The Pavesi-M\"obius-Artin Pinwheel: A Wild Geometric Object with Stratified Topology, Exotic Fundamental Group, and Quantum Invariants

We introduce the Pavesi-M\"obius-Artin pinwheel (GPMA), a compact, connected topological space obtained by attaching countably many M\"obius bands of rapidly decreasing size to a closed disk along a dense sequence of boundary arcs, and by identifying all their midpoints to a single wild cusp point. The resulting space is not a topological manifold, not a locally finite CW-complex, and exhibits a remarkable mixture of properties: local topological dimension alternates between $1$ and $2$, the fundamental group is an uncountable wild group with a double Hawaiian earring structure, while singular homology remains free abelian of countable rank. We compute its universal cover (a contractible $2$-dimensional tree of disks and strips), its orientable double cover (a pinwheel with cylindrical petals), and generalise the construction to higher dimensions, obtaining a $4$-dimensional wild space with trivial π₁ but wild π₂. Using the wild fundamental group, we build non-separable type II₁ von Neumann algebras and C^*-algebras that serve as quantum invariants. Finally, we define a deterministic geodesic flow on the wild core of the pinwheel, exhibiting finite-time dynamical traps and infinite topological entropy. The GPMA pinwheel thus provides a rich, new example connecting wild topology, non-commutative geometry, and chaotic dynamics.