I’m trying to understand Loday’s proof of Theorem 7.3.11 in “Cyclic Homology” (2nd ed 1998). I have a problem right at the beginning where it says:

The projection map $\text{proj}:\Gamma_\bullet G \to B_\bullet G$ onto the nerve of $G$, is given by $\text{proj}(g_0, \ldots, g_n) = (g_1, \ldots, g_n)$. It is obviously a continuous simplicial map, but it is even a continuous cyclic map provided that one puts on $B_\bullet G$ the cyclic structure of $B_\bullet(G, 1)$ as described in 7.3.3.

Recap of definitions: Given a topological group $G$, $\Gamma_\bullet G$ is the cyclic bar construction, which in this case is equivalent to the cyclic nerve $N_\bullet^\text{cy}(G)$. We have $\Gamma_n G \cong G^{n+1}$, the usual simplicial structure maps and the cyclic permutation $t_n:\Gamma_n G\to \Gamma_n G$ mapping $(g_0,\ldots,g_n)\mapsto (g_n, g_0, \ldots, g_{n-1})$.

The construction referenced is the “twisted nerve of $G$”: It is the fact that the simplicial space $B_\bullet G$ admits a cyclic structure as well, namely via

$$t_n:B_n G\to B_n G, \quad (g_1, \ldots, g_n) \mapsto ((g_1\cdots g_n)^{-1}, g_1, \ldots, g_{n-1}).$$

Now for the actual question: Is the assertion that $\text{proj}$ is a cyclic map actually true? It is certainly simplicial, but $\text{proj}$ and $t_n$ do not commute as far as I can see:

\begin{align} t_n\circ\text{proj}(g_0,\ldots,g_n) &= ((g_1\cdots g_n)^{-1},g_1,\ldots,g_{n-1})\\ \text{proj}\circ t_n(g_0,\ldots,g_n) &= (g_0, \ldots, g_{n-1}) \end{align}

If the two would commute, this would imply $g_0 = (g_1\cdots g_n)^{-1}$, which is not necessarily true for an element in $\Gamma_n G$ (else we would only need $\Gamma_n G \cong G^n$ if one element was pre-determined). But it is necessary that the map be cyclic, else its realization is not necessarily $S^1$-equivariant, and the proof cannot continue.

(It may be noteworthy that the map $i:B_n G\to \Gamma_n G$ sending $(g_1,\ldots,g_n)\mapsto ((g_1\cdots g_n)^{-1}, g_1, \ldots, g_n)$ is indeed cyclic, and $\text{proj}\circ i = \text{id}$.)

Any ideas or pointers to other references are greatly appreciated! Thank you.


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