The Freudenthal suspension theorem states that if a pointed CW-complex 
is 
-connected for an integer 
,
then for 
the suspension map 
is an isomorphism
for 
and a surjection for 
. It yields the stable range in which homotopy
groups are unchanged by suspension. Here 
-connected
means that 
is pathwise-connected
and 
for 
.
For 
,
this condition simply requires pathwise-connectedness.
Freudenthal Suspension Theorem
See also
Homotopy Group, SuspensionExplore with Wolfram|Alpha
References
Whitehead, G. W. Elements of Homotopy Theory. New York: Springer-Verlag, 1978.Cite this as:
Weisstein, Eric W. "Freudenthal Suspension Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/FreudenthalSuspensionTheorem.html