证明
Theorem 3.1 ([The presentation comparison theorem]). Suppose that Γ is…Lemma 1 ([Localized word-map degree]). Let m≥1 , and let f:H^m→H^m be a word…观点
Start with any two-dimensional K(Γ,1) . Collapse a maximal tree in its connected one-skeleton and replace the two-cell attaching loops by finite edge words. These operations preserve the homotopy type: a contractible CW subcomplex can be collapsed, and homotopic attaching maps give homotopy equivalent adjunction spaces . These facts apply to infinite CW complexes as well; a loop in a graph is homotopic to a finite edge path. The resulting presentation complex has one vertex, with some sets of generators and relators. Its universal cover is acyclic and has no cells above dimension two. Consequently its relator boundary map is an isomorphism onto the cycle module. The relator words also normally generate the kernel of the map from the free generator group to Γ . Adjoin the prescribed letters S , with one definition relator for each letter. Each new relator boundary has coefficient 1 on its own new edge and coefficient 0 on the other new edges. Subtracting these boundaries reduces any cycle uniquely to an old cycle, so the boundary basis property persists. For each former generator a , choose a word w_a(S) with value a in Γ , and replace a by t_a w_a(S) . This is an invertible change of free generators: its inverse sends t_a to a w_a(S)^ -1 and fixes S . The new letters t_a all have value 1 . On edge chains the substitution fixes the S -edges and sends e_a to e_ t_a +[w_a] , so it is a chain isomorphism and preserves the boundary basis property. Let T be the set of these new letters, R the current relator index set, and I= D T . Equation identifies the cycle module with Λ^ (I) . In these coordinates the relator boundary map is an isomorphism A:Λ^ (R) ⟶Λ^ (I) . We next arrange that the prescribed cycles themselves occur as relator boundaries. An auxiliary copy of the cycle module allows the three shears below to move this basis into a new relator block. Adjoin letters U=(u_i)_ i∈I and defining relators u_i . The stabilized boundary map is D_0= A, 0; 0, 1 : Λ^ (R) ⊕Λ^ (I) ⟶Λ^ (I) ⊕Λ^ (I) , where the second target summand consists of the U -edges. On its domain put P= 1, A^ -1 ; 0, 1 , Q= 1, 0; -A, 1 . These are blocks of homomorphisms of left modules; all products below mean composition. Direct multiplication gives PQP= 0, A^ -1 ; -A, 0 , D_0PQP= 0, 1; -A, 0 . We realize each of the successive precompositions P,Q,P by actual relator words. A shear leaves one block of words unchanged and adds to each boundary in the other block a finite sum ∑_k n_k g_k[b_k] of translates of unchanged boundaries. Choose a generator word v_k representing g_k and multiply the changing relator by copies of v_k b_k^ ±1 v_k^ -1 , with the indicated multiplicities and signs. All these words represent 1 in Γ ; the added boundary is precisely the stated sum. The unchanged words belong to both old and new relator lists, so the old changing word is recovered from the new one by a finite inverse multiplication. Thus the old and new lists have exactly the same normal closure. Every column of A and A^ -1 has finite support because their domains and targets are direct sums. Hence this construction uses finite words for every relator even when the index sets are uncountable. It does not require an infinite product of words or a limiting sequence of presentation moves. In the final second block, the relator indexed by i∈I has boundary (e_i,0) . Designate those indexed by z∈ D as ρ_z . Set T=T U . The remaining second-block relators are indexed by T , and the first-block relators are indexed by R ; these are the ordinary relators, with R=T R . Their projected boundary map is, in these orders, 1_T, 0; 0, -A : Λ^ (T) ⊕Λ^ (R) ≅ Λ^ (T) ⊕Λ^ (I) . Equation also shows that the complete boundary list remains a basis. Normal closure was preserved at every step, so these are the relators of an actual presentation of Γ .