Extended Adams-Hilton's construction (Q1070573)

Let \(F\to^{j}E\to^{p}B\) be a Hurewicz fibration. The homotopy lifting property defines (up to homotopy) an action of the H-space \(\Omega\) B on the fibre F which makes \(H_*(F)\) into a \(H_*(\Omega B)\)-module. Suppose B is connected. We prove that if \(E\to^{p}B\) is the cofibre of a map \(g: W\to E\) where W is a wedge of spheres, then the reduced homology of F, \(\tilde H_*(F)\), is a free \(H_*(\Omega B)\)-module generated by \(\tilde H_*(W)\). This result implies in particular a characterization of aspherical groups. The key point in the proof of this theorem is the following generalization of the Adams-Hilton construction. In their famous paper, Adams and Hilton construct for every simply connected c.w. complex B a graded differential algebra whose homology computes the algebra \(H_*(\Omega B)\). Extending their construction to any fibration p we construct a differential graded module C(F) whose homology computes the \(H_*(\Omega B)\)-module \(H_*(F)\). We suppose E is a subcomplex of B, then C(F) is a free \(H_*(\Omega B)\)-module generated by the cells of E. The differential is defined inductively on generators in accordance with the way the cells of E are attached. Our construction has many applications. For instance, let \(\tilde K\to^{p}K\) be a normal covering of a finite c.w. complex. \(\tilde K\) is the homotopy fibre of some classifying map \(K\to K(G,1)\). As \(H_*(\Omega K(G,1))\) is isomorphic to \({\mathbb{Z}}[G]\), our construction yields an explicit chain complex whose homotopy computes the homology \(\tilde K\) as a \({\mathbb{Z}}[G]\)-module. In particular, we establish some properties of infinite cyclic coverings in low dimensions.