STABLE SPLITTINGS FOR CLASSIFYING SPACES OF ALTERNATING, SPECIAL ORTHOGONAL
AND SPECIAL UNITARYGROUPS
Hans-Werner Henn and Huynh Mui
Abstract
Let $G(n)$ denote either the symmetric group $\Sigma(n)$, the orthogonal group
$O(n)$ or the unitary group $U(n)$ and let $SG(n)$ denote either the
alternating group $A(n)$, the special orthogonal group $SO(n)$ or the special
unitary group $SU(n)$. The classifying spaces $BG(n)$ are known to split
stably as $\displaystyle BG(n) \simeq \bigvee_{\ell=1}^n
BG(\ell)/BG(\ell - 1)$. We consider the case of $BSG(n)$ and prove that, after
localizing at any prime $p$, there are similar although somewhat coarser
splittings. E.g. we get a stable $2$-local splitting $\displaystyle
BA(4n) \simeq BA(4n)/BA(4n - 2) \vee \bigvee_{\ell=1}^{n-1}
BA(4\ell + 2)/BA(4\ell - 2)$. A crucial ingredient in our proof is a careful
study, for finite $p$- groups $P$, of the morphism sets
$\mathrm{mor}_{\mathcal A}(P;SG(n))$ in the ``Burnside category'' $\mathcal A$,
and in particular the effect of transfers on these sets.