Finite-index transfer on normalized bar chains

Use diagonal bar chains $C_{\ast }(G;M)$ for a left $G$-module $M$, with the DC and supplied-resolution convention when identifying their homology with derived homology. Let $H\le G$ have finite index. Choose representatives $T$ of the right cosets $Ht$, with $1\in T$. Write $x=r(x)t(x)$ with $r(x)\in H$ and $t(x)\in T$. Transfer is ${\mathrm{Tr}}_T[(g_0,\dots ,g_n)\otimes m]={\sum }_{t\in T}[(r(t{g}_0),\dots ,r(t{g}_n))\otimes tm]\mathrm{.}$ For trivial coefficients, in inhomogeneous coordinates set $t_0=t$, $t_i=t(t_{i-1}{g}_i)$ and $h_i=t_{i-1}{g}_i{t}_i^{-1}\in H$. Then ${\mathrm{Tr}}_T[g_1\mathrm{\mid }\cdots \mathrm{\mid }{g}_n]={\sum }_{t\in T}[h_1\mathrm{\mid }\cdots \mathrm{\mid }{h}_n]\mathrm{.}$ A tuple with an identity entry in inhomogeneous coordinates is zero. Well-definedness and the chain-map property are supplied by the following lemma.