Spaces with Lusternik–Schnirelmann category n and cone length n+1

We construct a series of spaces, X(n), for each n>0, such that cat(X(n))=n and cl(X(n))=n+1. We show that the Hopf invariants determine whether the category of a space goes up when attaching a cell of top dimension. We give a new proof of counterexamples to a conjecture of Ganea. Also we introduce some techniques for manipulating cone decompositions.