Rational homotopic type of Hermitian K-theory (Q2638594)

The authors establish a rational homotopy equivalence \[ \Omega \bar L(\Sigma X)\simeq_ Q\prod^{\infty}_{n=1}\Omega^{\infty}\Sigma^{\infty}(ED_ n+\Lambda_{D_ n}X^{(n)}), \] where \(\bar L(\cdot)\) represents the reduced hermitian K-theory of a topological space and \(ED_ n+\Lambda_{D_ n}X^{(n)}\) stands for the reduced Borel construction on the n-fold smash product \(X^{(n)}\) where the dihedral group \(D_ n\) acts by permuting the coordinates. The first step involves the identification of the symmetric part of the natural involution on the reduced algebraic K-theory \(\bar A(\Sigma X)\), in terms of a reduced Borel construction on the free loop space of \(\Sigma X\), viewed as an O(2)-space. The second step further stably splits this construction in terms of dihedral groups; this was also achieved by \textit{G. M. Lodder} [Proc. Lond. Math. Soc., III. Ser. 60, 201-224 (1990; Zbl 0691.55006)].