NovikovShubin

Novikov-Shubin invariants are a family of spectral invariants arising in L2-cohomology and index theory, named after Sergei Novikov and Vladimir Shubin. They provide refined information about the small-spectrum behavior of the Laplacian on the universal cover of a compact manifold or, more generally, a finite CW complex.

Setup and definition. Let X be a finite CW complex with fundamental group Γ and universal cover X̃. Denote by Δ_p the Laplacian on p-forms on X̃, which commutes with the Γ-action. Let N_p(λ) be the von Neumann spectral distribution function, defined as N_p(λ) = dim_Γ E_{Δ_p}(0, λ], where E_{Δ_p}(0, λ] is the spectral projection onto the interval (0, λ]. The L2-Betti number b_p is N_p(0) = dim_Γ ker Δ_p. The p-th Novikov-Shubin invariant α_p is defined by the small-λ asymptotics of N_p(λ) − b_p: for small λ>0, N_p(λ) − b_p ≈ C_p λ^{α_p/2}. Equivalently, one may describe α_p via the large-time behavior of the Γ-traced heat kernel, Tr_Γ e^{−tΔ_p} − b_p ∼ C′_p t^{−α_p/2} as t → ∞. In general, the limit may not exist and α_p is defined using limsup or liminf.

Properties and context. The invariants take values in (0, ∞] (with α_p = ∞ indicating a spectral gap above zero). They depend on the geometry of the covering and the chosen metric, and are not purely topological. They provide information beyond the L2-Betti numbers and are used in spectral geometry and related areas to distinguish geometric features of coverings.

See also. L2-Betti numbers, von Neumann dimension, L2-cohomology, spectral geometry, Novikov invariant.