On Realisability of Twisted Homology

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 ${\mathrm{M}}^{\mathrm{tw}}\mathrm{O}(n)$ over $\mathrm{BO}(1)$, 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 ${\mathrm{M}}^{\mathrm{tw}}\mathrm{O}(n)$ under a parametrised map $X\to {\mathrm{M}}^{\mathrm{tw}}\mathrm{O}(n)$ over $\mathrm{BO}(1)$. Finally, we construct the parametrised Postnikov tower of ${\mathrm{M}}^{\mathrm{tw}}\mathrm{O}(n)$ over $\mathrm{BO}(1)$ to derive obstructions to realisability and conclude by giving the first known examples of non-realisable integer homology classes in non-orientable manifolds.

arXiv commentmore details appear in the thesis of the second author; comments are welcome