Rational Lusternik–Schnirelmann category of fibrations

If F→E→B is a fibration, a classical result of Varadarajan asserts that cat Eless-than-or-equals, slantcat F+cat B(cat F+1), where cat S denotes the Lusternik–Schnirelmann category of S. We give improved upper bounds in the rational case of the formimagecat0 Eless-than-or-equals, slantcat0 F+cat0 B(cat0 F+2−r0F),where r0F is a new invariant, namely the rational retraction index of F satisfyingimagedepth Fless-than-or-equals, slantr0Fless-than-or-equals, slantcat0 F,so that we recover the classical formula when r0F=1. However, the retraction index is often larger than 1, and in particular, we prove that if image is a Poincaré duality algebra with at least 2 generators, then r0Fgreater-or-equal, slanted2, giving the bound of (Contemp. Math. 227 (1996) 177) without their dimension hypothesis. Moreover, if F is coformal, then r0F=cat0 F, which yields the much lower estimateimagecat0 Eless-than-or-equals, slantcat0 F+2cat0 B.