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- W3040339163 abstract "We consider orbit configuration spaces associated to finite groups acting freely by orientation preserving homeomorphisms on the $2$-sphere minus a finite number of points. Such action is equivalent to a homography action of a finite subgroup $Gsubset mathrm{PGL}(mathbb{C}^2)$ on the complex projective line $mathbb{P}^1$ minus a finite set $Z$ stable under $G$. We compute the cohomology ring and the Poincare series of the orbit configuration space $C_n^G(mathbb{P}^1 setminus Z)$. This can be seen as a generalization of the work of Arnold for the classical configuration space $C_n(mathbb{C})$ ($(G,Z)=({1},infty$)). It follows from the work that $C_n^G(mathbb{P}^1setminus Z)$ is formal in the sense of rational homotopy theory. We also prove the existence of an LCS formula relating the Poincare series of $C_n^G(mathbb{P}^1setminus Z)$ to the ranks of quotients of successive terms of the lower central series of the fundamental group of $C_n^G(mathbb{P}^1 setminus Z)$. The successive quotients correspond to homogenous elements of graded Lie algebras introduced by the author in an earlier work. Such formula is also known for classical configuration spaces of $mathbb{C}$, where fundamental groups are Artin braid groups and the ranks correspond to dimensions of homogenous elements of the Kohno-Drinfeld Lie algebras." @default.
- W3040339163 created "2020-07-10" @default.
- W3040339163 creator A5066725652 @default.
- W3040339163 date "2020-07-02" @default.
- W3040339163 modified "2023-09-27" @default.
- W3040339163 title "Algebraic invariants of orbit configuration spaces in genus zero associated to finite groups" @default.
- W3040339163 hasPublicationYear "2020" @default.
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