A twisted homology class is realised by a submanifold iff its Poincaré dual is pulled back from the twisted Thom class of M^tw O(n) via a map over BO(1); the first non-realisable examples in non-orientable manifolds are given.
Hermitian K-theory for stable∞-categories. II: Cobordism categories and additivity
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On Realisability of Twisted Homology
A twisted homology class is realised by a submanifold iff its Poincaré dual is pulled back from the twisted Thom class of M^tw O(n) via a map over BO(1); the first non-realisable examples in non-orientable manifolds are given.