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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
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