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Title:A construction of topological field theories
Author(s):Aramyan, Nerses
Director of Research:Ando, Matthew
Doctoral Committee Chair(s):Rezk, Charles
Doctoral Committee Member(s):McCarthy, Randy; Schenck, Henry
Department / Program:Mathematics
Discipline:Mathematics
Degree Granting Institution:University of Illinois at Urbana-Champaign
Degree:Ph.D.
Genre:Dissertation
Subject(s):topological field theories, cobordism hypothesis, Deligne complex, Dold-Kan construction
Abstract:We give an explicit construction of extended topological field theories over a manifold taking values in the deloopings of U(1) from the data of differential forms on the manifold. More specifically, for a manifold M, using a version of the Dold-Kan construction, we create from the Deligne complex a Kan complex, |D_n|^+(M), and show that there is a natural map into (Fun)^⊗(Bord_n^or (M),B^nU(1)). The main theorem asserts that thismap becomes a weak equivalence after restricting to the framed bordisms. The construction is on the point- set level, so immediately can be refined to the setting where the Deligne complex is considered discretely, which is known as a model for the differential cohomology. As a part of the proof we show that the geometric realization of the bordism (∞,n)-category is the n-fold delooping of the infinite loopspace of the Madsen-Weiss spectrum MTO(n). The direct attempts to generalize the Galatius-Madsen-Tillmann-Weiss proof of n = 1 case fail due the globularity restriction on the bordism (∞,n)-category. To overcome this, we use the abstract transversality argument of Bökstedt and Madsen, Gromov's theory of microflexible sheaves and Rezk's argument on realization fibrations.
Issue Date:2016-06-21
Type:Thesis
URI:http://hdl.handle.net/2142/92901
Rights Information:Copyright 2016 Nerses Aramyan
Date Available in IDEALS:2016-11-10
Date Deposited:2016-08


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