On the homotopy classification of sections in the free loop fibration (Q1313789)

The free-loop fibration \(\Omega X \to \Lambda X \to X\), where \(\Lambda X = \text{map} (S^1, X)\) and \(\Omega X\) is the space of based loops, has a canonical section. The author gives a classification of the set of sections in the case where \(\Omega X\) is a product of Eilenberg-MacLane spaces whose groups are vector spaces over a coefficient field \({\mathbf k}\), where \(\Lambda X\) has Hirsch model satisfying \[ \partial_h (C_* (X) \otimes \overline H_* (\Omega X) \subset C_* (X) \otimes \overline H_* (\Omega X), \] where \(H_* (\Omega X) = \pi_* (\Omega X) \oplus \overline H_* (\Omega X)\) and where \(\partial_h\) is the perturbed differential. The set of sections is then bijective with \(H^0 (L^* (\zeta), d_\nu)\), where \(L^i = C^{*+ i} (X, \pi_* (\Omega X))\) and \(d_\nu\) is a modified differential. A number of special cases and related results are given.