Abstract
The main aim of this thesis is to demonstrate new topological phenomena amongst manifolds with holonomy $\mathsf{G}_2$.This splits into two complementary aims: develop invariants that detect the topological phenomena,
and compute these invariants on a pool of $\mathsf{G}_2$-manifolds.
To address the first aim,
we discuss a general framework in which one can define invariants of structured manifolds via coboundaries.
We consider how previously defined invariants are constructed from this perspective.
The framework provides a transparent manner in which to generalise known invariants and define new ones.
We extend invariants of polarized spin $7$-manifolds,
and define new invariants of almost contact $7$-manifolds.
To address the second aim, we consider the Twisted Connected Sum construction for $\mathsf{G}_2$-manifolds.
We construct a suitably structured coboundary on which to compute invariants.
Using this we: present examples of smooth $7$-manifolds with disconnected $\mathsf{G}_2$-moduli space;
compute aforementioned invariants of polarized spin manifolds on several hundred examples;
and detect formality in the sense of rational homotopy theory.
To date we find only formal examples.
The TCS construction takes as input pairs of certain complex threefolds called building blocks,
together with some cohomological data called a `configuration'.
Most examples and mass production methods in the literature have used simple types of configurations.
Using simple types of configurations restricts the possible topology of the manifolds obtained.
To demonstrate that the invariants defined can be nontrivial it is necessary we consider more sophisticated configurations.
Although the theory for these configurations is available in the literature, it has not been developed.
For more sophisticated types of configurations one needs additional `genericity conditions' on the building block.
In general, this requires a greater understanding of their complex geometry.
Building blocks can often be derived from weak Fano threefolds.
We outline a systematic approach to producing genericity conditions for certain building blocks derived from weak Fanos.
| Date of Award | 1 Apr 2020 |
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| Original language | English |
| Awarding Institution |
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| Supervisor | Johannes Nordstrom (Supervisor) & Fran Burstall (Supervisor) |