Spaces of free loops on real projective spaces. (English) Zbl 1076.55004

Yamaguchi, Kohhei

Kyushu J. Math. 59, No. 1, 145-153 (2005).

Let \(L(X)\) be the free loop space on \(X\). For \(X= S^m\) or \(\mathbb{R} P^m\) there is a natural decomposition \(L_0(X)\coprod L_1(X)\) and natural maps \(Q^{n,d}_0(X)\to L_0(X)\), \(Q^{n,d}_1(X)\to L_1(X)\) where \(Q^{n,d}_{\varepsilon}(X)\), \(\varepsilon= 0,\,1\) are defined in terms of \(n\)-tuples of polynomials with real or complex coefficients, each polynomial having degree \(\leq d\), and these maps are \(D\)-homotopy equivalences for some \(D\) depending on \(n\) and \(d\). The author describes, for \(n\geq 3\) and \(d\geq 0\), the stable homotopy type of the polynomial approximations \(Q^{n,d}_\varepsilon(X)\).

Reviewer: Jean Claude Thomas (Angers)

MSC:

55P35	Loop spaces
55P15	Classification of homotopy type

Keywords:

free loop space; stable rank