Gorenstein spaces. (English) Zbl 0659.57011

Félix, Yves; Halperin, Stephen; Thomas, Jean-Claude

Adv. Math. 71, No. 1, 92-112 (1988).

A (not necessarily commutative) differential graded k-algebra R is Gorenstein if \(Ext_ R(k,R)\) is concentrated in a single degree and has k-dimension one. This definition generalizes the usual definition of a Gorenstein ring. In the context of spaces the DGA’s to which this definition is most often applied are \(C^*(X;k)\), \(C_*(\Omega X;k)\) and \(H^*(X;k)\). In particular, a space X is Gorenstein over k if \(C^*(X;k)\) is. It is shown that Gorenstein spaces are generalizations of Poincaré duality spaces. Indeed, if X has finite category, then it is Gorenstein over \({\mathbb{Z}}/p\) if and only if \(H^*(X;{\mathbb{Z}}/p)\) is finite dimensional and satisfies Poincaré duality. Furthermore, if X is finite, then an equivalent statement is that the Spivak normal fibration associated to X p-localizes to a sphere. This is exactly the p-local version of Spivak’s theorem about Poincaré duality spaces (see M. Spivak [Topology 6, 77-101 (1967; Zbl 0185.509)]). Similarly, a \({\mathbb{Q}}\)-Gorenstein space version is given of the fact that, for an appropriate fibration, the total space satisfies Poincaré duality if and only if the base and fibre do (see D. Gottlieb [Proc. Am. Math. Soc. 76, 148-150 (1979; Zbl 0423.57009)]). The concept of Gorenstein space thus extends many familiar notions about Poincaré duality to a much wider class of spaces. In particular, it is shown that, if \(\pi_*(X)\otimes {\mathbb{Q}}\) is finite dimensional, then X is Gorenstein over \({\mathbb{Q}}\). Hence, rational Postnikov pieces come within the purview of the Gorenstein framework.

Reviewer: J.Oprea

MSC:

57P10	Poincaré duality spaces
55P62	Rational homotopy theory
13H10	Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)

Keywords:

rational spaces with finite dimensional total homotopy; elliptic spaces; differential graded k-algebra; Gorenstein ring; Gorenstein spaces; Poincaré duality spaces; finite category; Spivak normal fibration

Citations:

Zbl 0185.509; Zbl 0423.57009

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