Freudenthal's suspension kit

The Freudenthal'sche Einhängungssatz is a set of the mathematical branch of algebraic topology , it forms a basis for the stable homotopy .

The statement is as follows:

Be and a coherent CW complex . Then is the image induced by the suspension ${\displaystyle n\geq 0}$ ${\displaystyle X}$ ${\displaystyle n}$

${\displaystyle \pi _{r}(X)\to \pi _{r+1}(\Sigma X)}$

for an isomorphism and for surjective . ${\displaystyle 1\leq r\leq 2n}$ ${\displaystyle r=2n+1}$

For the stable homotopy groups it follows that ${\displaystyle \pi _{*}^{s}(X)}$

${\displaystyle \pi _{r}(X)\to \pi _{r}^{s}(X)}$

for an isomorphism and for surjective . ${\displaystyle 1\leq r\leq 2n}$ ${\displaystyle r=2n+1}$

Generalization : Be and a -contiguous CW-complex . Let be a finite CW-complex with for . Then ${\displaystyle n\geq 0}$ ${\displaystyle X}$ ${\displaystyle n}$ ${\displaystyle Y}$ ${\displaystyle H^{q}(Y)=0}$ ${\displaystyle q>2n}$

${\displaystyle \left[Y,X\right]\to \left[\Sigma ^{k}Y,\Sigma ^{k}X\right]}$

for all a bijection between the sets of the homotopy classes. ${\displaystyle k\geq 0}$

literature

• Robert M. Switzer: Algebraic Topology - Homology and Homotopy. Springer, Berlin et al. 2002, ISBN 3-540-42750-3 ( Classics in Mathematics ).

Web links

• Tengren Zhang: Freudenthal Suspension Theorem

Individual evidence

1. ↑ Milnor, John; Spanier, Edwin: Two remarks on fiber homotopy type. Pacific J. Math. 10 1960 585-590.