PhD Defense Fadi Mezher

Title: Embedding calculus and automorphisms of high dimensional manifolds

Abstract: 

This thesis examines manifolds of dimension at least five, their automorphism groups, and moduli spaces thereof, by means of embedding calculus. The latter is a homotopy theoretic approximation of manifolds, which, informally, remembers the homotopy type of all spaces of framed configurations in a manifold 𝑀, together with the natural point-splitting and forgetful maps between them.                                                                                                                                                                                                               

In Paper A, we study the groups 𝜋0Homeo(𝑀) and 𝜋0Diff(𝑀), for a given smooth manifold 𝑀 of sufficiently high dimension, and under some additional technical conditions. In particular, we show that 𝜋0Homeo(𝑀) is a residually finite group, which, combined with a theorem of Sullivan, implies it is an arithmetic group. In contrast, we isolate the obstruction for residual finiteness of 𝜋0Diff(𝑀), which came out in the form of exotic spheres via the extension by identity morphism 𝜋0Diff𝜕 (𝐷𝑑) 𝜋0Diff(𝑀), for some embedded disc 𝐷𝑑 ⊂ 𝑀.
   In joint work with Manuel Krannich and Alexander Kupers, we investigate in Paper B whether the equivalence class of the truncated Disc-presheaf associated to a manifold is an invariant that can distinguish exotic spheres. We give a complete answer in the cases 𝑑 . 1(mod 4) and 𝑘 < ∞, in terms of the Kervaire-Milnor exact sequence.

Ask for a copy of the thesis here: https://www.math.ku.dk/english/programmes/ph.d/phd_theses/

Supervisors: Professor Søren Galatius, and Professor, Nathalie Wahl, University of Copenhagen

Assessment committee:

Professor, Jesper Grodal (chair), University of Copenhagen
Professor, Fabian Hebestreit, Bielefeld University
Professor, Oscar Randal-Williams, Cambridge University