[Paper Review] On Atiyah-Singer and Atiyah-Bott for finite abstract simplicial complexes

Oliver Knill|arXiv (Cornell University)|Aug 21, 2017

Topological and Geometric Data Analysis15 references3 citations

TL;DR

This paper establishes a discrete analogue of the Atiyah-Singer and Atiyah-Bott index theorems for finite abstract simplicial complexes by defining elliptic differential complexes via spectral symmetry (McKean-Singer property). It shows that the analytic index, defined as the supertrace of the heat kernel of the Hodge Laplacian, equals a topological index expressed as a curvature sum over vertices, generalizing Euler and Wu characteristics. The key contribution is a discrete fixed-point theorem via Brouwer-type indices, proven through heat flow and gradient deformation.

ABSTRACT

A linear or multi-linear valuation on a finite abstract simplicial complex can be expressed as an analytic index dim(ker(D)) -dim(ker(D^*)) of a differential complex D:E -> F. In the discrete, a complex D can be called elliptic if a McKean-Singer spectral symmetry applies as this implies str(exp(-t D^2)) is t-independent. In that case, the analytic index of D is the sum of (-1)^k b_k(D), where b_k(D) is the k'th Betti number, which by Hodge is the nullity of the (k+1)'th block of the Hodge operator L=D^2. It can also be written as a topological index summing K(v) over the set of zero-dimensional simplices in G and where K is an Euler type curvature defined by G and D. This can be interpreted as a Atiyah-Singer type correspondence between analytic and topological index. Examples are the de Rham differential complex for the Euler characteristic X(G) or the connection differential complex for Wu characteristic w_k(G). Given an endomorphism T of an elliptic complex, the Lefschetz number X(T,G,D) is defined as the super trace of T acting on cohomology defined by E. It is equal to the sum i(v) over V which are contained in fixed simplices of T, and i is a Brouwer type index. This Atiyah-Bott result generalizes the Brouwer-Lefschetz fixed point theorem for an endomorphism of the simplicial complex G. In both the static and dynamic setting, the proof is done by heat deforming the Koopman operator U(T) to get the cohomological picture str(exp(-t D^2) U(T)) in the limit t to infinity and then use Hodge, and then by applying a discrete gradient flow to the simplex data defining the valuation to push str(U(T)) to V, getting curvature K(v) or the Brouwer type index i(v).

Motivation & Objective

• To extend the Atiyah-Singer and Atiyah-Bott index theorems to finite abstract simplicial complexes using discrete differential geometry.
• To define elliptic complexes in the discrete setting via the McKean-Singer spectral symmetry condition.
• To show that the analytic index (supertrace of the heat kernel) equals a topological index (sum of curvature terms) on the 0-skeleton.
• To generalize the Lefschetz fixed-point theorem to discrete complexes using Brouwer-type indices.
• To provide a discrete, combinatorial framework that mirrors the continuum index theorems without assuming manifold structure.

Proposed method

• Define a differential complex $ D: E \to F $ on a finite simplicial complex $ G $, with $ D^2 $ being the Hodge Laplacian $ L $.
• Declare $ D $ elliptic if the supertrace $ \mathrm{str}(e^{-tL}) $ is independent of $ t $, implying spectral symmetry.
• Express the analytic index as $ \chi(G,D) = \sum_k (-1)^k b_k(D) $, where $ b_k(D) $ is the $ k $-th Betti number via Hodge theory.
• Represent the topological index as $ \sum_{v \in V} K(v) $, where $ K(v) $ is a curvature derived from the complex and $ D $.
• Use heat deformation to connect the Koopman operator $ U(T) $ to the cohomological supertrace $ \mathrm{str}(e^{-tL}U(T)) $, then apply Hodge theory.
• Apply a discrete gradient flow to deform the Lefschetz number $ \mathrm{str}(U(T)) $ to the 0-dimensional skeleton, yielding Brouwer-type indices $ i(v) $.

Experimental results

Research questions

• RQ1 Can the Atiyah-Singer index theorem be formulated in a purely discrete setting for finite simplicial complexes?
• RQ2 Does the McKean-Singer spectral symmetry condition serve as a valid discrete analogue of ellipticity?
• RQ3 Can the analytic index of a discrete differential complex be equated to a topological index via curvature summation?
• RQ4 How can the Atiyah-Bott fixed-point theorem be generalized to discrete simplicial complexes using discrete curvature and indices?
• RQ5 What is the role of heat flow and gradient deformation in connecting dynamical and cohomological invariants in the discrete?

Key findings

• The analytic index $ \chi(G,D) = \sum_k (-1)^k b_k(D) $ is equal to the topological index $ \sum_{v \in V} K(v) $, where $ K(v) $ is a curvature defined by the complex and the differential operator $ D $.
• For the de Rham complex, the topological index reduces to the Euler characteristic $ \chi(G) $, and for the connection complex, it yields the Wu characteristic $ \omega_k(G) $.
• The Lefschetz number $ \chi(T,G,D) $ of an endomorphism $ T $ equals the sum $ \sum_{v \in V} i(v) $, where $ i(v) $ is a Brouwer-type index associated to fixed simplices of $ T $.
• The proof relies on heat deformation of the Koopman operator $ U(T) $, followed by Hodge theory and discrete gradient flow to reduce the supertrace to the 0-skeleton.
• The curvature $ K(v) $ and index $ i(v) $ emerge naturally from the deformation process, providing a discrete analogue of the Atiyah-Bott fixed-point formula.
• The framework applies to general finite abstract simplicial complexes without requiring manifold or geometric structure, generalizing classical results combinatorially.