Co-H-spaces and the Ganea conjecture

A non-simply connected co-H-space X is, up to homotopy, the total space of a brewise-simply connected pointed brewise co-Hopf brant j: X! B1(X), which is a space with a co-action of B1(X) along j. We construct its homology decomposition, which yields a simple construction of its brewise localisation. Our main result is the construction of a series of co-H-spaces, each of which cannot be split into a one-point-sum of a simply connected space and a bunch of circles, thus disproving the Ganea conjecture. Problem 10 posed by Tudor Ganea [8], known as the Ganea conjecture (e.g, x6 in Arkowitz [1]), states: Does a co-H-space have the homotopy type of a one-point-sum of a bunch of circles (one-point-sum of S1’s or a point) and a simply connected space? If a CW complex X is a co-H-space, the co-H-structure gives a co-action (see Berstein and Dror [3] or Oda [16]) of the classifying space B1(X) of 1(X) along j: X! B1(X), the classifying map of the universal covering p(X) : eX! X. It is known by Eilenberg-Ganea [6] or [11], that 1(X) is free and B1(X) has the homotopy type of a bunch of circles, say B. Let i: B! X be a map representing a collection of generators of the free group 1(X) and c: X! C = X=B be the collapsing map from X to its cobre. Clearly, we may choose the map i so that ji 1X. It is also known by Corollary 3.4 and Theorem 3.3 in [11] that, for a given , a co-H-structure for X, there is a ‘natural ’ map s = s() : C = X=B! X which is a right homotopy inverse of c. More precisely, if f: (X;)! (X 0; 0) is a co-H-map, then f s() = s(0)f 0, where f 0: X=B! X 0=B0 is the unique map induced from f. Henc