Homology of configuration spaces of surfaces modulo an odd prime

For a compact orientable surface <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="normal">Σ</m:mi> <m:mrow> <m:mi>g</m:mi> <m:mo>,</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:math> Σg,1 of genus 𝑔 with one boundary component and for an odd prime number 𝑝, we study the homology of the unordered configuration spaces <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>C</m:mi> <m:mo>∙</m:mo> </m:msub> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi mathvariant="normal">Σ</m:mi> <m:mrow> <m:mi>g</m:mi> <m:mo>,</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo rspace="0.278em" stretchy="false">)</m:mo> </m:mrow> <m:mo rspace="0em">:</m:mo> <m:mo lspace="0em" rspace="0.111em">=</m:mo> <m:msub> <m:mo>∐</m:mo> <m:mrow> <m:mi>n</m:mi> <m:mo>≥</m:mo> <m:mn>0</m:mn> </m:mrow> </m:msub> <m:msub> <m:mi>C</m:mi> <m:mi>n</m:mi> </m:msub> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi mathvariant="normal">Σ</m:mi> <m:mrow> <m:mi>g</m:mi> <m:mo>,</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> C•(Σg,1):=n≥ 0Cₙ(Σg,1) with coefficients in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="double-struck">F</m:mi> <m:mi>p</m:mi> </m:msub> </m:math> Fₚ . We describe <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>H</m:mi> <m:mo>∗</m:mo> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:msub> <m:mi>C</m:mi> <m:mo>∙</m:mo> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi mathvariant="normal">Σ</m:mi> <m:mrow> <m:mi>g</m:mi> <m:mo>,</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>;</m:mo> <m:msub> <m:mi mathvariant="double-struck">F</m:mi> <m:mi>p</m:mi> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> H*(C•(Σg,1);Fₚ) as a bigraded module over the Pontryagin ring <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>H</m:mi> <m:mo>∗</m:mo> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:msub> <m:mi>C</m:mi> <m:mo>∙</m:mo> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>D</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>;</m:mo> <m:msub> <m:mi mathvariant="double-struck">F</m:mi> <m:mi>p</m:mi> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> H*(C•(D);Fₚ) , where 𝐷 is a disc, and compute in particular the bigraded dimension over <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="double-struck">F</m:mi> <m:mi>p</m:mi> </m:msub> </m:math> Fₚ . We also consider the action of the mapping class group <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="normal">Γ</m:mi> <m:mrow> <m:mi>g</m:mi> <m:mo>,</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:math> Γg,1 and prove that the mod-𝑝 Johnson kernel <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:msub> <m:mi mathvariant="script">K</m:mi> <m:mrow> <m:mi>g</m:mi> <m:mo>,</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>p</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>⊆</m:mo> <m:msub> <m:mi mathvariant="normal">Γ</m:mi> <m:mrow> <m:mi>g</m:mi> <m:mo>,</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:mrow> </m:math> Kg,1(p)⊆Γg,1 is the kernel of the action on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>H</m:mi> <m:mo>∗</m:mo> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:msub> <m:mi>C</m:mi> <m:mo>∙</m:mo> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi mathvariant="normal">Σ</m:mi> <m:mrow> <m:mi>g</m:mi> <m:mo>,</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>;</m:mo> <m:msub> <m:mi mathvariant="double-struck">F</m:mi> <m:mi>p</m:mi> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> H*(C•(Σg,1;Fₚ)) .