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Cochain model for thickenings and its application to rational LS-category. (English) Zbl 0964.55013
To any finite CW complex \(X\) there are associated compact manifolds (with boundary), known as thickenings. For example, if \(X\) is a smooth manifold of dimension \(k\), then one can smoothly embed \(X\) in Euclidean space, \({\mathbb R}^{n+1}\) for any sufficiently large \(n\), and then take a tubular neighbourhood to obtain a so-called \((n+1)\)-thickening of \(X\). This paper focusses on the boundary (as a manifold) of a thickening. This is again a compact manifold that is denoted \(\partial M\). Clearly \(\partial M\) reflects the topology of \(X\), although in general it is not even determined up to homotopy type by \(X\). The author shows that, subject to some mild technical hypotheses on the type of thickening, the cochain complex of \(\partial M\) with coefficients in a field is determined, as a module, by that of \(X\). We refer to the article for the precise statement.
A stronger result is proven in the rational homotopy setting: For a stable thickening (\(n \geq 2k\)), a rational model (in the sense of rational homotopy theory) for \(\partial M\) can be constructed directly from a certain type of rational model for \(X\). It follows from this that, for a stable thickening, the rational homotopy type of \(\partial M\) is determined by that of \(X\). The methods used to prove this result are the ‘semi-free DG module’ techniques, as summarized in [Y. Félix, S. Halperin and J.-C. Thomas, in A handbook of algebraic topology, 829-865 (1995; Zbl 0868.55016)].
An intriguing corollary is developed by combining the results of this paper with those of [Y. Félix, S. Halperin and J.-M. Lemaire, Topology 37, No. 4, 749-756 (1998; Zbl 0897.55001)]: Let \(X\) be a finite CW complex, and \(\partial M\) the boundary of a stable thickening of \(X\). Then \(\text{cat}_0(X) = \text{e}_0(\partial M) - 1\). Here, \(\text{cat}_0(X)\) denotes rational category and \(\text{e}_0(\partial M)\) denotes the so-called rational Toomer invariant. From another result of the last-mentioned paper, it is possible to replace \(\text{e}_0(\partial M)\) by \(\text{cat}_0(\partial M)\) in this identity. Some themes suggested by this corollary have been amplified in two subsequent papers, with particular emphasis on the relation between the cone-lengths of \(X\) and \(\partial M\). See P. Lambrechts and L. Vandembroucq [Bords homotopiques et modèles de Quillen, preprint] and P. Lambrechts, D. Stanley and L. Vandembroucq [Embeddings up to homotopy of two-cones into Euclidean spaces, preprint].

55P62 Rational homotopy theory
57P10 Poincaré duality spaces
55M30 Lyusternik-Shnirel’man category of a space, topological complexity à la Farber, topological robotics (topological aspects)
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