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I am reading Ravenel's Localization with Respect to Certain Periodic Homology Theories where he states;

For $n\ge2$, the spectra E(n) represent periodic homology theories which at present have no known geometric interpretation comparable to the description of K-theory in terms of vector bundles.

This is the paper where he gives his seven conjectures, all but one of which have since been proven. That would lead me to believe that this interpretation has been found in the process, but I am not aware of it.

Q: Is there a geometric interpretation of the Johnson-Wilson E(n) analogous to the vector bundle description of K-theory? If so, where I could I read about it in the literature?

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    $\begingroup$ People are actively working on such things. It is unlikely that there would be an answer for Johnson-Wilson theories. These things are built by much more homotopical means that seem to preclude geometric interpretations. If we had K-theory just from a homotopy theoretic construction then we would not know most of the things we do know about it. $\endgroup$
    – Sean Tilson
    May 30, 2014 at 23:31

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The answer is no, at least as far as I am aware of. For height 2 case, you can consider elliptic cohomology as some sort of geometric interpretation of $E(2)$ theory, but this is far less straight-forward than the vector bundle description of K-theory. Furthermore, the discovery by Igor Kriz of a group G with $K(n)^{odd}(BG)\neq 0$ makes it difficult to believe that one can find an $E(n)$-analogue of Atiyah-Segal completion theorem that relates $K$-theory and the representation ring.${}{}$

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