Corollary 12. The universal cover of X is contractible, and G is torsion-free.

The simply connected two-dimensional cover X has H_2=0 by Proposition 10 and Hurewicz, has H_1=0 , and has no cellular homology above dimension two. It is therefore contractible by the homological form of Whitehead's theorem~. For completeness, the torsion obstruction can be seen directly from resolutions; compare~. The augmented cellular chains of X give an exact free ℤG -resolution of ℤ of length two. If G had a nontrivial finite-order element, a power would generate a cyclic subgroup H=⟨g⟩ of prime order ℓ . Restricting the resolution to ℤH preserves freeness, since ℤG is free as a left ℤH -module on coset representatives. The cyclic group has the exact periodic resolution ⋯⟶ℤH \,N\,ℤH \,g-1\,ℤH \,N\,ℤH \,g-1\,ℤH ⟶ℤ⟶0, N=1+g+⋯+g^ ℓ-1 . Exactness follows by comparing the coefficients of 1,g,…,g^ ℓ-1 : the kernel of g-1 consists of the multiples of N , and the kernel of N is the augmentation ideal generated by g-1 . With trivial coefficients 𝔽_ℓ , all differentials in the corresponding cochain complex vanish. Hence H^k(H;𝔽_ℓ)≅𝔽_ℓ in every degree. The comparison theorem for projective resolutions makes this cohomology independent of the chosen resolution~. This contradicts the vanishing above degree two forced by the restricted finite-length free resolution.