# Transfers on Bloch groups and scissors congruence groups

I have a couple of questions concerning existence and description of transfers for Bloch groups and scissors congruence groups/pre-Bloch groups.

To fix notation and recall definitions:

From the general algebraic K-theory machinery, we get transfers on $K_3$. In particular, for a finite field extension $E/F$, we get a map $\operatorname{tr}_{E/F}:K_3(E)\to K_3(F)$ such that the composition $K_3(F)\to K_3(E)\stackrel{\operatorname{tr}_{E/F}}{\longrightarrow}K_3(F)$ is multiplication with the degree $[E:F]$.

Now by the work of Bloch, Dupont-Sah, Suslin and others, we have another description of $K_3(F)^{\operatorname{ind}}=K_3(F)/K_3^M(F)$ in terms of an exact sequence $$0\to \widetilde{\operatorname{Tor}}(\mu(F),\mu(F))\to K_3(F)^{\operatorname{ind}}\to B(F)\to 0.$$ In the above, the Bloch group $B(F)$ is defined as $$B(F)=\ker\left(\mathcal{P}(F)\to \Lambda^2(F^\times):[x]\to x\wedge(1-x)\right)$$ and the group $\mathcal{P}(F)$ is the pre-Bloch group or scissors congruence group $$\mathcal{P}(F)=\left(\bigoplus_{x\in F^\times\setminus\{1\}}\mathbb{Z}[x]\right) /\left([x]-[y]+[y/x]-[(1-x^{-1})/(1-y^{-1})]+[(1-x)/(1-y)]\right),$$ the relation coming from the five-term relation satisfied by the dilogarithm. In particular, elements of the Bloch group $B(F)$ can be written down as linear combinations of symbols $[x], x\in F^\times\setminus\{1\}$ satisfying certain relations.

Now I can formulate my questions on transfers on Bloch groups and scissors congruence groups.

1. Does the transfer on $K_3$ induce a transfer on the Bloch group? I think that this is not the case in general. The element $[x]+[1-x]\in B(F)$ is independent of $x$, and is typically denoted by $c_F$. The element $c_{\mathbb{R}}$ has exact order $6$ in $B(\mathbb{R})$, and the element $c_{\mathbb{C}}$ is trivial in $B(\mathbb{C})$. This seems to contradict transfers for the Bloch group (it does not contradict transfers for $K_3^{\operatorname{ind}}$ because the torsion moves from $B(F)$ to $\widetilde{\operatorname{Tor}}(\mu(F),\mu(F))$). Are there more torsion elements like this, in particular with other odd orders? Are there further obstructions to the existence of transfers on Bloch groups? If $F$ contains an algebraically closed fields, it follows from work of Suslin and Levine that $B(F)$ is uniquely $\ell$-divisible for $\ell$ different from the characteristic - in particular $c_F=0$. Does the $K$-theory transfer induce a transfer on the Bloch group in this situation? What would be a good reference?

2. Is there an explicit description of what the transfer map on $K_3$ does on the Bloch group? I would be interested in a description that only uses the definition of the Bloch group via points on $\mathbb{P}^1(F)\setminus\{0,1,\infty\}$ given above.

3. More generally, are there transfers known on scissors congruence groups/pre-Bloch groups? As written above, these groups are defined in terms of points on $\mathbb{P}^1\setminus\{0,1,\infty\}$ modulo the five-term relation. A very naive approach to the definition of transfers for pre-Bloch groups in an extension $E/F$ would be to sum over $E$-points lying over $F$-points of $\mathbb{P}^1\setminus\{0,1,\infty\}$. Has anyone ever tried to work this out, or are there known obstruction why this cannot provide a transfer? Assuming it works, how would one relate such a naive definition to the definition of transfers for algebraic K-theory? In a related direction, what torsion elements besides those in $B(F)$ are known in the scissors congruence groups over fields which are not algebraically closed (in the algebraically closed case, the scissors congruence groups are uniquely divisible)?

Concerning 1: The torsion in $B(F)$ is cyclic, and is related to the roots of unity which are roots of irreducible polynomials of degree $2$ over $F$. For example, $K_3(\mathbb{F}_q)^{\operatorname{ind}}\cong\mathbb{Z}/(q^2-1)$ where $\mathbb{Z}/(q-1)$ comes from the roots of unity in $\mathbb{F}_q$, and $\mathbb{Z}/(q+1)$ comes from the Bloch group. Torsion elements in the Bloch group (such as $[x]+[x^{-1}]$ or $[x]+[1-x]$) seem to be related to automorphisms of $\mathbb{P}^1\setminus\{0,1,\infty\}$. However, I do not know of a general way to write down explicit generators for the torsion in $B(F)$. These torsion elements do obstruct the existence of transfers for $B(F)$ because the torsion element corresponding to $\zeta_n$ is killed in the extension $F(\zeta_n)$.
Concerning 2 and 3: If $E/F$ is a finite Galois extension with group $G$, then $K_3(E)^{\operatorname{ind}}$ is a $G$-module, and by the work of Levine, $K_3(F)^{\operatorname{ind}}=(K_3(E)^{\operatorname{ind}})^G$. I guess the transfers are given by summing over Galois orbits, and that this procedure should also work for the Bloch group.
In the case of a radical extension the relevant facts can be found in papers of Dupont and Sah: assume $F$ is such that $\zeta_n\in F$, let $a\in F$ and consider the Galois extension $F(\sqrt[n]{a})/F$. Then Dupont-Sah prove the following equality in the scissors congruence group: $$\frac{[a]}{n}=\sum_{0\leq i\leq n-1}[\zeta_n^i a].$$ In particular, the sum over elements of the Galois orbit in $F(\sqrt[n]{a})$ has a representative over $F$.
Ok, as said before: this is only partial information. I guess, what I want to know is, if the above works generally for arbitrary Galois extensions. How could one generally obtain $F$-representatives of sums over Galois orbits in $E/F$?
I have worked on this question for some time (in the $$\mathbb{Q}$$-coefficients case), trying to construct norm maps on $$B_2(F)$$ in a way, similar to Milnor $$K-$$theory. Eventually, I was able to construct norm maps on the middle cohomology groups of complexes $$B_3(F) \otimes \Lambda^{n-3} F^{\times} \longrightarrow B_2(F) \otimes \Lambda^{n-2} F^{\times} \longrightarrow \Lambda^n F^{\times},$$ which conjecturally coincide with $$H^{n-1}_{M}(F,\mathbb{Q}(n)),$$ for which existance of norm maps is known (see https://arxiv.org/abs/1511.00520). Unfortunately, I had to use the fact that the kernel of the map $$B_2(F)\longrightarrow \Lambda^2 F^{\times}$$ does not change under a simple transcendental extension of the field, so the construction of the norm is not completely explicit at the end of the day.