On Atiyah-Singer and Atiyah-Bott for finite abstract simplicial complexes (1708.06070v1)

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).

• The paper presents a discrete adaptation of the Atiyah-Singer and Atiyah-Bott theorems that unifies analytic, cohomological, and topological indices in finite abstract simplicial complexes.
• It leverages discrete differential forms and calculus to extend classical results such as Stokes' theorem and the fundamental theorem of calculus into combinatorial settings.
• The framework offers new computational methods for evaluating topological invariants, potentially advancing research in computational topology and graph theory.

An Analysis of "On Atiyah-Singer and Atiyah-Bott for Finite Abstract Simplicial Complexes"

This paper presents a discrete perspective on the classical theorems of Atiyah-Singer and Atiyah-Bott, adapting these concepts for finite abstract simplicial complexes. The author, Oliver Knill, situates this exploration within the framework of discrete differential geometry, leveraging concepts from graph theory and combinatorial topology.

Simplicial Complexes and Valuations

The work begins with a formal definition of finite abstract simplicial complexes and introduces the concept of valuations. A simplicial complex is a collection of simplices that adheres to certain closure properties, thus allowing for geometrical and topological properties to be analyzed discretely. Knill emphasizes valuations, which are integer-valued functions on the Boolean lattice of sub-complexes, showcasing their role in linking combinatorial properties with topological invariants such as the Euler characteristic.

The discrete Hadwiger theorem is referenced as providing the theoretical foundation for conceptualizing a linear space of valuations in dimensions equal to the maximal dimension of the complex plus one. Examples include the Euler characteristic and more complex bilinear valuations like the Wu intersection number. The exposition on valuations highlights their utility in expressing topological invariants combinatorially, thus casting traditional continuous properties into the discrete field.

Discrete Calculus Framework

Knill further extends the classical notions of calculus into the discrete setting. The paper delineates the construction of discrete differential forms and the integration theory over simplicial complexes, paralleling geometric measure theory and geometric probability theory. These constructs enable the formulation of discrete analogs to classical theorems, such as Stokes' theorem and the fundamental theorem of calculus, within the context of simplicial complexes.

The approach to defining the discrete exterior derivative introduces a complex D that encapsulates both boundary and coboundary operations. The integration of signed valuations and their differentiation forms a backbone for discrete differential geometry, offering insights into the combinatorial nature of traditional geometric concepts.

Elliptic Complexes and Index Theory

Central to this paper is the discussion on elliptic complexes within the context of discrete settings. Knill introduces discrete differential complexes characterized by sequences of linear maps that satisfy specific kernel dimension properties. This structure enables the exploration of analytic, cohomological, and topological indices, which serve as discrete counterparts to their continuous analogs. The paper argues for the applicability of these discrete constructions, asserting their capacity to retain essential properties of traditional continuum-based differential operators.

The author adapts a McKean-Singer spectral symmetry concept to justify the definition of elliptic complexes in the discrete setting. By establishing symmetry in the non-zero spectrum of the discrete Laplacians, the paper lays the groundwork for demonstrating index theorems analogous to those in continuous elliptic complexes.

Discrete Atiyah-Singer and Atiyah-Bott Theorems

At the theoretical apex, this paper articulates discrete versions of the Atiyah-Singer and Atiyah-Bott theorems. These results link the analytic index, cohomological index, and topological index, demonstrating equality through the use of heat kernel deformations and combinatorial transpositions. The discrete Atiyah-Singer theorem is validated by equating analytic and cohomological indices, whereas the Atiyah-Bott theorem extends this to account for automorphism-induced fixed points within the complexes. These results exemplify the power of discrete methods in capturing the complexity of topological transformations and invariants.

Implications and Future Directions

The implications of Knill's work are profound, both practically and theoretically. In practice, these results offer new methods for computing and analyzing topological and geometric properties in networks and discrete spaces, providing potential applications in computational topology and graph theory. Theoretically, the work stimulates further research into the intersection of discrete mathematics and classical differential geometry, suggesting possible extensions into other areas, such as quantum geometry or algebraic topology.

In conclusion, while the paper deepens our understanding of discrete geometry's potential to reflect and extend continuous concepts, it also opens new frontiers for exploration, particularly in how these discrete analogs might inform or transform our approach to complexity in mathematical and physical systems.