A Homological Calculus for Unstable Homotopy Groups of Spheres

The computation of unstable homotopy groups of spheres, denoted $pi_i(S^n)$, stands as one of the most challenging and historically significant problems in algebraic topology. Unlike their stable counterparts, unstable homotopy groups exhibit highly intricate and often unpredictable structures. This paper introduces a novel approach, termed a "homological calculus," designed to systematically investigate and potentially unravel the complexities of these groups. Our methodology leverages advanced techniques from homological algebra, including spectral sequences and the action of the Steenrod algebra, to translate homotopical information into algebraic invariants that are more amenable to computation. We review the foundational challenges, survey existing methods such as the Adams and May spectral sequences, and then elaborate on the theoretical construction of this homological calculus. The proposed calculus aims to provide a framework for detecting and classifying elements, understanding group extensions, and identifying structural patterns within the unstable range. While a complete algorithmic computation remains elusive, this paper presents a rigorous theoretical foundation and outlines a programmatic approach to derive significant insights into the algebraic structure of these elusive groups, offering a fresh perspective on a classic problem.