The 𝑝-exponent of the 𝐾(1)_{*}-local spectrum Φ𝑆𝑈(𝑛)

Let p p be a fixed odd prime. In this paper we prove an exponent conjecture of Bousfield, namely that the p p -exponent of the spectrum Φ S U ( n ) \Phi SU(n) is ( n − 1 ) + ν p ( ( n − 1 ) ! ) (n-1) + \nu _p((n-1)!) for n ≥ 2 n \geq 2 . It follows from this result that the p p -exponent of Ω q S U ( n ) ⟨ i ⟩ \Omega ^{q} SU(n) \langle i \rangle is at least ( n − 1 ) + ν p ( ( n − 1 ) ! ) (n-1) + \nu _p((n-1)!) for n ≥ 2 n \geq 2 and i , q ≥ 0 i,q \geq 0 , where S U ( n ) ⟨ i ⟩ SU(n) \langle i \rangle denotes the i i -connected cover of S U ( n ) SU(n) .