Decomposition of the diagonal map

This paper presents a new method for using cup product information to draw conclusions about the Lusternik–Schnirelmann category of a space. The key idea is that of the Hopf set in X of a map f : Sn−1→L; if K=L∪fDn⊆X, then catX(K)=catX(L) if and only if ∗ is in the Hopf set in X of f. The main result explicitly constructs elements of the Hopf set in X of f in terms of members of the Hopf set in X of the attaching maps of lower dimensional cells. Applications include: a calculation of the category of Sp(2) without higher order cohomology operations; new, easily used upper bounds for Lusternik–Schnirelmann category that apply to any space; and new information about the category of the CW skeleta of loop spaces and free loop spaces on even-dimensional spheres.