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Axioms and general theorems
Homology theory of simplicial complexes
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abelian group admissible category algebraic barycentric subdivision boundary operator c-category called Cech groups Cech homology Cech theory chain complex Chapter cochain coefficient group cofinal commutativity holds compact pairs compact space compactification Consider consisting Cov(X,A defined Definition deformation retract denoted diagram dimension direct sum direct system directed set element exact sequence Excision axiom exists finite number follows function g e G g-simplex group G Hence homology and cohomology homology cohomology homology groups homology sequence homology theory homotopy equivalence identity map implies inclusion map indexed induced homomorphisms induces isomorphisms inverse limit inverse system kernel Lemma Let X,A linear lower sequence map f map g Mayer-Vietoris sequence morphism nonvacuous obtained open sets pair X,A Proof proper triad prove relative homeomorphism satisfying Show simplex simplicial complex simplicial map singular homology subcomplex subgroup Theorem triangulable pairs unique vertex vertices X,ir Xa,Aa yields zero