Operational Mathematics of Topological Operations: Extending the Iteration Count of Topological Operations and Their Inverses to the Complex Domain and Infinity

This paper systematically transplants the core methodology of Operational Mathematics—the extension of the repetition count of fundamental operations from natural numbers to integers, rational numbers, real numbers, complex numbers, and even infinity—onto a new class of binary operations: topological operations and their inverses.Topological operations include, but are not limited to, fundamental group multiplication, higher homotopy group addition, Whitehead products, Toda brackets, homology direct sums and tensor products, cohomology cup products, Steenrod squares, characteristic class operations (Chern classes, Pontryagin classes, Stiefel Whitney classes, Euler classes), spectral sequence differentials, K-theory operations (direct sum, tensor product, exterior powers, Bott periodicity), bordism operations,simplicial set operations, and homological algebra operations (Ext, Tor).A complete set of nine independent axioms is established, explicitly incorporating functoriality (interaction with continuous maps), compatibility with spectral sequences, and infinite-order extensions. Integer-order, fractional-order, real-order,complex-order, and infinite-order iterations are rigorously defined, and the existence and uniqueness of iterative roots at each level are proved by means of Schröder’s equation, Abel’s equation, Kneser-type constructions adapted to topological settings,matrix logarithm-exponential methods, and limit arguments for spectral sequences.The singularity structure of complex-order topological iterations is analyzed in depth. For most forward iterations (group multiplications in Lie groups, cup powers with nilpotent elements, spectral sequence differentials), the complex iteration is an entire function of the iteration count, possessing no finite singularities. For certain non-nilpotent Whitehead products, the iteration develops a finite radius of convergence, yielding a natural boundary. Moreover, we show that the spectral sequence page number can be analytically continued via polynomial interpolation, and that no natural barrier exists in the finite plane; instead, an essential singularity may appear at infinity. The local monodromy groups are classified. A fundamental structural discovery is rigorously proved: the topological operational hierarchy collapses completely for all levels n ≥ 2, leaving only the base operations at level n = 1 and the collapsed family at level n = 2. The form of the collapsed family—exponential, linear, polynomial, or mixed—is uniquely determined by the algebraic type of the underlying operation (Abelian group, commutative ring, nilpotent operator, Lie algebra action). Non-idempotent operations and weight-parameterized families are shown to preserve the collapse, while enriching the dynamics.Fractional calculus and the calculus of variations with topological kernels(Chern classes,Pontryagin classes)are shown to be special cases of the topological operational framework,thereby unifying discrete topological hyperoperations with continuous analysis. A categorical duality between the mathematics of numbers and the mathematics of topological operations is established, yielding a field isomorphism between the topological hyperfield and the complex numbers. The connection between topological iteration values and characteristic numbers (Euler characteristic,Chern numbers,Pontryagin numbers) is examined,showing that the iteration count precisely parametrizes continuous families of characteristic classes.