Deformations and the rational homotopy of the monoid of fiber homotopy equivalences
A space is (rationally) elliptic if its rational homotopy and rational cohomology are finite dimensional. The alternating sum of the dimensions of the rational homotopy groups is called the homotopy Euler characteristic \(\chi_{\pi}\). Halperin showed that \(\chi_ \pi = 0\) implies that the cohomology algebra \(A_ 0\) has the special form of a polynomial algebra (on even degree generators) \(P\) modulo an ideal \(I_ 0\) generated by a regular sequence of elements. Because these spaces (which the author calls \(F_ 0\)-spaces) have such a rigid structure, there is a real possibility of understanding their fibration classifying spaces. The computation of the rational homotopy groups of the classifying space was accomplished by \textit{W. Meier} [Math. Ann. 258, 329-340 (1982; Zbl 0466.55012) and Math. Z. 183, 473-481 (1983; Zbl 0517.55005)]. In this paper, for an oriented fibration with fibre an \(F_ 0\)-space and base a formal space with vanishing odd cohomology, the author extends Meier's work by computing the rational homotopy groups of (the identity component of) the monoid of fibre homotopy equivalences. The hypotheses on the fibre and base, via the homotopy theory of commutative differential graded algebras, allow fibre homotopy equivalence to be placed within the context of deformation theory of algebras. The author makes optimal use of this theory and computes the required homotopy groups in terms of derivations and (infinitesimal) deformations of the cohomology of the total space of the fibration.
scientific article; zbMATH DE number 561198