# Suspension

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 .${\ displaystyle \ pi _ {k} ^ {s} (S ^ {n}) = \ pi _ {kn} ^ {s}}$ ${\ displaystyle \ pi _ {k} ^ {s}}$ • 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 .