Date of Degree

9-2015

Document Type

Dissertation

Degree Name

Ph.D.

Program

Mathematics

Advisor(s)

Martin Bendersky

Keywords

algebraic topology; geometric homology; massey product; string topology

Abstract

Given a manifold M, Massey triple products exist in ordinary homology h*(M) for triples of classes a, b, c when the pairwise intersection products ab and bc vanish. This means that the triple product abc vanishes for two different reasons and this is captured by the Massey product. In this thesis we make a similar construction in string topology where we use the Chas-Sullivan product instead of the intersection product; we call it the Chas-Sullivan-Massey product. Our constructions use the Geometric Homology Theory description of String Topology and this allows us to obtain a very geometric picture of both the ordinary Massey Product and the Chas-Sullivan-Massey product. We then turn our attention to the Loop Homology Spectral Sequence and prove that it can be used to calculate Chas-Sullivan-Massey products. We find examples of non-zero products for even dimensional spheres and for complex projective spaces. Finally, we indicate how the defintion can be extended to free sphere spaces, to generalised homology theories and to higher Massey-type products. We prove that the Loop Homology Spectral sequence can be used to calculate these products even in the most general case.

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