Braid groups of non-orientable surfaces and the Fadell-Neuwirth short exact sequence
نویسنده
چکیده
Let M be a compact, connected non-orientable surface without boundary and of genus g ¥ 3. We investigate the pure braid groups PnpMq of M, and in particular the possible splitting of the Fadell-Neuwirth short exact sequence 1 ÝÑ PmpMz tx1, . . . , xnuq ãÝÑ Pn mpMq p ÝÑ PnpMq ÝÑ 1, where m, n ¥ 1, and p is the homomorphism which corresponds geometrically to forgetting the last m strings. This problem is equivalent to that of the existence of a section for the associated fibration p : Fn mpMq ÝÑ FnpMq of configuration spaces, defined by pppx1, . . . , xn, xn 1, . . . , xn mqq px1, . . . , xnq. We show that p and p admit a section if and only if n 1. Together with previous results, this completes the resolution of the splitting problem for surfaces pure braid groups.
منابع مشابه
Braid groups of surfaces and one application to a Borsuk Ulam type theorem
During initial lectures we present the full and pure Artin braid groups. We give presentations of these groups and study several of their properties. We compute their centers, de ne a special element called Garside and study its properties. For the pure braid groups, we show how to write them as iterated product of free groups. Then we move on to the study of the full and pure braid groups of s...
متن کاملThe braid groups of the projective plane and the Fadell-Neuwirth short exact sequence
We study the pure braid groups Pn(RP) of the real projective plane RP2, and in particular the possible splitting of the Fadell-Neuwirth short exact sequence 1 −→ Pm(RP \ {x1, . . . , xn}) −֒→ Pn+m(RP 2) p∗ −→ Pn(RP) −→ 1, where n ≥ 2 and m ≥ 1, and p∗ is the homomorphismwhich corresponds geometrically to forgetting the last m strings. This problem is equivalent to that of the existence of a sect...
متن کاملThe braid groups of the projective plane
Let Bn(RP ) (respectively Pn(RP )) denote the braid group (respectively pure braid group) on n strings of the real projective plane RP 2 . In this paper we study these braid groups, in particular the associated pure braid group short exact sequence of Fadell and Neuwirth, their torsion elements and the roots of the ‘full twist’ braid. Our main results may be summarised as follows: first, the pu...
متن کاملOrdering pure braid groups on closed surfaces
We prove that the pure braid groups on closed, orientable surfaces are bi-orderable, and that the pure braid groups on closed, non-orientable surfaces have generalized torsion, thus they are not bi-orderable.
متن کاملOrdering Pure Braid Groups on Compact, Connected Surfaces
The purpose of this paper is to answer the following question: Are pure braid groups on compact, connected surfaces bi-orderable? We will prove that the answer is positive for orientable surfaces, and negative for the non-orientable ones. In this section we give the basic definitions and classical results. We also explain what is known about orders on braid groups, and finally we state our resu...
متن کامل