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Discriminants, resultants and a conjecture of S. Halperin. (English) Zbl 1018.55008
The Halperin conjecture states that for every orientable fibration \(X\to Y\to B\) with fibre \(X\) a space of type \(F_0\) (i.e. \(\dim \pi_* (X)\otimes Q<\infty\), \(H^{\text{odd}}(X,\mathbb{Q})=0\), and \(\dim H^*(X,\mathbb{Q}) <\infty)\) the Serre spectral sequence degenerates at the term \(E_2\). The conjecture has been proved for a large family of spaces of type \(F_0\), for example for homogeneous spaces \(G/H\) of rank zero by Shiga and Tezuka, and in the case of \(H^*(X,\mathbb{Q})\) being generated by \(\geq 3\) elements by Hao Chen. The paper under review contains a very nice survey on the subject. The author also introduces differential calculus and algebraic geometry around the conjecture. Consider all the algebras of the form \(\mathbb{Q}[a_1, \dots,a_n]/(f_1, \dots,f_n)\) where \(n\), the degrees of the \(a_i\) and the degrees of the homogeneous polynomials \(f_i\) are fixed. Each sequence \((f_1,\dots,f_n)\) corresponds to a point in some \(\mathbb{Q}^N\). The sequence is regular if some resultant does not vanish. Therefore regular sequences form a Zariski open set \(V\). The author proves that the sequences \((f_1,\dots,f_n)\) that satisfy the Halperin conjecture form an open subset of \(W\).

MSC:
55P62 Rational homotopy theory
55T10 Serre spectral sequences
13D10 Deformations and infinitesimal methods in commutative ring theory
13E10 Commutative Artinian rings and modules, finite-dimensional algebras
14B07 Deformations of singularities
14B12 Local deformation theory, Artin approximation, etc.
14M10 Complete intersections
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