A Gap Cohomology Group

Dan Talayco has recently defined the gap cohomology group of a tower in p(ω)/fin of height ω₁. This group is isomorphic to the collection of gaps in the tower modulo the equivalence relation given by two gaps being equivalent (cohomologous) if their levelwise symmetric difference is not a gap in the tower, the group operation being levelwise symmetric difference. Talayco showed that the size of this group is always at least 2^N0 and that it attains its greatest possible size, 2^N1, if ? holds and also in some generic extensions in which CH fails, for example on adding many Cohen or random reals. In this paper it is shown that there is always some tower whose gap cohomology group has size 2^N1. It is still open as to whether there are models in which there are towers whose gap cohomology group has size less than 2^ω1.