Whitehead-Serre theorem

The set of Whitehead Serre is a mathematical theorem from algebraic topology , particularly the homotopy .

Homotopy groups of topological spaces are notoriously difficult to calculate, while there are simple algorithms for calculating the homology groups of CW complexes . However, the Whitehead-Serre Theorem states that for simply connected spaces the rational homotopy groups can be calculated just as easily as the rational homology groups.

It is named after JHC Whitehead and Jean-Pierre Serre .

Whitehead-Serre theorem

Be

${\displaystyle f\colon X\to Y}$

a continuous mapping between simply connected spaces. Then the following conditions are equivalent:

• ${\displaystyle f_{*}\otimes id\colon \pi _{*}(X)\otimes \mathbb {Q} \to \pi _{*}(Y)\otimes \mathbb {Q} }$is an isomorphism .
• ${\displaystyle f_{*}\colon H_{*}(X;\mathbb {Q} )\to H_{*}(Y;\mathbb {Q} )}$ is an isomorphism.

Related sentences

Closely related to Whitehead-Serre's theorem is Whitehead's theorem that a continuous mapping between simply connected spaces is a weak homotopy equivalence if and only if it induces an isomorphism of the singular homology groups .

Furthermore, for -contiguous spaces , the homomorphism given by Hurewicz's theorem is an isomorphism ${\displaystyle r}$ ${\displaystyle r\geq 2}$ ${\displaystyle \pi _{r}(X)\to H_{r}(X;\mathbb {Z} )}$

${\displaystyle \pi _{r}(X)\otimes \mathbb {Q} \simeq H_{r}(X;\mathbb {Q} )}$

induced.

literature

• Yves Félix, Steve Halperin, JC Thomas: `` Rational Homotopy Theory '', Graduate Texts in Mathematics, 205, Springer-Verlag, 2000.

Individual evidence

1. ↑ Whitehead, Combinatorial Homotopy I, Bulletin AMS, Vol. 55, 1949, pp. 213-245, online