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There is a well-known operation known as cabling for knots, and also for braid groups, where it is a homomorphism $$\beta_k \times \beta_\ell \longrightarrow \beta_{k\ell}$$ given by thickening up the $k$-strand braid and putting one copy of the $\ell$-strand braid inside each of its $k$ tubes. This arises from a map of unordered configuration spaces $C_k(\mathbb{R}^2)\times C_\ell(\mathbb{R}^2) \to C_{k\ell}(\mathbb{R}^2)$, and more generally one has an operation $C_k(M)\times C_\ell(\mathbb{R}^n) \to C_{k\ell}(M)$ for any parallelized manifold $M^n$. More generally again, if one has chosen $a$ linearly independent vector fields on $M$, there is an operation $C_k(M)\times C_\ell(\mathbb{R}^a) \to C_{k\ell}(M)$. In particular, if $M$ admits at least one non-vanishing vector field (either $M$ is non-compact or $\chi(M)=0$) there is a map $C_k(M) \to C_{k\ell}(M)$. This can be thought of as flowing each point of the configuration along the vector field and taking its image at $\ell$ different small time values to obtain the new configuration of $k\ell$ points.

This map has been important in a joint project http://arxiv.org/abs/1406.4916 which I have worked on recently, where we call this the "$\ell$-replication map".

We would be interested in knowing about anywhere in the literature where this has appeared before, as it seems a very natural operation to consider. We would be especially interested in the case of closed manifolds $M$.

One reference which we have come across is the paper http://arxiv.org/abs/math/0701189, which is concerned with this replication map in the case $M=\mathbb{R}^2$.

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Simpler maps $C_k(M)\to C_{k+1}(M)$ which "double a strand" appear in the work of Berrick, Cohen, Wong and Wu, where the induced maps on fundamental groups are shown to induce the degeneracies in a simplicial group structure on the braid groups of $M$. See Section 3 of http://www.ams.org/journals/jams/2006-19-02/S0894-0347-05-00507-2/

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    $\begingroup$ Thanks for the reference! I think the maps they construct only exist at the level of fundamental groups though, and are not induced by maps of configuration spaces. You can't single out one point in an unordered configuration space to double, but you can single out one strand to double when you go to $\pi_1$, since you can fix an ordering of the basepoint configuration. It's still interesting to know about this though, since it uses similar ideas in its construction. $\endgroup$ – Martin Palmer Jul 11 '14 at 13:30

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