Year: 2026
Type: preprint
Abstract: 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 $\operatorname{M^\mathrm{tw}O}(n)$ over $\operatorname{BO}(1)$, 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 $\operatorname{M^\mathrm{tw}O}(n)$ under a parametrised map $X \to \operatorname{M^\mathrm{tw}O}(n)$ over $\operatorname{BO}(1)$. By showing that $\operatorname{M^\mathrm{tw}O}(n)$ is parametrised homotopy equivalent to the unbased homotopy orbits of $\operatorname{MSO}(n)$ 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.
Source: arXiv (Cornell University)
Authors: Grant, Mark, Jung, Michael, Schutte, Baylee
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7171511262