Lower bounds¶for the relative Lusternik–Schnirelmann category

Abstract:

We give estimates of numerical homotopy invariants of the pair (X,X×S ^p) in terms of homotopy invariants of X. More precisely, we prove that σ^{p +1} cat(X) + 1 ≤ cat(X,X×S ^p}), that and that e(X,X×S<^p)=e(X)+1, where σ^{p +1} cat is the (relative) σ category of Vandembroucq and e is the (relative) Toomer invariant. The proof is based on an extension of Milnor's construction of the classifying space of a topological group to a relative setting (due to Dold and Lashof).