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
S.
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/
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∼
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{\ 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.
X
×
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{\ displaystyle X \ times I}
Suspension is a functor that increases the dimension of a room by one:
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{\ 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:
(
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{\ displaystyle (X, x_ {0})}
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{\ displaystyle x_ {0}}
X
{\ displaystyle X}
Σ
X
{\ displaystyle \ Sigma X}
X
{\ displaystyle X}
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{\ 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 :
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{\ displaystyle \ Sigma X \ cong S ^ {1} \ wedge X}
,
more generally, the -fold iterated reduced suspension is essentially the smash product with the - sphere :
n
{\ displaystyle n}
n
{\ displaystyle n}
Σ
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{\ 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
X
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{\ displaystyle X, Y}
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{\ displaystyle [\ Sigma X, Y] = [X, \ Omega Y].}
In particular,
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{\ displaystyle \ pi _ {n + 1} (Y) = \ pi _ {n} (\ Omega Y).}
The functoriality of the suspension induces images
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{\ 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
n
{\ displaystyle n}
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{\ displaystyle X}
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{\ displaystyle k \ leq 2n}
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{\ displaystyle k = 2n + 1}
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{\ 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.
k
{\ displaystyle k}
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{\ displaystyle X}
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{\ displaystyle X = S ^ {0}}
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{\ displaystyle m \ geq k + 2}
π
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{\ 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 .
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{\ displaystyle \ pi _ {k} ^ {s} (S ^ {n}) = \ pi _ {kn} ^ {s}}
π
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{\ displaystyle \ pi _ {k} ^ {s}}
For all true
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{\ displaystyle n \ geq 1}
H
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{\ 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
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{\ displaystyle n \ geq 0}
H
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{\ 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
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