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Acat invariant of quasi-commutative algebras. (L’invariant Acat des algèbres quasi-commutatives.) (French) Zbl 0938.55020
The author proves that different algebraic invariants approximating the Lyusternik-Shnirel’man category of a topological space are in fact equal (namely: \(\text{lMcat}= \text{rMcat}= \text{biMcat}= \text{Acat}\)). To describe these invariants lMcat, rMcat, biMcat, Acat, recall that Félix and Halperin have associated to any commutative graded differential algebra \(A\) a numerical invariant, denoted by \(\text{cat}(A)\), with the property that if \(A\) is quasi-isomorphic to the singular cochain algebra \(C^*(X;\mathbb Q)\) then \(\text{cat}(A,d)=\text{cat}(X_{\mathbb Q})\) where \(X_{\mathbb Q}\) is the rationalization of \(X\) [Y. Félix and S. Halperin, Trans. Am. Math. Soc. 273, 1-37 (1982; Zbl 0508.55004)]. In [S. Halperin and J.-M. Lemaire, Lect. Notes Math. 1318, 138-154 (1988; Zbl 0656.55003)] an analogous invariant was introduced for (not necessarily commutative) differential graded algebra \(A\) over a field \(\mathbb F_p\) of characteristic \(p\); this invariant is denoted by \(\text{Acat}(A)\). They proved that \(\text{Acat}(C^*(X;\mathbb F_p))\leq \text{cat}(X)\) but inegality can occur. Some variations of this invariant were also introduced: lMcat, rMcat, and biMcat. Later Hess proved that if \(p=0\) and \(A\) is commutative then \(\text{cat}(A)=\text{lMcat}(A)= \text{rMcat}(A)= \text{biMcat}(A)= \text{Acat}(A)\), and this was a key step in the proof of Ganea’s conjecture for rational spaces [K. P. Hess, Topology 30, No. 2, 205-214 (1991; Zbl 0717.55014)]. On the other hand, Idrissi proved that for noncommutative DGA’s these invariants are not always equal, even when the cohomology algebra is commutative [E. H. Idrissi, Ann. Inst. Fourier 41, No. 4, 989-1004 (1991; Zbl 0702.55002)]. In the paper under review, the author introduces a notion of quasi-commutative DGA and he proves the two following results: (a) the singular cochain algebra \(C^*(X;\mathbb{F}_p)\) of a space is a quasi-commutative DGA; (b) if \(A\) is a quasi-commutative DGA then \(\text{lMcat}(A)= \text{rMcat}(A)= \text{biMcat}(A)= \text{Acat}(A)\).
From (a) and (b) it is clear that these invariants are the same for topological spaces.
MSC:
55P62 Rational homotopy theory
55M30 Lyusternik-Shnirel’man category of a space, topological complexity à la Farber, topological robotics (topological aspects)
55S05 Primary cohomology operations in algebraic topology
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