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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?

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Contractible. The sphere spectrum is initial as an $E_{\infty}$-ring.

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  • 4
    $\begingroup$ Oh yes - right... Maybe I should have thought a little longer on that one :). $\endgroup$ – Thomas Kragh May 28 '13 at 19:11

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