2
$\begingroup$

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?

$\endgroup$

1 Answer 1

15
$\begingroup$

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

$\endgroup$
1
  • 4
    $\begingroup$ Oh yes - right... Maybe I should have thought a little longer on that one :). $\endgroup$ May 28, 2013 at 19:11

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.