Explicit formulae for the rational L–S category of some homogeneous spaces

Abstract

Explicit formulae for rational Lusternik–Schnirelman (L–S) category (cat 0) are rare, but some are available for a class of spaces which includes homogeneous spaces G/ H when H is a product of at most 3 rank 1 groups, and rank G− rank H⩽1 . We extend the applicability of these formulae to the case when rank G=5 and H is a 4-torus or ( SU 2) 4. With a Sullivan minimal model as data, implementing the formula requires the selection of a regular subsequence of length 4 from a sequence f 1,…, f 5 of homogeneous polynomials in 4 variables satisfying dim Q [x 1,…,x 4]/(f 1,…,f 5)<∞. Such subsequences are readily obtainable, and the ease of computation is in contrast to most available methods for determining rational L–S category, which usually involve both upper and lower bounds and a good measure of luck. The proof of the formula is a pretty application of ideal class groups in algebraic topology. We also present some examples to illustrate our result.