I want to know what is known about the cofibre of the $n$-fold transfer map $\mathbb{R}P^{\wedge n}_+\to S^0$, for $n>1$. I am happy to know of any specific example worked out. The case $n=1$ is known, due to James I think, to be $\Sigma\mathbb{R}P_{-1}$.
It appears that the below strategy does not work. But, I keep it for the comments below to make sense.
In particular, if it was possible to find some representation $U$ of $O(1)^{\times n}$ with a $O(1)^{\times n}$ invariant metric so that we have a homeomorphism $O(1)^{\times n}\to S(U)$ as $O(1)^{\times n}$-spaces, then I would be able to identify the cofibre (in some sense at least). Here, $S(U)$ is the sphere within $U$ with respect to the metric of $U$. Again, for the case $n=1$ we have $O(1)\to S(\mathbb{R})$. If you think such a representation $U$ cannot exist, which I think is the case, then how do you see this?
EDIT: I still, look forward other methods of identifying cofibre of the transfer maps, specifically those transfer maps associated to the diagonal embedding $1<G^{\times n}$ where we allow $G$ to be a compact Lie group. As far as I know, there is not much known about this? But, I appreciate any reference to any work or any unpublished work.
Addendum. For $n=2$, if I am not mistaken, I think playing with cofibrations, one can show that there exists a fibration of spectra as $$\mathbb{R}P_{-1}\to \mathbb{R}P_+\wedge\Sigma\mathbb{R}P_{-1}\to C_{\lambda^2}$$ where $\lambda^2:\mathbb{R}P^{\wedge 2}_+\to S^0$ is the double transfer. I think one can play around to find similar fibrations to fit in the coufibre of the other folded transfer maps.