Periodic Reporting for period 2 - FIBRING (Fibring of manifolds and groups)
Período documentado: 2022-01-01 hasta 2023-06-30
Groups are algebraic objects that measure symmetry, similarly to the way numbers measure quantity. It turns out that a group can also fibre, which again allows us to view it as a simpler object evolving in time.
The overall objectives are to understand all the ways a given manifold or a group can fibre. Understanding this question will bring us closer to understanding all manifolds, including our own 4-dimensional space time. On the group level, it is important to have tools to comprehend a group at the ready, since all aspects of science that exhibit or utilise symmetries use group theory in a fundamental way.
Studying properties that are determined by knowing only the extra bit of symmetry, but for all possible finite covers, is known as looking for "profinite invariants". We have shown that in a large number of situations, being fibred is a profinite invariant. We have also shown that a variety of groups and manifolds have finite covers that fibre, and in the groups case we obtained a characterisation of many different flavours of fibring using numbers associated to a group, called Betti numbers. We have also looked at how these Betti numbers behave when one decomposes a manifold in a way similar to a fibring, but in which the time parameter is allowed to be more complicated than a simple line.
In the remaining 2.5 years of the project, we intend to focus on the dynamical aspect of fibring: the way the smaller space actually changes. One line of enquiry is studying the properties of various invariants associated to the dynamical system arising this way. We intend to focus on a analytic invariant known as a zeta function. The other aspect we want to understand is: which properties of the dynamical system stay the same for all the various ways our initial object fibred?