Rational cohomology of the free loop space of a simply connected 4-manifold

Abstract

The purpose of this paper is to calculate the rational cohomology \({H^{\ast}(X^{{S}^{1}} ; \mathbb{Q})}\) of the free loop space for a simply connected closed 4-manifold X. We use minimal models, so the starting point is the cohomology algebra \({H^{\ast}(X; \mathbb{Q})}\) which depends only on the second Betti number b2 and the signature of X itself. Calculations of \({H^{\ast}(X^{{S}^{1}} ; \mathbb{Q})}\) for b2 ≤ 2 are known. We study the case b2 > 2. We obtain an explicit formula for Poincare series of the space \({X^{{S}^{1}}}\), with the second Betti number b2 as a parameter.