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1-connected acyclic algebraic argument assertion axiom base point called canonical neighborhood chain chain-complex chain-homotopy closed simplex coefficients cohomology commutative compact connected consider continuous Corollary covering space define definition denote diagram differential groups dimension edge loops edge path element Ep,q equivalence relation exact couple example exists fact filtration finite finitely-generated follows free abelian group function functor fundamental group given graded H-space Hilton-Wylie homology groups homology theory homomorphism homotopy class homotopy invariance Horn A,B induced integer isomorphism lemma loop on xQ map f morphism natural chain-maps natural transformation Observe obtained open neighborhood open sets path-connected Plainly polyhedra polyhedron Proof Proposition push-out reader Rees system Remark short exact sequence simplex simplicial approximation simplicial complex simplicial map singular homology singular theory spectral sequence subcomplex subdivision subgroup subset Suppose topological group topological space tt(X unique vertex vertices X,xQ