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 or susp(X).
There is a variant of the suspension for a 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.
Contents
Free suspension
The (free) suspension
S
X
{\displaystyle SX}
of a topological space
X
{\displaystyle X}
can be defined in several ways.
1.
S
X
{\displaystyle SX}
is the quotient space
(
X
×
[
0
,
1
]
)
/
(
X
×
{
0
}
)
/
(
X
×
{
1
}
)
.
{\displaystyle (X\times [0,1])/(X\times \{0\}){\big /}(X\times \{1\}).}
In other words, it can be constructed as follows:
Construct the cylinder
X
×
[
0
,
1
]
{\displaystyle X\times [0,1]}
.
Consider the entire set
X
×
{
0
}
{\displaystyle X\times \{0\}}
as a single point ("glue" all its points together).
Consider the entire set
X
×
{
1
}
{\displaystyle X\times \{1\}}
as a single point ("glue" all its points together).
2. Another way to write this is:
S
X
:=
v
0
∪
p
0
(
X
×
[
0
,
1
]
)
∪
p
1
v
1
=
lim
→
i
∈
{
0
,
1
}
(
(
X
×
[
0
,
1
]
)
↩
(
X
×
{
i
}
)
→
p
i
v
i
)
,
{\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
v
0
,
v
1
{\displaystyle v_{0},v_{1}}
are two points, and for each i in {0,1},
p
i
{\displaystyle p_{i}}
is the projection to the point
v
i
{\displaystyle v_{i}}
(a function that maps everything to
v
i
{\displaystyle v_{i}}
). That means, the suspension
S
X
{\displaystyle SX}
is the result of constructing the cylinder
X
×
[
0
,
1
]
{\displaystyle X\times [0,1]}
, and then attaching it by its faces,
X
×
{
0
}
{\displaystyle X\times \{0\}}
and
X
×
{
1
}
{\displaystyle X\times \{1\}}
, to the points
v
0
,
v
1
{\displaystyle v_{0},v_{1}}
along the projections
p
i
:
(
X
×
{
i
}
)
→
v
i
{\displaystyle p_{i}:{\bigl (}X\times \{i\}{\bigr )}\to v_{i}}
.
3. One can view
S
X
{\displaystyle SX}
as two cones on X, glued together at their base.
4.
S
X
{\displaystyle SX}
can also be defined as the join
X
⋆
S
0
,
{\displaystyle X\star S^{0},}
where
S
0
{\displaystyle S^{0}}
is a discrete space with two points.
5. In Homotopy type theory,
S
X
{\displaystyle SX}
is defined as a higher inductive type generated by
S:
S
X
{\displaystyle SX}
N:
S
X
{\displaystyle SX}
M
e
r
i
d
:
(
X
)
→
(
N
=
S
)
{\displaystyle Merid:{\bigl (}X{\bigr )}\to (N=S)}
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
f
:
X
→
Y
,
{\displaystyle f:X\rightarrow Y,}
there is a continuous map
S
f
:
S
X
→
S
Y
{\displaystyle Sf:SX\rightarrow SY}
defined by
S
f
(
[
x
,
t
]
)
:=
[
f
(
x
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:
Σ
X
=
(
X
×
I
)
/
(
X
×
{
0
}
∪
X
×
{
1
}
∪
{
x
0
}
×
I
)
{\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).
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
Ω
{\displaystyle \Omega }
taking a pointed space
X
{\displaystyle X}
to its loop space
Ω
X
{\displaystyle \Omega X}
. In other words, we have a natural isomorphism
Maps
∗
(
Σ
X
,
Y
)
≅
Maps
∗
(
X
,
Ω
Y
)
{\displaystyle \operatorname {Maps} _{*}\left(\Sigma X,Y\right)\cong \operatorname {Maps} _{*}\left(X,\Omega Y\right)}
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:
The suspension of an n-ball is homeomorphic to the (n+1)-ball.
Desuspension
Desuspension is an operation partially inverse to suspension.