PhD Defense Zhipeng Duan

Title: On Equivariant Euler Characteristics and Spaces of Trees

This thesis contains two parts. The first part studies general properties of equivariant Euler characteristics and several concrete calculations. I first review the basic notions of equivariant Euler characteristics introduced by Atiyah and Segal[AS89] and some useful properties and explanations of it. Then I calculate the equivariant Euler characteristics of Grothendieck constructions of G-functors and apply it to study the dfferences between the equivariant Euler characteristic of the centralizer CSp+G(λ) and the subposet Sp+CG(λ). Moreover, I generalize the Tamanoi's result regarding the equivariant Euler characteristicof product of manifolds to poset cases. Lastly I determine the equivariant Euler characteristics of all subgroup complexes of symmetric groups in many cases.

The second part is a joint work with Greg Arone. We study the equivariant homotopy equivalence between a kind of space of trees and double suspension of the complex of not 2-connected graphs. This project was motivated by an easy observation that the homology of these two spaces as n-modules are same up to a sign representation. The way we prove it is by constructing a third space via a special homotopy colimits as a bridge. We show that these two spaces are both n-equivarianthomotopy equivalent to the third space.

Zhipeng Duan's Thesis

You are invited to an online defense

Zoom Link: https://ucph-ku.zoom.us/j/66709486659 

Pwd: 942541 

Principal supervisor:

Professor Jesper Michael Møller, University of Copenhagen.

Assessment committee:

Professor (chairman), Jesper Grodal, University of Copenhagen

Professor, Jerome Scherer, EPFL, Lausanne

Prof. Wojciech Chacholski, KTH, Stockholm