HAIRED AND MULTI-ORIENTED GRAPH COMPLEXES WITH APPLICATIONS TO ALGEBRA AND GEOMETRY

In this thesis we study graph complexes and their applications to homotopy algebra, differential geometry, and the cohomology of the moduli space of algebraic curves. The first main topic of this thesis is the study of a new higher dimensional incarnation of one of the most mysterious mathematical structures, the Grothendieck-Teichm\"uller group, using methods and ideas of multi-oriented props and graph complexes. In Chapter \ref{chap:props}, we fix our notation and conventions. We recall the definitions of operads and props following J.-L. Loday and B.\ Vallette \cite{c:LoVa}, \cite{c:Vaprops}. and D.V. Borisov and Y.I. Manin \cite{c:GenGraphs}. We also recall the basic tools of the deformation theory of props developed by B. Vallette and S. Merkulov in \cite{c:Defofprop}, and generalize them, rather straightforwardly, to the multi-oriented setting. In Chapter \ref{chap:Multior}, we study a family of multi-oriented properads of multi-oriented homotopy Lie bialgebras $\hoLieB^{k\uparrow l}_d$ parametrized by the valued parameter $d$ and equipped with $k$ directions, $l$ of them being oriented. Multi-oriented properads were introduced by S. Merkulov in \cite{c:MultiProps}, where it is shown that they admit representations in vector spaces with branes, and provide us with a far reaching generalization of Drinfeld's notion of a {\em Manin triple} \cite{c:Drinfeld2}. The multi-oriented properads $\hoLieB^{k\uparrow l}_d$ of homotopy Lie bialgebras, also introduced in \cite{c:MultiProps}, are from a combinatorial point of view a natural extension of the ordinary properad of Lie bialgebras. Their representations on vector spaces with branes are not yet fully understood. However, it is shown in \cite{c:MultiProps} that representations of a two oriented operad $\Lie^{2\uparrow}$, obtained from a similar 'combinatorial lifting' of the Lie operad $\Lie$, do really govern Manin triples as defined in \cite{c:Drinfeld2}. In Section \ref{sec:Def}, we prove that the derivation complex of the $k$-oriented prop of $l$-oriented homotopy Lie bialgebras is quasi-isomorphic to the $k$-directed, $l$-oriented graph complex. This is a generalization of the result in \cite{c:DTLBP}, where the deformation complex of the ($1$-oriented) prop of homotopy Lie bialgebras is studied. Combining this with the results of T. Willwacher \cite{c:GCandGRT} and M. Živković \cite{c:Multior}, we get that the Grothendieck-Teichm\"uller group acts on the multi-oriented properads of Lie bialgebras. In Section \ref{sec:Minmod}, we turn our attention to the (co)homology of the first non-trivial multi-oriented properad, the 2-oriented properad $\hoLieB^{2\uparrow 2}_d$. We prove that $\hoLieB^{2\uparrow 2}_d$ is indeed a minimal model of the $2$-oriented properad of $2$-oriented Lie bialgebras $\LieB^{2\uparrow 2}_d$. It is worth emphasizing that this proof does {\em not} follow the scenario of the famous constructions of the minimal resolutions of the ordinary {\em 1-oriented}, props of Lie bialgebras \cite{c:KontLetter}, \cite{c:Vaprops}, \cite{c:propped_up}, \cite{c:MePois}. The main problem is that the key idea of using spectral sequences associated with path filtrations does not completely work in the multi-oriented case. \smallskip The second main topic of this thesis is the study of hairy graph complexes and their applications to the theory of cohomology groups of moduli spaces of genus $g$ algebraic curves with $n\geq 1$ punctures. In Chapter \ref{chap:HairySource}, we show that the hairy graph complex $(\HGC_{d,d},\delta)$, studied in e.g. \cite{c:AroneTurchin}, \cite{c:DGC2}, can be understood as an associated graded complex of the oriented graph complex $(\OGC_{d+1}, \delta)$, subject to a filtration on the number of target vertices, or equivalently source vertices, called the \emph{fixed source graph complex}. The fixed source graph complex $(\OGC_1,\delta_0)$ maps into the ribbon graph complex $\RGC$ \cite{c:MWRibbon}, which models the moduli space of Riemann surfaces with marked points \cite{c:KontModSpace}. The full differential $\delta$ on the oriented graph complex $\OGC_{d+1}$ corresponds to the deformed differential $\delta+ \chi$ on the hairy graph complex $\HGC_{d,d}$, where $\chi$ adds a hair. This deformed complex $(\HGC_{d,d},\delta+\chi)$ is already known to be quasi-isomorphic to the standard Kontsevich's graph complex $\GC_d$ \cite{c:DGC2}. This chapter is based on joint work with M. Živković \cite{c:HairySource}. \smallskip The third main topic of this thesis is a new application of the remarkable theory of differential forms with logarithmic singularities developed in \cite{c:Loganddef} for constructing a new universal transcendent formula for an exotic Lie $\infty$-automorphism of the Schouten-Nijenhuis Lie algebra of polyvector fields. In Chapter \ref{chap:Exotic}, we develop a new (regularized) De Rham field theory based on a two parameter propagator with logarithmic singularities. We use this for constructing a new two parametric family of exotic Lie $\infty$-automorphisms of Schouten-Nijenhuis Lie algebra of polyvector fields on an arbitrary affine space $$\cF^{t,\lambda}: T_{poly}(\R^d) \rightsquigarrow T_{poly}(\R^d).$$ This universal formula involves all odd Riemann zeta values $\frac{1}{2\pi } \zeta(2n+1)$, $n\geq 1$. This is a new application of the regularized Stokes formula, introduced by A. Alekseev, C. A. Rossi, C. Torossian and T. Willwacher in \cite{c:Loganddef}, in order to prove a statement by M. Kontsevich \cite{c:KontMot} regarding the existence of a formality morphism $$\cU^{\log}: T_{poly}(\R^d) \rightsquigarrow D_{poly}(\R^d)$$ with weights obtained by integrating logarithmic forms.