On the Rational Retraction Index

Abstract

If X is a simply connected CW complex, then it has a unique (up to isomorphism) minimal Sullivan model. There is an important rational homotopy invariant, called the rational Lusternik–Schnirelmann of X, denoted cat0(X), which has an algebraic formulation in terms of the minimal Sullivan model of X. We study another such numerical invariant called the rational retraction index of X, denoted r0(X), which is defined in terms of the minimal Sullivan model of X and satisfies 0 ≤ r0(X) ≤ cat0(X). It was introduced by Cuvilliez et al. as a tool to estimate the rational Lusternik–Schnirelmann category of the total space of a fibration. In this thesis we compute the rational retraction index on a range of rationally elliptic spaces, including for example spheres, complex projective space, the biquotient Sp(1) \ Sp(3) / Sp(1) × Sp(1), the homogeneous space Sp(3)/U(3) and products of these. In particular, we focus on formal spaces and formulate a conjecture to answer a question posed in the original article of Cuvilliez et al., “If X is formal, what invariant of the algebra H∗(X;Q) is r0(X)?”