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Aug 16, 2020 at 20:58 comment added geometricK Although I completely agree with your answer, I was secretly hoping for an approach without Poincare duality, as (although I didn't mention this) Poincare duality may not be available to me in the setting I'm actually working in.
Aug 16, 2020 at 20:09 history answered Nicholas Kuhn CC BY-SA 4.0