A remark on odd-primary components of special unitary groups (Q760689)

Let M be a (p,q)-framed manifold with involution [\textit{E. G. Bredon}, Bull. Am. Math. Soc. 73, 269-273 (1967; Zbl 0152.218); \textit{P. S. Landweber}, Ann. Math., II. Ser. 89, 125-137 (1969; Zbl 0174.262)] and let \([M]\in \pi^ s_{p+q}\) be the class obtained by forgetting the involution. The author shows: For odd p the order of [M] is a power of 2; moreover, under certain conditions on (p,q), the real e-invariant of [M] vanishes. This is applied to left-invariant framings on certain Lie groups, in particular to SU(4n) and SU(4n\(+3)\).