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Homology of the universal covering of a co-H-space. (English) Zbl 0936.55006

It is the intention of the authors to make a contribution to the following conjecture of Ganea: A connected CW-complex \(X\) with a co-\(H\)-structure has the homotopy type \(B\vee E\), where \(B\) is a wedge of circles and \(E\) is a simply connected space. Affirmative answers have been given under additional assumptions on the co-\(H\)-structure. It should be pointed out that in the meantime the first author has produced counterexamples.
The approach in the present paper is to analyze the homology of the universal covering \(\widetilde X\) of \(X\). A co-action of a space \(B\) on a space \(A\) along a map \(f:A \to B\) is a map \(\mu:A\to B\vee A\), whose projections to \(B\) and \(A\) are \(f\) and \(id_A\) respectively. The authors prove: Let \(X\) be a finite CW-complex and \(j:X\to B\pi\) the classifying map of the universal covering \(\widetilde X\to X\), \(\pi= \pi_1(X)\). If there exists a co-action of \(B\) on \(X\) along \(j\), then \(H_*(\widetilde X;\mathbb{Z}) \cong\mathbb{Z} \pi\otimes H_* (X;\mathbb{Z})\) for \(*>1\). From this they deduce:
If \(X\) is a finite co-\(H\)-space, whose homology is concentrated in dimensions 1, \(n>1\), and \(n+1\) with \(H_{n+1} (X;\mathbb{Z})\) torsion free, then \(X\) splits into a wedge of 1-, \(n\)-, and \((n+1)\)-spheres and Moore spaces \(S^n\cup_le^{n+1}\).
In particular, the Ganea conjecture is true for complexes whose homology is concentrated in dimensions \(\leq 3\).

MSC:

55P45 \(H\)-spaces and duals
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