On Realisability of Twisted Homology

@misc{grant2026realisability,
  title = {{On Realisability of Twisted Homology}},
  author = {Mark Grant and Michael Jung and Baylee Schutte},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2607.24462},
  eprint = {2607.24462},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.24462},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - On Realisability of Twisted Homology
AU  - Mark Grant
AU  - Michael Jung
AU  - Baylee Schutte
PY  - 2026
DA  - 2026/11/04
PB  - arXiv
DO  - 10.48550/arxiv.2607.24462
UR  - https://arxiv.org/abs/2607.24462
AB  - We discuss the question of when a homology or cohomology class with twisted integer coefficients of a manifold X is realised by a submanifold. While this question is classical in nature, providing an answer requires relatively modern techniques from parametrised homotopy theory. More specifically, we introduce cobordism classes twisted by a coefficient system and then define a twisted Thom space M^twO(n) over BO(1), which serves as the classifying object for this cobordism theory under a twisted Pontryagin-Thom construction. As a result, a twisted homology class is realisable if and only if its Poincaré dual is the image of the twisted Thom class in M^twO(n) under a parametrised map X → M^twO(n) over BO(1). By showing that M^twO(n) is parametrised homotopy equivalent to the unbased homotopy orbits of MSO(n) under the orientation-reversing involution, we are able to construct its parametrised Postnikov tower to derive obstructions to realisability. We conclude by giving the first known examples of non-realisable integer homology classes in non-orientable manifolds.
ER  -