Suspension

Suspension of a circle. The original space is blue, the collapsed endpoints are green.

In topology , the suspension or suspension SX of a topological space X denotes the quotient space

${\displaystyle SX=(X\times I)/\{\forall x,y\in X:(x,0)\sim (y,0),\ (x,1)\sim (y,1)\}}$

of the product of X with the unit interval I = [0, 1].

Clearly, X is first expanded into a " cylinder "  , the ends of which are then combined into points, and X is viewed as "suspended" between these end points. The suspension can also be viewed as two geometric cones over X , which are glued together on their base. A third possibility is to consider it as the quotient of the topological cone over X , in which the points on the base side are summarized as equivalent. ${\displaystyle X\times I}$

Suspension is a functor that increases the dimension of a room by one: ${\displaystyle \forall n\in \mathbb {N} _{0}\colon S(S^{n})\cong S^{n+1}.}$

Reduced suspension

If there is a dotted space (with a base point ), there is a modified suspension of , which is again dotted: The reduced suspension of is the quotient space: ${\displaystyle (X,x_{0})}$ ${\displaystyle x_{0}}$ ${\displaystyle X}$ ${\displaystyle \Sigma X}$ ${\displaystyle X}$

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

The construction collapses the straight line ( x ₀ × I ) in SX , whereby the ends are combined to one point. The base point of Σ X is the equivalence class of ( x ₀ , 0). Σ is an endo-functor in the category of dotted spaces.

One can show that the reduced suspension of X is homeomorphic to the smash product of X with the unit circle S ¹ :

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

more generally, the -fold iterated reduced suspension is essentially the smash product with the - sphere : ${\displaystyle n}$ ${\displaystyle n}$

${\displaystyle \Sigma ^{n}X\cong S^{n}\wedge X}$.

For CW complexes , the reduced suspension is homotopy equivalent to the usual one.

properties

• The reduced suspension is left adjoint to the formation of the loop space : If they are compact , there is a natural isomorphism ${\displaystyle X,Y}$

${\displaystyle [\Sigma X,Y]=[X,\Omega Y].}$

In particular,

${\displaystyle \pi _{n+1}(Y)=\pi _{n}(\Omega Y).}$

• The functoriality of the suspension induces images

${\displaystyle \pi _{k}(X)=[S^{k},X]\to [\Sigma S^{k},\Sigma X]=\pi _{k+1}(\Sigma X).}$

between homotopy groups . The Freudenthalsche Einhängungssatz states that these figures for - contiguous spaces in the field isomorphisms and epimorphisms are. The direct Limes ${\displaystyle n}$ ${\displaystyle X}$ ${\displaystyle k\leq 2n}$ ${\displaystyle k=2n+1}$

${\displaystyle \pi _{k}^{s}(X)=\operatorname {colim} _{m}\pi _{k+m}(\Sigma ^{m}X)}$

above these maps is the -th stable homotopy group of . In particular , the inductive system is essentially constant for, i.e. H. ${\displaystyle k}$ ${\displaystyle X}$ ${\displaystyle X=S^{0}}$ ${\displaystyle m\geq k+2}$

${\displaystyle \pi _{2k+2}(S^{k+2})=\pi _{2k+3}(S^{k+3})=\ldots =\pi _{k}^{s}(S^{0})=:\pi _{k}^{s};}$

Because of this, the groups are simply called stable homotopy groups of the spheres . ${\displaystyle \pi _{k}^{s}(S^{n})=\pi _{k-n}^{s}}$ ${\displaystyle \pi _{k}^{s}}$

• For all true ${\displaystyle n\geq 1}$

${\displaystyle H_{n}(X)=H_{n+1}(SX),H^{n}(X)=H^{n+1}(SX).}$

If you use reduced homology or reduced cohomology, it even applies to all ${\displaystyle n\geq 0}$

${\displaystyle {\tilde {H}}_{n}(X)={\tilde {H}}_{n+1}(SX),{\tilde {H}}^{n}(X)={\tilde {H}}^{n+1}(SX).}$

This suspension isomorphism (or suspension isomorphism ) also applies to all generalized cohomology theories .

See also

• Doll series