Combinatorial Fixed-Point Theorems

• Combinatorial fixed-point theorems are defined by analyzing self-maps on discrete models, relaxing traditional f-invariance and openness requirements.
• The framework extends classical Lefschetz invariants by introducing a localized index, i_c, computed via interior localization and additive axioms.
• Robust homotopy invariance and commutativity properties support an integration theory over fixed-point measures, enabling applications in topology and applied fields.

A combinatorial fixed-point theorem provides a method to count or certify the existence of fixed points of a self-map by directly analyzing its action on a combinatorial model of a space, often through labeling principles, chain-level arguments, or invariance formulas. These theorems bridge discrete, topological, and algebraic frameworks, extending beyond the classical Lefschetz number to novel index-theoretic invariants, and underlie a powerful integration theory with minimal structural restrictions.

1. The Combinatorial Fixed Point Index: Definition and Motivation

Traditional fixed-point theory focuses on global invariants such as the Lefschetz number,

$$\Lambda(f) = \sum_{i=0}^n (-1)^i \operatorname{trace}(f_{*} : H_i(X) \to H_i(X)),$$

which counts fixed points algebraically via induced actions on homology. These invariants are robust under homotopy, but practical computation at the level of subspaces A ⊆ X has previously required restrictive conditions: openness, f-invariance, or definability (e.g., the subspace is invariant under f and well-behaved from a logical or geometric viewpoint).

The combinatorial fixed point index $i_c(X, f, A)$ is formulated by localizing the fixed point information on the interior $\mathring{A}$ and extending by additivity: $i_c(X, f, A) := i(X, f, \mathring{A}),$ where $i(X, f, U)$ is the classical fixed point index (as defined in, e.g., Brown's or Lefschetz's theory) on open sets. This approach allows one to ignore the boundary where fixed points might induce nonlocal artifacts and relaxes constraints on the type of subspace under consideration, making it applicable even when A is not f-invariant or open, as long as $f$ is fixed-point-free on $\partial A$.

2. Extension of the Combinatorial Lefschetz Number

This new combinatorial index generalizes the combinatorial Lefschetz number $\Lambda(A, f)_X$ in the following precise sense: If $f$ is a homeomorphism and $A$ is a definable f-invariant subspace, then

$i_c(X, f, A) = \Lambda(A, f)_X.$

Thus, $i_c$ preserves and extends the primary algebraic and topological invariance properties that characterize the classical Lefschetz framework, while dispensing with the need for openness, f-invariance, or definability ([Thm: igualdad indice numero combinatorio, (López et al., 30 May 2025)]).

3. Axiomatic Properties and Structural Invariance

The combinatorial fixed point index is uniquely determined by a set of axioms (CI1–CI5) that mirror the classical fixed point index. The principal properties are:

Axiom	Statement
(CI1) Localization	The index depends only on the germ of $f$ near $\mathring{A}$
(CI2) Homotopy Invariance	The index is invariant under homotopies that avoid creating fixed points on the boundary
(CI3) Additivity	$i_c(X, f, A) = \sum_j i_c(X, f, A_j)$ for disjoint $A = \bigsqcup_j A_j$ (with fixed point-free boundaries)
(CI4) Normalization	$i_c(X, f, X) = \Lambda(f)$
(CI5) Commutativity	The index is compatible under coordinate changes and products

These formal properties ensure that $i_c$ is topologically and homotopically well-behaved, robust under subdivision or passage to homeomorphic images, and consistent with classical invariants wherever all notions are defined.

4. Integration and the Fixed Point Index Measure

An essential application of the combinatorial fixed point index is to integration over a space with respect to fixed points. For a step function $h: X \to \mathbb{Z}$, supported on subspaces $U_j$ (with $f$ fixed-point-free on $\partial U_j$),

$$\int_X h \, d i_c(f) = \sum_j d_j \cdot i_c(X, f, U_j)$$

for $h = \sum_j d_j \cdot \mathbf{1}_{U_j}$. This definition lifts directly to measurable functions via Riemann-type sums: $$\int_X h \left\lfloor d i_c(f) \right\rfloor = \lim_{n\to\infty} \frac{1}{2^n} \int_X \lfloor 2^n h \rfloor\, d i_c(f).$$ Because additivity and invariance are retained, the resulting measure supports product formulas and Fubini-type statements, allowing integration of real-valued functions against the fixed point index.

5. Improvements Over Classical Approaches

The new index $i_c$ eliminates the need for restrictive assumptions that pervaded earlier work:

• f-invariance: No longer necessary; one may now use arbitrary subspaces A, as long as fixed points are absent on the boundary.
• Openness: Subspaces A need not be open, as interior localization suffices.
• Definability: Assumptions related to o-minimality or semi-algebraic structure are not required. These improvements allow application to a broad class of spaces and self-maps, including those encountered in applied topology, sensor networks, and o-minimal geometry.

6. Relationship to Topological and Combinatorial Invariance

The combinatorial fixed point index ensures topological invariance (unchanged under homeomorphisms) and homotopical invariance (unchanged under fixed point-preserving homotopies). Additivity and commutativity under coordinate change establish that one may compute in one complex and transfer results to another — a crucial property for both theoretical development and computational applications.

In particular, if $f$ and $g$ are homotopic (in the sense that the fixed points on boundaries remain absent throughout the homotopy), the index remains constant: $i_c(X, f, A) = i_c(X, g, A).$ Similarly, passing to homeomorphic copies of X preserves the index. This invariance is critical for deploying $i_c$ in algebraic-topological computations and for defining meaningful integrations.

7. Implications and Applications

The combinatorial fixed point index $i_c$ provides the foundation for a broad new integration theory over fixed point data, supporting operations that were previously impossible under the Lefschetz or classical combinatorial paradigms. The extension to real-valued integrands, the ability to work with arbitrary subspaces, and robust invariance properties make this framework suitable for a wide spectrum of applications, including those in topological data analysis, non-invariant combinatorial decompositions, and geometric measure theory.

By blending combinatorial, topological, and measure-theoretic ideas, the new fixed point index positions itself as the core tool for both computation and theoretical development in modern fixed point theory, as it relates to integration, homological invariants, and the paper of minimal and non-invariant substructures (López et al., 30 May 2025).