\(\Omega\)SU(3) splits after four suspensions, but not two
Let \(J_ k\subset\Omega S^ 3\) be the \(k\)-th James filtration (consisting of all words of length \(\leq k\) in the free monoid over \(S^ 2\)). The Hopf line bundle \(S^ 2\to BU(1)\) extends to a map \(\Omega S^ 3\to BU\), whose restriction to \(J_ k\) defines a \(k\)-dimensional complex vector bundle \(\gamma_ k\). The Thom space \(X_ k\) of \(\gamma_ k\) is the Barratt-Mahowald construction. Ten years ago the first author proved that \(\Omega SU(3)\) splits stably into a wedge \(\bigvee_{k\geq 1}X_ k\). In the present paper the authors prove that this splitting can be achieved after four but not after two suspensions. For the splitting result (which was also obtained but not published by F. Cohen, M. Hopkins and F. Peterson) their task is facilitated considerably by S. Mitchell's description of the \(X_ k\) (explained in section 1) which provides a strictly associative multiplication \(X_ k\wedge X_ \ell\to X_{k+\ell}\). Using this they construct inductively cofibrations \(S^ 6\wedge X_{k-1}\to X_ k\wedge X_ 1\to X_{k+1}\). By obstruction theory they obtain first a map \(S^ 4\wedge X_ 2\to S^ 4\wedge \Omega SU(3)\); using the Hopf construction on the multiplication map of \(\Omega SU(3)\) they are then able to extend this inductively to the desired splitting of \(S^ 4\wedge \Omega SU(3)\). For the non-splitting result it is shown that \(S^ 2\wedge X_ 2\) does not split from \(S^ 2\wedge \Omega SU(3)\). This involves some delicate unstable homotopy theory relying on T. Ganea's relative EHP sequence. The paper is very carefully written, giving complete details, and makes for pleasant reading.
scientific article; zbMATH DE number 54904
scientific article