Abstract

A fundamental question in geometric topology, known as the Steenrod problem, asks whether a given homology class can be realized as the image of the fundamental class of a smooth manifold. While this was largely settled by Thom in the 1950s for constant coefficients, the case for local coefficients remained unanswered. In this talk, we introduce the twisted Thom space and sketch the twisted Pontryagin-Thom construction adapted for local coefficients to provide a criterion for realizability. We will discuss the resulting obstructions by building the Postnikov tower of the twisted Thom space within the slice category over BO(1), and highlight how this framework differs from the classical setting. This is joint work with Baylee Schutte and Mark Grant.