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The complex \(J\)-homomorphism is a well-known map \(J: BU\to BSG\). However, in addition to being a map of spaces, \(J\) is a map of infinite loopspaces. If \(\psi^r:BU\to BU\) is the \(r\)-th Adams operation the following diagram is known to be homotopy commutative when localized away from \(r\).
\[\tag{*}\begin{tikzcd} BU\ar[rr, "{\psi^r}"] \ar[dr, "J"] & & BU \ar[dl, "J"] \\ & BSG & \end{tikzcd}\] Homotopy commutativity results from the affirmative solution of the Adams conjecture. Localized away from \(r, \psi^r\) is also an infinite loop map. The author shows that (*) is actually homotopy commutative as a diagram of infinite loopspaces.
Precisely Theorem: For any integer \(r >0\) (*), localised away from \(r\), is the first stage of a triangle of connected 0-spectra. Several proofs of the classical Adams conjecture are known. However the adaptation of one of them to the infinite loopspace category seems to be a very difficult task. To quote the author, “view this as an excellent challenge to the methods used in the various proofs of the Adams conjecture and to the various approaches to infinite loopspace theory.” In fact the author adapts D. Sullivan’s proof [Ann. Math. (2) 100, 1–79 (1974; Zbl 0355.57007)]. This proof consisted of the construction of unstable versions of in a homotopy commutative diagram of spherical fibrations as follows.
\[\tag{**}\begin{tikzcd} S^{2n-i} \ar[r, "="] \ar[d] & S^{2n-i} \ar[d] \\ BU(n-1) \ar[r,"\psi^r"] \ar[d] & BU(n-1) \ar[d] \\ BU(n) \ar[r,"\psi^r"] & BU(n) \end{tikzcd}\]
From (**) the Adams conjecture for spaces follows by appeal to the fact that BSG classifies stable spherical fibrations while (**) shows that \(\psi^r\) preserves the spherical fibration of the universal complex \(n\)-plane bundle. Therefore the author has to analyze precisely what BSG, BU etc. classify in the infinite loopspace sense – hence the title. Incidentally, the theorem has as a corollary the fact that \(J\) splits \(J\)-theory from BSG-theory, thereby giving a piece of the latter which can be understood. Although BSG-theory of a sphere is just the stable homotopy of spheres we know little of BSG-theory of a general space, \(X\). Certain, BSG-theory of \(X\) and \(\pi^s_*(X)\) have little resemblance. For example, when \(X=HP^\infty\), \(\pi_{15}^*(HP^\infty)\) is too small to contain \(J_{15}^*(HP^\infty)\) as a summand.
Editorial remark (2022): In [P. Bhattacharya and N. Kitchloo, Ann. Math. (2) 195, No. 2, 375–420 (2022; Zbl 1487.55022)], it was shown that this proof is incorrect and a corrected version given.