An explicit classification of three-stage Postnikov towers

Peter I. Booth

Homology, homotopy, and applications, Tome 8 (2006) no. 2, pp. 133-155.

Résumé

The problems of classifying Hurewicz fibrations whose fibres have just two non-zero homotopy groups and classifying 3-stage Postnikov towers are substantially equivalent. We investigate the case where the fibres have the homotopy type of $K(G, m) \times K(H, n)$, for $1 m n$. Our solution uses a classifying space $M_{\infty}$, i.e. a mapping space whose underlying set consists of all null homotopic maps from individual fibres of the path fibration $PK(G, m+1) \to K(G, m+1)$ to the space $K(H, n+1)$, and the group ${\cal E}(K(G, m) \times K(H, n))$ of based homotopy classes of based self-homotopy equivalences of $K(G, m)\times K(H, n)$. If $B$ is a given space, then a group action \[ {\cal E}(K(G, m) \times K(H, n))\times [B, M_{\infty}]^0\;\, \to\;\, [B, M_{\infty}]^0 \] is defined, and the orbit set $[B, M_{\infty}]^0\,/\,{\cal E}(K(G, m) \times K(H, n)) $ is shown to classify the above fibrations over $B$ up to fibrewise homotopy type. Our explicit definitions of the classifying spaces, together with our computationally effective group actions, are advantageous for computations and further developments. Two stable range simplifications are given here, together with a classification result for cases where $B$ is a product of spheres.

Détail

DOI : 10.4310/HHA.2006.v8.n2.a8

Classification : 55P20, 55R15, 55R35, 55S45
Keywords: fibration, Postnikov system, classifying space, Eilenberg-MacLane space