Volume 6, issue 2 (2006)

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Categorical sequences

Rob Nendorf, Nick Scoville and Jeffrey Strom

Algebraic & Geometric Topology 6 (2006) 809–838

arXiv: 0904.0619


We define and study the categorical sequence of a space, which is a new formalism that streamlines the computation of the Lusternik–Schnirelmann category of a space X by induction on its CW skeleta. The kth term in the categorical sequence of a CW complex X, σX(k), is the least integer n for which catX(Xn) k. We show that σX is a well-defined homotopy invariant of X. We prove that σX(k + l) σX(k) + σX(l), which is one of three keys to the power of categorical sequences. In addition to this formula, we provide formulas relating the categorical sequences of spaces and some of their algebraic invariants, including their cohomology algebras and their rational models; we also find relations between the categorical sequences of the spaces in a fibration sequence and give a preliminary result on the categorical sequence of a product of two spaces in the rational case. We completely characterize the sequences which can arise as categorical sequences of formal rational spaces. The most important of the many examples that we offer is a simple proof of a theorem of Ghienne: if X is a member of the Mislin genus of the Lie group Sp(3), then cat(X) = cat(Sp(3)).

categorical sequence, Lusternik–Schnirelmann category, CW skeleta, rational homotopy
Mathematical Subject Classification 2000
Primary: 55M30
Secondary: 55P62
Forward citations
Received: 5 January 2006
Accepted: 23 April 2006
Published: 21 June 2006
Rob Nendorf
Department of Mathematics
Northwestern University
Evanston, IL 60208-2730
Nick Scoville
Department of Mathematics
Dartmouth College
Hanover, NH 03755-3551
Jeffrey Strom
Department of Mathematics
Western Michigan University
Kalamazoo, MI 49008