What question did this study set out to answer?

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.

March 25, 2026Open Access

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.

Abstract

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 RP² R P 2 with 3 strings, as well as that of the integral group ring of the mapping class group of RP² R P 2 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 3 k ≥ 3.

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