The spectrum $(P\wedge \textrm {BP}\langle 2\rangle )_ {-\infty }$

Abstract:

The spectrum ${(P \wedge {\text {BP}}\langle {\text {2}}\rangle )_{ - \infty }}$ is defined to be the homotopy inverse limit of spectra ${P_{ - k}} \wedge {\text {BP}}\langle {\text {2}}\rangle$, where ${P_{ - k}}$ is closely related to stunted real projective spaces, and ${\text {BP}}\langle {\text {2}}\rangle$ is formed from the Brown-Peterson spectrum. It is proved that this spectrum is equivalent to the infinite product of odd suspensions of the $2$-adic completion of the spectrum of connective $K$-theory. An odd-primary analogue is also proved.