On the topology of fibrations with section and free loop spaces (Q2766423)

Given a fibration \(F\to E\to B\) with a section, the Whitehead product on \(\pi_*(E)\) induces the brace product \(\{-,-\}\): \(\pi_p(B) \times \pi_q(F) \to\pi_{p+q-1} (F)\). The first main result is:NEWLINENEWLINENEWLINELet \(F\to E\to S^p\) with \(p>1\) be a fibration with a section. Then in the homology Serre spectral sequence of this fibration the brace product \(\pi_p(S^p) \otimes \pi_q(F) @>\{-,-\}>> \pi_{p+q-1} (F)\)is naturally related to the differential \(d^p: E^p_{p,q} \to E^p_{0,p+q-1}\). For general fibrations with section and \(B\) simply connected, similar (long) differentials exist on the product of spherical classes that survive to the \(E^p_{p,q}\) term. A particularly interesting application of this result occurs for the evaluation fibration \(\Omega^kX\to {\mathcal L}^kX @>\text{ev}>> X\) where \({\mathcal L}^kX\) is the space of all continuous maps from \(S^k\) to \(X\). The authors illustrate their techniques by calculating the homology (with field coefficients) of the free loop space of a finite bouquet of spheres \(V_iS^{n_{i+1}}\), \(n_i>0\).