Cone length and Lusternik-Schnirelmann category in rational homotopy
Paul-Eugène Parent, Daniel Tanré
TL;DR
The paper addresses the rational-case relationship between Lusternik–Schnirelmann category and cone-length by generalizing Dupont’s counterexample: for every $k\ge3$ it constructs rational spaces with $cat X_k=k$ and $Cl X_k=k+1$. This is achieved through explicit Quillen models ${\mathscr L}_{k}$ built from carefully designed differential graded Lie algebras, using a connected-sum construction, attachment arguments that preserve $cat$, and a minimal-model analysis to obstruct shorter cone-lengths. The main technical contribution is showing that ${\mathscr L}_{k}$ has a decomposition of length $k+1$ but no isomorphism can shorten it to length $k$, thereby yielding the desired separation. Altogether, the results extend known rational examples of discrepancy between $cat$ and $Cl$, furnishing concrete algebraic models for all $k\ge3$.
Abstract
Lusternik-Schnirelmann category (LS-category) of a topological space is the least integer $n$ such that there is a covering of $X$ by $n+1$ open sets, each of them being contractible in $X$. The cone length is the minimum number of cofibations necessary to get a space in the homotopy type of $X$, starting from a suspension and attaching suspensions. The LS-category of a space is always less than or equal to its cone length. Moreover, these two invariants differ by at most one. In 1981, J.-M. Lemaire and F. Sigrist conjectured that they are always equal for rational spaces. This conjecture is clearly true for spaces of LS-category 1 and, in 1986, Y. Félix and J-C. Thomas verify it for spaces of LS-category 2. But, in 1999, the general conjecture is invalidated by N. Dupont who built a rational space of cone-length 4 and LS-category 3. In this work, we provide examples of rational spaces of cone-length $(k+1)$ and LS-category $k$ for any $k>2$.