Whitehead tower

Summary

The Whitehead tower of a space $X$ is a sequence of spaces $\cdots \to X\langle 2 \rangle \to X\langle 1 \rangle \to X$. The stage $X\langle n \rangle$ is $n$-connected and has the same homotopy groups as $X$ above degree $n$. It is dual to the Postnikov tower: instead of discarding the high homotopy groups, it discards the low ones. Its first stage is the universal cover, and each further stage is a fibration over the previous one with fibre an Eilenberg–MacLane space. It is used to compute homotopy groups with spectral sequences and to define string structures through the stage $BO\langle 8 \rangle$.