Beta Phase: Square45 is currently in beta testing. Expect some features or content to be incomplete or missing.
45

Algebraic Topology

Field: Topology

The study of topological spaces using algebraic methods.

Sequence of Expressions

Let XX be a path-connected topological space with base point x0x_0. Let πn(X,x0)\pi_n(X, x_0) denote the nn-th homotopy group and Hn(X;G)H_n(X; G) denote the nn-th singular homology group with coefficients in an abelian group GG. The Hurewicz Theorem states that if πk(X,x0)=0\pi_k(X, x_0) = 0 for all k<nk < n (i.e., XX is (n−1)(n-1)-connected), then the Hurewicz homomorphism \rho_n: \pi_n(X, x_0) \to H_n(X; \bb{Z}) is an isomorphism, and furthermore, the image of ρn\rho_n generates H_n(X; \bb{Z}) as a Z\mathbb{Z}-module. Specifically, for n≠1n \neq 1, we have the isomorphism:\n\nρn:πn(X,x0)⟶≅Hn(X;Z)if πk(X,x0)=0 for k<n. \rho_n: \pi_n(X, x_0) \stackrel{\cong}{\longrightarrow} H_n(X; \mathbb{Z}) \quad \text{if } \pi_k(X, x_0) = 0 \text{ for } k < n.
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.$
Intermediate
Every continuous function from a closed ball to itself has at least one fixed point.\nf:Dn→Dn  ⟹  ∃x∈Dn,f(x)=xf: D^n \to D^n \implies \exists x \in D^n, f(x) = x