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Important theorems

Key results such as the Brouwer Fixed Point Theorem and the Seifert-van Kampen Theorem.
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The statement of the theorem

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.
Source: Wikipedia