The 𝐿₂-localization of 𝑊(𝑛)

Thompson, Robert

doi:10.1090/S0002-9947-98-02149-7

The $L_2$-localization of $W(n)$
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by Robert D. Thompson

Trans. Amer. Math. Soc. 350 (1998), 1931-1944

DOI: https://doi.org/10.1090/S0002-9947-98-02149-7

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Abstract:

In this paper we analyze the localization of $W(n)$, the fiber of the double suspension map $S^{2n-1}\to \Omega ^{2}S^{2n+1}$, with respect to $E(2)$. If four cells at the bottom of $D_pM^{2np-1}$, the $p$th extended power spectrum of the Moore spectrum, are collapsed to a point, then one obtains a spectrum $C$. Let $QM^{2np-1}\to QC$ be the James-Hopf map followed by the collapse map. Then we show that the secondary suspension map $BW(n)\to QM^{2np-1}$ has a lifting to the fiber of $QM^{2np-1}\to QC$ and this lifting is shown to be a $v_2$-periodic equivalence, hence an $E(2)$-equivalence.

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