Homotopy groups are algebraic structures that capture the topological features of a space, specifically focusing on the different ways in which loops and higher-dimensional spheres can be transformed into one another without leaving the space. These groups, denoted as \( \pi_n(X) \) for a space \( X \) and dimension \( n \), provide crucial information about the space's shape, allowing for insights into its connectivity and higher-dimensional holes. The study of homotopy groups is essential for understanding the Bott periodicity theorem and its implications in various fields.
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