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May 23, 2022 at 7:19 comment added user482036 see Integral generators for the cohomology ring of moduli spaces of sheaves over Poisson surfaces, around formula (12)
Mar 5, 2022 at 23:46 comment added Z. M The vanishing conditions seem to make the Atiyah–Hirzebruch spectral sequence degenerates at $E_2$. Since you mentioned the Poincaré duality, it might also be interesting to compare this with the Atiyah duality (one could find a proof in a note by Rezk). In particular, (almost) complex manifolds admit a $\operatorname{Spin}^c$-structure, therefore are $\operatorname{KU}$-orientable.
S Mar 5, 2022 at 18:02 history suggested The Amplitwist CC BY-SA 4.0
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