathrm{,0})\sim \perp$ for $a$ in $X$;
• $(a\mathrm{,1})\sim \top$ for $a$ in $X$.

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

If $X$ has a point $p$, then we can define $SX$ as the quotient space of $X\times I$ by the equivalence relation generated by:

• $(a\mathrm{,0})\sim (b\mathrm{,0})$ for $a,b$ in $X$;
• $(a\mathrm{,1})\sim (b\mathrm{,1})$ for $a,b$ in $X$.

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

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

Examples

The suspension of the $n$-cube is the $(n+1)$-cube, probably best visualised as a diamond. This gives a recursive? definition of cube, starting with the $0$-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 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 $1$) automatically by using a more complicated quotienting process.

Up to topological structure, the suspension of the $n$-sphere is the $(n+1)$-sphere; note that the $n$-sphere is (topologically) the boundary of the $(n+1)$-cube. The coincidence that ‘sphere’ and ‘suspension’ both begin with ‘s’ has not been ignored; we can write $S^n\cong S^n(2)$, where $S^n$ on the left is the $n$-sphere and $S^n$ on the right is the $n$-fold composite of the suspension functor. (Actually, you should start with the $(-1)$-sphere as the empty space, which is not the suspension of anything; then the $0$-sphere is $S\varnothing =2$.) 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 $n$-globe is the $(n+1)$-globe.

Up to topological structure, the suspension of the $n$-simplex is the $(n+1)$-simplex, but now this is not very useful. To study simplices, you should use the cone functor instead, which is $\Lambda X=X\star 1$, where $1$ 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 $X\star Y$ 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 $S\varnothing =2$, although even it regards this as an exception.