Robin Stoll: Homotopy automorphisms, graph complexes, and modular operads

Abstract

This licentiate thesis consists of two 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 \(3 \leq k < l \leq 2k − 2\). We express the result in terms of Lie graph complex homology. In Paper II we show that modular operads are equivalent to modules over a certain simple properad which we call the Brauer properad. Furthermore we show that, in this setting, 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 sketch a Koszul duality theory for modular operads.

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