On genus and cancellation in homotopy

Abstract

Thegenus is determined for spaces of the homotopy type of aCW complex with one cell each in dimensions 0, 2n and 4n (and no other cells), such spaces providing the only cases of spaces with two non-trivial cells such that the homotopy class of the attaching map for the top cell is of infinite order and the genus of the space is non-trivial. The genus is characterised completely by two well understood invariants: theHopf invariant of the attaching map of the 4n-cell and the genus of thesuspension of the space. The algebraic tools are developed for the investigation of the ν-cancellation behaviour of these spaces and a cancellation theorem is proved: the homotopy type of a finite wedge of such spaces determines the homotopy type of each of the summands as long as the attaching maps of the 4n-cells all represent homotopy classes of infinite order. Comparing this result to known results aboutfinite co-H-spaces shows that the Hopf invariant is the single obstruction to such spaces admitting a co-H structure.