From the long exact sequence of homology groups associated to the short exact sequence of chain complexes \(0 \to C_i(X) \xrightarrow{n} C_i(X) \to C_i(X; \mathbb{Z}_n) \to 0\) deduce immediately that there are short exact sequences
\[0 \to H_i(X)/nH_i(X) \to H_i(X; \mathbb{Z}_n) \to n\text{-Torsion}(H_{i-1}(X)) \to 0\]
where \(n\)-Torsion\((G)\) is the kernel of the map \(G \xrightarrow{n} G\), \(g \mapsto ng\). Use this to show that \(\tilde{H}_i(X; \mathbb{Z}_p) = 0\) for all \(i\) and all primes \(p\) iff \(\tilde{H}_i(X)\) is a vector space over \(\mathbb{Q}\) for all \(i\).
Solution
The short exact sequence of chain complexes \[ 0\longrightarrow C_*(X) \xrightarrow{\,n\,}C_*(X) \longrightarrow C_*(X;\mathbb Z_n) \longrightarrow0 \] gives the long exact sequence \[ \cdots\to H_i(X)\xrightarrow{n}H_i(X) \to H_i(X;\mathbb Z_n) \to H_{i-1}(X)\xrightarrow{n}H_{i-1}(X)\to\cdots. \]
1. Exactness yields a short exact sequence\[ 0\to H_i(X)/nH_i(X) \to H_i(X;\mathbb Z_n) \to {}_nH_{i-1}(X) \to0, \]where\[ {}_nG=\ker(n:G\to G). \]
The kernel of the map into \(H_i(X;\mathbb Z_n)\) is the image \(nH_i(X)\), so the first map factors injectively through the cokernel \[ H_i(X)/nH_i(X). \] The image of the connecting map is exactly the kernel of multiplication by \(n\) on \(H_{i-1}(X)\). These two observations give the displayed short exact sequence.
2. If\[ \widetilde H_i(X;\mathbb Z_p)=0 \]for every \(i\) and every prime \(p\), then every integral reduced homology group\[ G=\widetilde H_i(X) \]is a \(\mathbb Q\)-vector space.
Applying 1 with \(n=p\) gives simultaneously \[ G/pG=0 \] and \[ {}_pG=0 \] for every prime \(p\). Thus multiplication by \(p\) is both surjective and injective on \(G\). Hence multiplication by every nonzero integer is bijective. There is therefore a unique division operation by each nonzero integer, defining a unique \(\mathbb Q\)-vector-space structure on the abelian group \(G\).
3. Conversely, if every \(\widetilde H_i(X)\) is a \(\mathbb Q\)-vector space, then\[ \widetilde H_i(X;\mathbb Z_p)=0 \]for all \(i\) and all primes \(p\).
On a \(\mathbb Q\)-vector space, multiplication by \(p\) is an automorphism. Hence both \[ G/pG \] and \({}_pG\) vanish. The short exact sequence in 1 then forces the homology with \(\mathbb Z_p\) coefficients to vanish.
Thus \[ \boxed{ \widetilde H_i(X;\mathbb Z_p)=0\text{ for all }i,p \iff \widetilde H_i(X)\text{ is a }\mathbb Q\text{-vector space for all }i.} \]