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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].

##### MSC:
 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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