The following example arose out of an unsuccessful attempt to improve a result by Ganea and Hilton (5), according to which

implies that for each field of coefficients F one at least of the factors Xi is acyclic; here cat X is the Lusternik-Schnirelmann category of X. It was natural to ask whether this assumption also implies that one at least of the Xi is simply connected. This could be proved if it were possible to obtain spaces of arbitrarily large 1-dimensional category ((3), (4)) by forming Cartesian products of sufficiently many 2-dimensional pseudoprojective spaces (1) corresponding to different primes. However, we prove that this is impossible:

Theorem. If P1, …, Pn are 2-dimensional pseudo-projective spaces corresponding to different primes p1, …, pn, then

Before proving this theorem, we recall the definitions involved. The 1-dimensional category cat1X of a space X is the least number of open sets covering X and such that each closed path in any of them is contractible in X. A 2-dimensional pseudo-projective space P corresponding to the prime p is the complex obtained from the disk |z| ≤ 1 by identifying the points ei[θ+(2mπ|p)], m = 0, 1, …, p − 1. We have (1)

for k ≥ 2 and cat1P = 3 since π1(P) is not free and cat1P ≤ 1 + dim P (4).