2012/11/21 by Youssef Rami, Youssef Ramı, Rami, Youssef
Mathematics · Medicine · #55M30 (Secondary) #55P62 (Primary) #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #Cancer Treatment and Pharmacology #Commutative Algebra (math.AC) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #math.AC #math.AT #msc:55M30 #msc:55P62
paper · pdf · doi:10.48550/arxiv.1211.5068
21 pages
openalex publication_date 2012/11/21 · arxiv created 2015/03/12 · arxiv updated 2015/03/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let X be a simply connected CW-complex of finite type and \mathbbK any field. A first known lower bound of LS-category cat(X) is the Toomer invariant e_\mathbbK (X) (\citeToo). In 1980's Félix et al. introduced the concept of \it depth in algebraic topology and proved the depth theorem: depth (H_*(ΩX, \mathbbK)) ≤ cat(X). In this paper, we use the Eilenberg-Moore spectral sequence of X to introduce a new numerical invariant, denoted by r(X, \mathbbK), and show that it has the same properties as those of e_\mathbbK (X). When the evaluation map (\citeFHT88) is non-trivial and char(\mathbbK)\not = 2, we prove that r(X, \mathbbK) interpolates depth(H_*(ΩX, \mathbbK)) and e_\mathbbK (X). Hence, we obtain an improvement of L. Bisiaux theorem (\citeBis99) and then of the depth theorem. Motivated by these results, we associate to any commutative differential graded algebra (A,d), a purely algebraic invariant r(A,d) and, via the theory of minimal models, we relate it with our previous topological results. In particular, if (ΛV,d) is a Sullivan minimal algebra such that d=∑i≥ kdi and di(V)⊆ ΛiV, a greater lower bound is obtained, namely e0(ΛV, d)≥ r(ΛV, d) + (k-2).