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 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.

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.