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Algebraic Topology (Definition)

The branch of topology that uses tools from abstract algebra (homology, homotopy) to study topological spaces.
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The statement of the theorem

Algebraic Topology is the study of invariants derived from a topological space X by associating algebraic structures, typically modules or groups, to its homotopy or homology groups.\text{Algebraic Topology is the study of invariants derived from a topological space } X \text{ by associating algebraic structures, typically modules or groups, to its homotopy or homology groups.} \\text{Formally, given a topological space } X \text{ and a coefficient ring } R, \text{ the } k\text{-th singular homology group } H_k(X; R) \text{ is defined via the singular chain complex } C_*(X; R). \\text{The chain groups are defined as } C_k(X; R) = \left\{ \sum_{i} r_i \sigma_i \mid r_i \in R, \sigma_i \text{ is a singular } k\text{-simplex in } X \right\}. \\text{The boundary map } \partial_k: C_k(X; R) \to C_{k-1}(X; R) \text{ is the linear map defined on a singular } k\text{-simplex } \sigma \text{ by:}\ \\partial_k(\sigma) = \sum_{j=0}^{k} (-1)^j \sigma|_{[v_0, \dots, \hat{v}_j \dots, v_k]} \\text{The homology group is then the quotient module:}\ \\text{H}_k(X; R) = \frac{\ker(\partial_k)}{\text{im}(\partial_{k+1})}. \\text{The core principle is that } H_k(X; R) \text{ is a topological invariant, meaning that if } X \text{ and } Y \text{ are homotopy equivalent, then } H_k(X; R) \cong H_k(Y; R) \text{ for all } k \text{ and } R.$