The dynamical moduli space of rational maps of degree d, defined as the space of Möbius conjugacy classes of degree d holomorphic self-maps of the Riemann sphere, is a ubiquitous object in complex and arithmetic dynamics. Using the techniques of Geometric Invariant Theory, Silverman constructs this orbit space as an affine variety of dimension 2d-2 which admits a model over the rationals. In the case of degree two, Milnor identifies this space with the affine plane. I will present the results of a joint work with Maxime Bergeron and Sam Nariman regarding the topology of these moduli spaces. We compute the fundamental group of the dynamical moduli space and show that the space is rationally acyclic while its cohomology groups with finite coefficients could be non-trivial. As an application, the ranks of certain rational homotopy groups of the parameter space of rational maps (within the unstable range) will be computed.

Reference:
M. Bergeron, K. Filom and S. Nariman, Topological aspects of the dynamical moduli space of rational maps, arXiv: 1908.10792 (2019).

[IPM Youth Seminars on Topology and Dynamics]
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