Homotopy classification of maps into homogeneous spaces (Q847601)

The author gives an alternative description of Postnikov's homotopy classification of maps in a particular case. More precisely, let \(M\) be a 3-dimensional CW-complex, \(G/H\) be a compact simply connected homogeneous space and \(b_G\in H^3(G;\pi_3(G))\) be the basic class of \(G\). If \(\varphi: M\to G/H\) is given, we denote by \({\mathcal O}_G=\{\omega^*(b_G)\mid \omega \circ\varphi=\varphi \text{ for } \; \omega: M\to G\}\subset H^3(M;\pi_3(G))\). The main result states as follows: Two maps \(\varphi,\,\psi: M\to G/H\) are homotopic if, and only if, there exists a map \(u: M\to G\) such that \(\psi=u\circ\varphi\) and \(u^*(b_G)\in {\mathcal O}_G\). Moreover, within the homotopy classes of the restriction to the 2-skeleton of \(M\), the homotopy classes are in one-to-one correspondence with the quotient \(H^3(M;\pi_3(G))/{\mathcal O}_G\). Extension of this result to Sobolev maps is done in a different paper.