Free simplicial groups and the second relative homotopy group of an adjunction space (Q1063287)

A proof is given using free simplicial groups of the theorem of \textit{J. H. C. Whitehead} (1939-46) that \(\pi_ 2(X\cup \{e^ 2_{\lambda}\}, X)\) is a free crossed \(\pi_ 1X\)-module. \{It is interesting that such methods have only with some work been able to prove this special case of the Van Kampen type theorem for crossed modules proved by the reviewer and \textit{P. J. Higgins} [Proc. Lond. Math. Soc., III. Ser. 36, 192-212 (1978; Zbl 0405.55015)].\}