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May 23, 2016 at 4:01 comment added HuiRong Thank you Marco, your reply exactly answered my question. For the case that the first betti number is non-zero, Danny (see below) showed me the references. So for non-torsion spin-c structures, the infinity-version is zero, and for torsion ones, it is given by the "cup homology".
May 16, 2016 at 6:30 history answered Marco Golla CC BY-SA 3.0