Autoreducibility, mitoticity, and immunity

We show the following results regarding complete sets.
• NP-complete sets and PSPACE-complete sets are polynomial-time many–one autoreducible.
• Complete sets of any level of PH, MODPH, or the Boolean hierarchy over NP are polynomial-time many–one autoreducible.
• EXP-complete sets are polynomial-time many–one mitotic.
• If there is a tally language in NP∩coNP−P, then, for every ϵ>0, NP-complete sets are not 2n(1+ϵ)-immune.These results solve several of the open questions raised by Buhrman and Torenvliet in their 1994 survey paper on the structure of complete sets.