The space of free loops on a real projective space
For a finite dimensional real projective space \(V\not= 0\), let \(\text{O}(\mathbb R^2,V)\) denote the Stiefel manifold consisting of all orthogonal \(2\)-frames on \(V\) and let \(\text{PO}(\mathbb R^2,V)\) be its projective manifold \(\text{O}(\mathbb R^2,V)/\{\pm 1\}\). In this paper, the authors study the topology of the space \({\mathcal L}P(V)\) consisting of all continuous free loops on the projective space \(P(V)\) from the point of view of homotopy theory. The action of the orthogonal group \(\text{O}(V)\) on \(V\) induces an action of \(\text{PO}(V)\) on \(P(V)\) and this action induces the action on \({\mathcal L}P(V)\) as well. Then they show that there is a \(\text{PO}(V)\)-equivariant stable splitting of \({\mathcal L}P(V)\) by using a technique of an equivariant fibrewise stable splitting for projective bundles. Although their method is completely homotopy theoretic, their result may suggest that a very interesting geometric (more precisely, Morse theoretic) principle would hold for \({\mathcal L}P(V)\). For explaining this, we recall the energy functional \(E:M\to \mathbb R\) given by \(E(\omega)=\frac{1}{2}\int_0^{2\pi}\|\omega'(t)\|^2 dt\), where \(M\) denotes the space consisting of all smooth loops \(\omega:S^1\to P(V)\). \(E\) is a Morse-Bott function with critical submanifolds \(C_l\) (\(l\in \mathbb N\)), where \(C_0\) is the space of constant loops and \(C_l\) (\(l\geq 1\)) is the space of closed geodesics of multiplicity \(l\) (of length \(\pi l\)). So one can easily identify \(C_l\) with \(\text{PO}(\mathbb R^2,V)\) for \(l\geq 1\). So that their main result may also be very interesting and be valuable from the Morse theoretic point of view.NEWLINENEWLINEFor the entire collection see [Zbl 0960.00050].
scientific article; zbMATH DE number 1639894