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<p>We describe the lower algebraic <i>K</i>-theory of the integral group ring of both the pure and full braid groups of the real projective plane <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb {R}P^2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">R</mi> <msup> <mi>P</mi> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> with 3 strings, as well as that of the integral group ring of the mapping class group of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathbb {R}P^2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">R</mi> <msup> <mi>P</mi> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> with 3 marked points. In addition, we give a general formula for the algebraic <i>K</i>-theory groups of the group ring of the mapping class group of non-orientable surfaces with <i>k</i> marked points, where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(k\ge 3\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>

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

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  • 发表日期：

    2026年3月23日

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  • John Guaschi,
  • Daniel Juan-Pineda

摘要

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 \(\mathbb {R}P^2\) R P 2 with 3 strings, as well as that of the integral group ring of the mapping class group of \(\mathbb {R}P^2\) 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 \(k\ge 3\) k ≥ 3 .