The braid groups of the projective plane

Let B n (RP 2 ) (respectively P n (RP 2 )) 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 pure braid group short exact sequence 1 → P m - n (RP 2 \ {x 1 ,...,x n }) → P m (RP 2 ) → P n (RP 2 ) → 1 does not split if m > 4 and n = 2, 3. Now let n > 2. Then in B n (RP 2 ), there is a k-torsion element if and only if k divides either 4n or 4(n - 1). Finally, the full twist braid has a k t h root if and only if k divides either 2n or 2(n-1).

[1]  D. Gonçalves,et al.  The roots of the full twist for surface braid groups , 2004, Mathematical Proceedings of the Cambridge Philosophical Society.

[2]  Vladimir Lin.  Braids and Permutations , 2004, math/0404528.

[3]  D. Gonçalves,et al.  On the structure of surface pure braid groups , 2004 .

[4]  Frederick R. Cohen,et al.  On loop spaces of configuration spaces , 2002 .

[5]  Jeffrey Wang.  On the braid groups for RP2 , 2002 .

[6]  J. M. Boardman,et al.  FIBREWISE HOMOTOPY THEORY (Springer Monographs in Mathematics) , 1999 .

[7]  Hans J. Baues,et al.  Obstruction Theory: On Homotopy Classification of Maps , 1977 .

[8]  G. P. Scott.  Braid groups and the group of homeomorphisms of a surface , 1970, Mathematical Proceedings of the Cambridge Philosophical Society.

[9]  J. V. Buskirk,et al.  The braid groups of $E^2$ and $S^2$ , 1962 .

[10]  E. Fadell.  Homotopy groups of configuration spaces and the string problem of Dirac , 1962 .

[11]  J. Whitehead,et al.  Combinatorial homotopy. II , 1949 .

[12]  Emil Artin,et al.  Theorie der Zöpfe , 1925 .

[13]  B. Dundas,et al.  DIFFERENTIAL TOPOLOGY , 2002 .

[14]  D. Rolfsen,et al.  Geometric subgroups of surface braid groups , 1999 .

[15]  Ioan Mackenzie James,et al.  Fibrewise Homotopy Theory , 1998 .

[16]  Albert Schwarz,et al.  Elements of Homotopy Theory , 1993 .

[17]  Kunio Murasugi,et al.  Seifert Fibre Spaces and Braid Groups , 1982 .

[18]  K. Brown,et al.  Graduate Texts in Mathematics , 1982 .

[19]  J. V. Buskirk.  Braid groups of compact 2-manifolds with elements of finite order , 1966 .

[20]  E. Artin.  The theory of braids. , 1950, American scientist.

[21]  Robert Lipshitz,et al.  Algebraic & Geometric Topology , 2023 .