## Invariants for effective homotopy classification and extension of mappings |

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1-simplex abelian group addition formula algebraic argument base point change of base coboundary operator cocycle cohomology operations commutativity completes the proof Corollary 16.2 Corollary 7.5 corresponding coset cup product defined definition 6.6 denote difference homomorphism dimension q Eilenberg and MacLane Eilenberg-MacLane element epimorphism extension problem f and g f*en follows at once g re1 group G H TT Hn(X Hn(Y homotopy classification homotopy enumeration homotopy group homotopy invariant homotopy problem homotopy rel Hq(A Hq(X Hq(Y Hq+1(TT hypothesis injection isomorphism j*xn kq+1 Lemma Let f mapping cylinder mapping f Math Mayer-Vietoris Mayer-Vietoris sequences monomorphism n-sphere non-abelian notation obstruction theory Olum pathwise connected PAUL OLUM polyhedral pair proper triad properties prove S(Af S(xQ S(yQ satisfies 10.2 second level invariants second obstruction semi-simplicial mapping Similarly for homotopy subgroup suppose given Theorem 7.1 tion wel1-defined wq+1 h z(pu