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

Key Points

• This work aims to explore the algebraic K-theory of braid groups and mapping class groups associated with the real projective plane and non-orientable surfaces.
• Analyzed algebraic K-theory of integral group rings for pure and full braid groups of real projective plane with 3 strings.
• Investigated the integral group ring of mapping class groups of the real projective plane with 3 marked points.
• Derived a general formula for the algebraic K-theory groups of mapping class groups of non-orientable surfaces with k marked points where k equals or exceeds 3.
• Identified the lower algebraic K-theory of the integral group rings for specific braid groups.
• Presented a general formula for calculating algebraic K-theory groups of mapping class groups in non-orientable surfaces.
• Demonstrated connections between braid groups, mapping class groups, and algebraic structures.