what is the effect on $F_p$-homology groups of the cofiber sequence $S_{hC_p} \to S^{hC_p} \to S^{tC_p}$? i think i'm getting thrown off by the fact that $BC_p$ isn't finite, so e.g. the middle term isn't literal cohomology
That cofiber sequence splits after p completing (and the last term is the p-complete sphere) so it just looks like almost an equivalence on mod p homology.
