Let $S$ be the spheres spectrum and $X$ a finite CW-complex. Then $X\times S \to X$ is the trivial fibered spectrum over $X$.


The spaces of fiber preserving equivalences $X\times S \to X\times S$ is (up to homotopy) given by the mapping space


maps$(X,G)$,


where $G=$hocolim$\strut\_{n\to\infty} (\Omega^nS^n)\_{\pm 1} $ is the stable group of self homotopy equivalences of spheres.


These guys are fiber-wise $S$-module equivalences.


**Question:** What is the space of equivalences which fiber-wise are $E\_\infty$ ring-spectrum equivalences?



Auto equivalences of the spheres spectrum as a ring spectrum
Contractible. The sphere spectrum is initial as an $E\_{\infty}$-ring.


