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The Fundamental Group
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AH'I algebraic base point based map chain complex chain homotopy chain map Chapter Cn(K coefficients commutative diagram composite consider constant map construct contained continuous map Corollary corresponding CW-complex Deduce defined Definition Duality Theorem element exact sequence Example Exercise exists finite number follows free abelian group fundamental group Given group G Hausdorff Hence Hn(C Hn(K Hn(Mn Hn(X homology groups homology n-manifold homomorphism homotopy classes homotopy equivalence homotopy F identification map identified identity isomorphism identity map inclusion map induces integer interior inverse irn(X isomorphism l-l)-correspondence l)-simplex Lemma manifolds map f map of pairs Moreover n-manifold with boundary n-simple open set path component path-connected polyhedra polyhedron properties Proposition prove result Section Similarly simplex Simplicial Approximation Theorem simplicial complex simplicial map simplicial pair space obtained subcomplex subgroup subset subspace suppose topological space triangulation vector space vertex vertices