A GEOMETRIC REPRESENTATIVE FOR THE FUNDAMENTAL CLASS IN KK-DUALITY OF SMALE SPACES

The paper proves—constructively and with full technical control—that for every irreducible Smale space there is a θ-summable Fredholm module representing the KK-duality fundamental class, giving an explicit ρ and an isometry V built from dynamical partitions of unity; see the extension’s Busby invariant (equation (17)) and the geometric compression identity (equation (19)), culminating in Theorem 6.2 that establishes θ-summability on a dense Lipschitz subalgebra . By contrast, the model’s argument crucially assumes one can choose a completely positive lift and its Stinespring dilation so that the commutator control needed for Li_{1/2}-summability holds; this is precisely the delicate geometric step the paper supplies but the model does not justify. The model therefore relies on unproven claims about selecting a cp-lift/dilation with the required summability properties and does not provide the concrete construction the paper uses via δ-enlarged Markov partitions and averaging isometries .