# zbMATH — the first resource for mathematics

Spaces of free loops on real projective spaces. (English) Zbl 1076.55004
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 $$Q^{n,d}_\varepsilon(X)$$.
##### MSC:
 55P35 Loop spaces 55P15 Classification of homotopy type
##### Keywords:
free loop space; stable rank
Full Text: