The full braid group on $n$ strands $B_n$ admits a surjective homomorphism $p\colon\thinspace B_n\to \Sigma_n$ onto the symmetric group on $n$ letters, which takes a braid to the induced permutation of its ends. The kernel $P_n$ is well understood; it is the pure braid group on $n$ strands.

What about $p^{-1}(A_n)$, where $A_n$ is the alternating group on $n$ letters? Let me call this group $E_n$ for now, because I think it should be called the *even braid group*. However an internet search using this name (and others such as "orientation preserving braids", "positive braids" and so on) came up blank.

> Does this group $E_n$ have a name, and has it been studied anywhere in the literature?  

I would be particularly interested in computations of the cohomology rings of these groups.