Multiplicative indecomposable splittings of \(MSp_{[2]}\) (Q1284353)

The author decomposes the symplectic cobordism Thom spectrum \(MSp_{[2]}\) localized away from \(2\) as a wedge of suspensions of an indecomposable ring spectrum retract \(LSp\). This is achieved by giving a power series description of multiplicative and idempotent multiplicative self-maps of \(MSp_{[2]}\). In particular, the homotopy type of \(LSp\) is defined by the multiplicative summand split off by certain idempotent multiplicative self-maps of \(MSp_{[2]}\), which in turn are described in terms of their corresponding power series. Furthermore, \(MSp_{[2]}\) is a ring spectrum retract of the complex cobordism Thom spectrum \(MU_{[2]}\), and \textit{V. M. Bukhshtaber} [Math. USSR, Izv. 12, 125-177 (1978); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 42, 130-184 (1978; Zbl 0384.57018)] described a multiplicative idempotent of \(MU_{[2]}\) with \(MSp_{[2]}\) as image. The author studies the relationship between the power series descriptions of multiplicative maps in \(MU_{[2]}\)- and \(MSp_{[2]}\)-theory and \(MU_{[2]}\) has a decomposition in terms of suspensions of \(LSp_{[2]}\) extending the one of \(MSp_{[2]}\). When localized at an odd prime, \(MSp_{(p)}\) and \(MU_{(p)}\) split into wedge sums of suspensions of copies of \(BP\), and the result of this paper gives that when only \(2\) is inverted, copies of \(BP\) are bound together in \(LSp\), and they cannot be separated since \(LSp\) is indecomposable.