The $pi_1$-Filtration of Homotopy Spheres: A Covering Space Resolution

This paper introduces and develops the concept of the $pi_1$-filtration applied to manifolds that are candidates for homotopy spheres, and presents a novel resolution of this filtration through the lens of covering space theory. While classical homotopy spheres for dimensions $n ge 2$ possess trivial fundamental groups, understanding the precise structure of the fundamental group is paramount in distinguishing genuine homotopy spheres from other manifold types. The $pi_1$-filtration, derived from the lower central series of the fundamental group, provides a hierarchical decomposition of its non-abelian structure. We demonstrate how a sequence of appropriately constructed normal covering spaces can geometrically realize each stage of this algebraic filtration. Each covering space in the sequence corresponds to a specific subgroup from the filtration, effectively resolving the complex group structure into more accessible components. This geometric resolution offers new insights into the topological properties of such manifolds, facilitating a deeper analysis of their fundamental group and its implications for their homotopy type. The methodology leverages established results from algebraic topology, including the lifting correspondence between subgroups of the fundamental group and covering spaces, to establish a rigorous framework for this resolution. The findings contribute to a more comprehensive understanding of the structure of manifolds potentially homeomorphic or homotopy equivalent to spheres, and offer a powerful tool for their classification and characterization.