The Eilenberg-Moore spectral sequence and the $\mathrm{mod}\,2$ cohomology of certain free loop spaces

Abstract

There has been considerable interest in computing the cohomology of the space A(X) of free loops on X, at least since the theorem of Gromoll and Meyer, connecting the unboundedness of the Betti numbers {dim H(AX; Q)} with the existence of infinitely many closed geodesics on X when X is a Riemannian manifold.There have been however relatively few explicit calcu- lations.The minimal model theory of Sullivan has been used in the rational case to obtain a few results.However with finite coefficients, aside from [12-1, which only contains Betti number estimates, there seems nothing known, apart from those facts that are easily knowable.In [9-1 we observed that the free loop space sits in a fibre square for any connected space X, where A is the diagonal map" if(x) A(X) X X ,XxXThis observation makes available the Eilenberg-Moore spectral sequence (see for example [8]) as a tool for computing H*(A(X); k) for simply connected X.In 1-9] we dealt with the case where the coefficient field k was of charac- teristic zero.In this note we take up the case of k Z/2, and derive the following not so easily knowable result.THEOREM.Let X be a simply connected space, and suppose Sq vanishes on H*(X; Z/2) and (*) H*(X; Z/2) -P[xl,..., x.]/(x]',..., where e en is a power of 2, and P[ the Eilenberg-Moore spectral sequence ] denotes a polynomial algebra.Then

Locations

• Illinois Journal of Mathematics - View - PDF