Robin Stoll: Relative self-equivalences and graph complexes

Abstract.

This thesis consists of three papers.

In Paper I, we identify the cohomology of the stable classifying space of homotopy automorphisms (relative to an embedded disk) of connected sums of \(S^k \times S^l\), where \(3 \le k < l \le 2k - 2\). We express the result in terms of Lie graph complex homology.

In Paper II, we construct a rational model for the classifying space \(\mathrm{Baut}_A(X)\) of homotopy automorphisms of a simply connected finite CW-complex \(X\) relative to a simply connected subcomplex \(A\). Using this model, we provide a purely algebraic description of the cohomology of this classifying space. This constitutes an important input for the results of Paper I.

In Paper III, we show that modular operads are equivalent to modules over a certain simple properad which we call the Brauer properad. Furthermore we show that the Feynman transform corresponds to the cobar construction for modules of this kind. To make this precise, we extend the machinery of the bar and cobar constructions relative to a twisting morphism to modules over a general properad. As an application, we provide the foundations of a Koszul duality theory for modular operads.