$T$-stable vs. $S^1$-stable motivic homotopy category: which sorts of transfers are available?

Question

I would like to have some sort of transfers in motivic stable homotopy categories in order to adapt Voevodsky's split standard triple argument to cohomology theories that can be factorized through these categories. So, I wonder:

1. What sorts of 'homotopy transfers' are known? At the moment I found http://folk.uio.no/paularne/rigidityma.pdf; probably I can find something in Morel's papers.

2. What are the main differences between the (functoriality?) properties of the categories $SH$ and $SH^{S^1}$ (in the latter we do not stabilize wrt. the Tate twists)?

3. Do there exist interesting 'motivic' theories that cannot be factorized through $SH$?

4. What about transfers for $MGl$-module theories, i.e. for $Y$ being finite over $X$ what can one say about $SH^{S^1}$-morphisms from $MGl\wedge X_+\to MGl\wedge Y_+$, where $MGl$ is the spectrum that represents Voevodsky's algebraic cobordism?

5. Will these questions become easier if one will take $\mathbb{Z}[1/2]$-coefficients?

Answer 1

To your first question: There is sort of generalized transfer structure on $\pi_{p,q}(E)$ as $GW$-module in $SH$, which you can find in Morel's papers. In fact, the composition $$\pi_{p,q}(E)(k) \rightarrow \pi_{p,q}(E)(L) \rightarrow \pi_{p,q}(E)(k)$$

is the multiplication with $Tr^L_k (\langle a \rangle)$ for some $a \in L^{\times}$, where $L/k$ is a finite extension and $\langle a \rangle$ denotes the class of $a$ in $GW(L)$.