Braid groups of the projective plane, mapping class groups of non-orientable surfaces, and algebraic K-theory of their group rings

Abstract We describe the lower algebraic K -theory of the integral group ring of both the pure and full braid groups of the real projective plane $$\mathbb {R}P^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>R</mml:mi> <mml:msup> <mml:mi>P</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> with 3 strings, as well as that of the integral group ring of the mapping class group of $$\mathbb {R}P^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>R</mml:mi> <mml:msup> <mml:mi>P</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> with 3 marked points. In addition, we give a general formula for the algebraic K -theory groups of the group ring of the mapping class group of non-orientable surfaces with k marked points, where $$k\ge 3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> .