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Poincaré duality algebras and the rational classification of differentiable manifolds. (English) Zbl 0555.55013

Homotopie algébrique et algèbre locale, Journ. Luminy/France 1982, Astérisque 113/114, 268-272 (1984).
[For the entire collection see Zbl 0535.00017.]
This is an announcement of results contained in two related papers. In [Geom. Dedicata 17, 199-205 (1984; Zbl 0548.55011)] the isomorphism types of Poincaré duality algebras of fixed formal dimension n and fixed minimal system of graded algebra generators are described as the orbits of a linear algebraic group action on a constructible subset of the nonzero linear functionals on the free commutative graded polynomials of degree n in the generators. It should be mentioned that the same proof gives the result over an arbitrary field.
The second paper contains the list of the \({\mathbb{Q}}\)-homotopy types of homologically 1-connected closed manifolds either, in low dimensions or in the homogeneously generated case (i.e. with an equal degree system of algebra generators of the cohomology), in the maximal range of intrinsic formality (dim\(\leq 6\), resp. cup-length\(\leq 3)\). Details may be found in ”The rational homotopy classification of differentiable manifolds in some intrinsic formal cases” [preprint, INCREST 11 (1984)].

MSC:

55P62 Rational homotopy theory
55P15 Classification of homotopy type
55R20 Spectral sequences and homology of fiber spaces in algebraic topology