## An introduction to algebraic topology |

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algebraic topology antipodal points assigns barycentric coordinates barycentric subdivision Borsuk-Ulam Theorem boundary Br(K called chain complex chain homotopy chain transformation Chapter coefficient group commutative completes the proof Consequently consider contained contiguity class continuous function covariant functor Cq(K Cr(K cycle defined denoted diagram element elementary integral r-chain equation equivalence class example EXERCISE exists face of sk fixed-point property Fixed-Point Theorem follows geometrically independent subset graded group graded homology group group G group Hr(X homo homology groups homology theory homomorphism homotopy classes Hr(K hyperplane identity map induced homomorphism integer l)-simplex Lefschetz Lemma morphism n-simplexes nonzero distribution nullhomotopic oriented simplex plex polytope map polytope pair proof of Theorem Prove Theorem r-dimensional chains real numbers represent retract sequence set of vertices Show simplicial approximation simplicial map simplicial pair star related subcomplex Suppose surjective symbol Theorem 11 Theorem 20 topological pairs topological spaces triangulation vector distribution vector space vertex winding number zero Zr(K