2021/02/12 by Bergfalk, Jeffrey, Hrušák, Michael, Lambie-Hanson, Chris · 1 citation
#03E35 #03E75 #18E25 #55N07 #Algebraic Topology (math.AT) #FOS: Mathematics #Logic (math.LO)
paper · doi:10.48550/arxiv.2102.06699
A question dating to Sibe Mardešić and Andrei Prasolov's 1988 work Strong homology is not additive, and motivating a considerable amount of set theoretic work in the ensuing years, is that of whether it is consistent with the ZFC axioms for the higher derived limits limn (n>0) of a certain inverse system A indexed by ωω to simultaneously vanish. An equivalent formulation of this question is that of whether it is consistent for all n-coherent families of functions indexed by ωω to be trivial. In this paper, we prove that, in any forcing extension given by adjoining \bethω-many Cohen reals, limn A vanishes for all n > 0. Our proof involves a detailed combinatorial analysis of the forcing extension and repeated applications of higher dimensional Δ-system lemmas. This work removes all large cardinal hypotheses from the main result of arXiv:1907.11744 and substantially reduces the least value of the continuum known to be compatible with the simultaneous vanishing of limn A for all n > 0.