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The University of Southampton
Mathematical Sciences

Research project: Massey products in toric topology

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This work is about understanding properties of Massey products in the cohomology of moment-angle complexes from a combinatorial perspective. Moment-angle complexes, one of the main objects in toric topology, have a natural underlying combinatorial structure that allows us to study these topological spaces by understanding their corresponding simplicial complex. There are many applications of Massey products in topology; they determine differentials in spectral sequences, are obstructions to formality of spaces, etc. As higher cohomology operations, Massey products are extremely difficult to compute. Fortunately, the combinatorial structure of moment-angle complexes (or more generally, polyhedral products) lends nicely to the study of Massey products in the cohomology of such spaces.

One goal of this research is to find, and better understand, non-formal manifolds (manifolds with a non-trivial Massey product in its cohomology). To date, there are many known examples of formal manifolds but so far not many of non-formal manifolds. In special cases, moment-angle complexes are manifolds, and so one avenue is to create non-trivial Massey products in moment-angle complexes to build new non-formal manifolds. For example, currently this work involves investigating combinatorial operations, such as joins or stellar subdivision, on the underlying simplicial complexes to create new non-trivial Massey products, and thus new non-formal manifolds.

Related research groups

Pure Mathematics
Topology
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