Depth and Toomerʹs invariant

We show that for every finite, simply connected CW complex X, and for any field K, depth H∗(ΩX,K)≤eK(X). In fact, we prove the same result under a weaker assumption, namely X is a simply connected CW complex of finite type with non-zero evaluation map. This is a strong improvement of the depth theorem which states that depth H∗(ΩX,K)≤cat(X).