Extension of Torsors and Curved Maurer Cartan Equation

In this thesis we will study the extension problem of (nilpotent) G-torsors.In chapter 1, we will review the Maurer Cartan equation of a DGLA and go through someexamples of Maurer Cartan equations in deformation of different algebraic structures.In chapter 2, we will define L infinity algebras, which are generalizations of DGLAs. We will alsostate the homotopy transfer of structure theorem and the formal Kuranishi theorem, whichare used throughout the whole thesis.In chapter 3, we will briefly go through Getzler's result on the unique horn filling of Deligne-Getzler infinite-groupoids, which gives us a generalization (on L infinity algebras) of the Baker-Campbell-Hausdorff formula for a Lie algebra.In chapter 4, we will state Hinich's result on descent of Deligne groupoid and go throughFiorenza, Manetti, and Martinengo's result on the special case when we have a semicosimplicialLie algebra, i.e. the solutions to the group valued cocycle condition are exactly theMaurer Cartan solutions on the L infinity total complex and equivalence of cocycles are exactlyequivalence of Maurer Cartan solutions. We will then show an example of this result ondeformation of (nilpotent) G-torsors.In chapter 5, we will apply Fiorenza, Manetti, and Martinengo's result on the extensionproblem of (nilpotent) G-torsors and show that solutions to the curved cocycle conditionthat gives the G-torsors extensions are exactly the curved Maurer Cartan solutions of acurved L infinity algebra and equivalence of extensions are exactly equivalence of curved MaurerCartan solutions