Suspension (topology)

In topology, a branch of mathematics, the suspension of a topological space X is intuitively obtained by stretching X into a cylinder and then collapsing both end faces to points. One views X as "suspended" between these end points. The suspension of X is denoted by SX^[1] or susp(X).^[2]: 76

Suspension of a circle. The original space is in blue, and the collapsed end points are in green.

There is a variation of the suspension for pointed space, which is called the reduced suspension and denoted by ΣX. The "usual" suspension SX is sometimes called the unreduced suspension, unbased suspension, or free suspension of X, to distinguish it from ΣX.

Free suspension

The (free) suspension ${\displaystyle SX}$ of a topological space ${\displaystyle X}$ can be defined in several ways.

1. ${\displaystyle SX}$ is the quotient space ${\displaystyle (X\times [0,1])/(X\times \{0\}){\big /}(X\times \{1\}).}$ In other words, it can be constructed as follows:

• Construct the cylinder ${\displaystyle X\times [0,1]}$.
• Consider the entire set ${\displaystyle X\times \{0\}}$ as a single point ("glue" all its points together).
• Consider the entire set ${\displaystyle X\times \{1\}}$ as a single point ("glue" all its points together).

2. Another way to write this is:

${\displaystyle SX:=v_{0}\cup _{p_{0}}(X\times [0,1])\cup _{p_{1}}v_{1}\ =\ \varinjlim _{i\in \{0,1\}}{\bigl (}(X\times [0,1])\hookleftarrow (X\times \{i\})\xrightarrow {p_{i}} v_{i}{\bigr )},}$

Where ${\displaystyle v_{0},v_{1}}$ are two points, and for each i in {0,1}, ${\displaystyle p_{i}}$ is the projection to the point ${\displaystyle v_{i}}$ (a function that maps everything to ${\displaystyle v_{i}}$). That means, the suspension ${\displaystyle SX}$ is the result of constructing the cylinder ${\displaystyle X\times [0,1]}$, and then attaching it by its faces, ${\displaystyle X\times \{0\}}$ and ${\displaystyle X\times \{1\}}$, to the points ${\displaystyle v_{0},v_{1}}$ along the projections ${\displaystyle p_{i}:{\bigl (}X\times \{i\}{\bigr )}\to v_{i}}$.

3. One can view ${\displaystyle SX}$ as two cones on X, glued together at their base.

4. ${\displaystyle SX}$ can also be defined as the join ${\displaystyle X\star S^{0},}$ where ${\displaystyle S^{0}}$ is a discrete space with two points.^[2]: 76

5. In Homotopy type theory, ${\displaystyle SX}$ be defined as a higher inductive type generated by

S: ${\displaystyle SX}$

N:${\displaystyle SX}$

${\displaystyle Merid:{\bigl (}X{\bigr )}\to (N=S)}$^[3]

Properties

In rough terms, S increases the dimension of a space by one: for example, it takes an n-sphere to an (n + 1)-sphere for n ≥ 0.

Given a continuous map ${\displaystyle f:X\rightarrow Y,}$ there is a continuous map ${\displaystyle Sf:SX\rightarrow SY}$ defined by ${\displaystyle Sf([x,t]):=[f(x),t],}$ where square brackets denote equivalence classes. This makes ${\displaystyle S}$ into a functor from the category of topological spaces to itself.

Reduced suspension

If X is a pointed space with basepoint x₀, there is a variation of the suspension which is sometimes more useful. The reduced suspension or based suspension ΣX of X is the quotient space:

${\displaystyle \Sigma X=(X\times I)/(X\times \{0\}\cup X\times \{1\}\cup \{x_{0}\}\times I)}$.

This is the equivalent to taking SX and collapsing the line (x₀ × I ) joining the two ends to a single point. The basepoint of the pointed space ΣX is taken to be the equivalence class of (x₀, 0).

One can show that the reduced suspension of X is homeomorphic to the smash product of X with the unit circle S¹.

${\displaystyle \Sigma X\cong S^{1}\wedge X}$

For well-behaved spaces, such as CW complexes, the reduced suspension of X is homotopy equivalent to the unbased suspension.

Adjunction of reduced suspension and loop space functors

Σ gives rise to a functor from the category of pointed spaces to itself. An important property of this functor is that it is left adjoint to the functor ${\displaystyle \Omega }$ taking a pointed space ${\displaystyle X}$ to its loop space ${\displaystyle \Omega X}$. In other words, we have a natural isomorphism

${\displaystyle \operatorname {Maps} _{*}\left(\Sigma X,Y\right)\cong \operatorname {Maps} _{*}\left(X,\Omega Y\right)}$

where ${\displaystyle X}$ and ${\displaystyle Y}$ are pointed spaces and ${\displaystyle \operatorname {Maps} _{*}}$ stands for continuous maps that preserve basepoints. This adjunction can be understood geometrically, as follows: ${\displaystyle \Sigma X}$ arises out of ${\displaystyle X}$ if a pointed circle is attached to every non-basepoint of ${\displaystyle X}$, and the basepoints of all these circles are identified and glued to the basepoint of ${\displaystyle X}$. Now, to specify a pointed map from ${\displaystyle \Sigma X}$ to ${\displaystyle Y}$, we need to give pointed maps from each of these pointed circles to ${\displaystyle Y}$. This is to say we need to associate to each element of ${\displaystyle X}$ a loop in ${\displaystyle Y}$ (an element of the loop space ${\displaystyle \Omega Y}$), and the trivial loop should be associated to the basepoint of ${\displaystyle X}$: this is a pointed map from ${\displaystyle X}$ to ${\displaystyle \Omega Y}$. (The continuity of all involved maps needs to be checked.)

The adjunction is thus akin to currying, taking maps on cartesian products to their curried form, and is an example of Eckmann–Hilton duality.

This adjunction is a special case of the adjunction explained in the article on smash products.

Applications

The reduced suspension can be used to construct a homomorphism of homotopy groups, to which the Freudenthal suspension theorem applies. In homotopy theory, the phenomena which are preserved under suspension, in a suitable sense, make up stable homotopy theory.

Examples

Some examples of suspensions are:^{[4]: 77, Exercise.1}

• The suspension of an n-ball is homeomorphic to the (n+1)-ball.

Desuspension

Main article: Desuspension

Desuspension is an operation partially inverse to suspension.^[5]

See also

• Double suspension theorem
• Cone (topology)
• Join (topology)

References

1. Allen Hatcher, Algebraic topology. Cambridge University Presses, Cambridge, 2002. xii+544 pp. ISBN 0-521-79160-X and ISBN 0-521-79540-0
2. Matoušek, Jiří (2007). Using the Borsuk-Ulam Theorem: Lectures on Topological Methods in Combinatorics and Geometry (2nd ed.). Berlin-Heidelberg: Springer-Verlag. ISBN 978-3-540-00362-5. Written in cooperation with Anders Björner and Günter M. Ziegler
3. "suspension type in nLab". ncatlab.org. Retrieved 2024-08-20.
4. Matoušek, Jiří (2007). Using the Borsuk-Ulam Theorem: Lectures on Topological Methods in Combinatorics and Geometry (2nd ed.). Berlin-Heidelberg: Springer-Verlag. ISBN 978-3-540-00362-5. Written in cooperation with Anders Björner and Günter M. Ziegler , Section 4.3
5. Wolcott, Luke. "Imagining Negative-Dimensional Space" (PDF). forthelukeofmath.com. Retrieved 2015-06-23.

• This article incorporates material from Suspension on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.