Hurewicz theorem

The Hurewicz theorem provides a bridge between two ways of understanding the shape of a space: its topological complexity (homotopy groups) and its algebraic structure (homology groups). Specifically, for spaces that are sufficiently connected (meaning they have no "holes" up to a certain dimension), the theorem states that the first non-trivial homology group accurately reflects the corresponding homotopy group. In simple terms, if a space is nicely connected, then its first significant algebraic measurement of shape (homology) gives you information about its more nuanced forms of connectivity (homotopy), helping mathematicians analyze and classify spaces more effectively.