= Homological transfer for a finite covering
{title2=$f_*\operatorname{tr}=n\,\mathrm{id}$}
For an $n$-sheeted <covering space> $f:X\to Y$, define the transfer of a <singular simplex> as the sum of its $n$ lifts. Restriction of these lifts to any face gives exactly the lifts of that face, so transfer is a <chain map>. Its induced map $\operatorname{tr}:H_i(Y;R)\to H_i(X;R)$ satisfies $f_*\operatorname{tr}=n\,\mathrm{id}$ for any commutative coefficient ring $R$. If $n$ is invertible in $R$, then $n^{-1}\operatorname{tr}$ is a right inverse of $f_*$. In particular a finite covering surjects onto the base's rational <homology>.
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