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# Homology, Homotopy and Applications

## Volume 8 (2006)

### Number 2

### An explicit classification of three-stage Postnikov towers

Pages: 133 – 155

DOI: http://dx.doi.org/10.4310/HHA.2006.v8.n2.a8

#### Author

#### Abstract

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.

#### Keywords

fibration; Postnikov system; classifying space; Eilenberg-MacLane space

#### 2010 Mathematics Subject Classification

55P20, 55R15, 55R35, 55S45