The classical Euler-Poincar´e theorem Pni=0(−1)iβi = χ(M) holds in threedimensional space, but its higher-dimensional generalizations exhibit inconsistent expressions across different dimensions and fail to establish a complete correspondence with the expansion coefficients of the binomial theorem. This paper aims to construct a unified characteristic expression applicable to topological spaces of arbitrary dimensions and reveal the profound isomorphism between topological dimension and exponent, while establishing a dual theoretical system of alternating sums and total sums with parameters p, q ∈ C. By introducing the concept of negative-one-dimensional topological cell N(−1) =1 and formal (N + 1)-dimensional cell N(N+1) = 1, we establish the characteristic expression PN+1k=0 (−1)kN(N−k) = 0 and the total sum expression PN+1k=0 N(N−k) = 2N+1, and prove their one-to-one correspondence with the expansion coefficients of (a−b)N+1 and (a+b)N+1, where a, b can be arbitrary complex numbers. From this, we propose the Dimensional Extension Principle: the exponent m corresponds to m − 1 dimensional topological space, and dimensions can be extended to negative integers, real numbers, and even complex numbers. The alternating sum corresponds to the binomial power difference expansion (a − b)m, and the total sum corresponds to the binomial power sum expansion (a + b)m, with the parameters a, b selecting different topological invariants. We employ combinatorial topology to construct negative-dimensional simplicial complexes, establish a rigorous algebraic foundation for negative-dimensional topological spaces through analytic continuation and Gamma functions, and extend topological invariants such as Betti numbers, homology groups, and characteristic classes to real and complex dimensions using the generalized binomial theorem. Core topological theorems including the Gauss-Bonnet theorem, the Riemann-Roch theorem, the Atiyah-Singer index theorem, Poincar´e duality, Morse theory, characteristic class theory, cobordism theory, knot theory, and spectral sequence theory are incorporated into a unified framework, establishing a dual formulation of alternating sums and total sums, where the alternating sum corresponds to the parameter selection (1, −1) and the total sum corresponds to (1, 1), while more general parameters (a, b) correspond to more general topological invariants.1. We prove that all topological formulas expressible as alternating sums, after introducing the negative-one-dimensional structure, can be unified into a vanishing alternating sum form, while all topological formulas expressible as total sums can be unified into the form 2dim X+1, with parameters (a, b) extendable to arbitrary complex numbers;2. We rigorously construct chain complexes and homology groups for negative integer-dimensional topological spaces, prove that their extended Euler characteristics satisfy ˜χ(−n) = 0 and exhibit duality relations with positive-dimensional spaces, and prove that their extended total sums satisfy Σ( ˜ −n) = 2−n+1;3. We extend the Gauss-Bonnet theorem to real and complex dimensional manifolds, establish the dual relationship between alternating integrals and total integrals, and generalize to parameterized forms;4. We extend the Riemann-Roch theorem to complex dimensional algebraic varieties, establish the generalized Hirzebruch-Riemann-Roch formula in dual alternating and total forms;5. We extend the Atiyah-Singer index theorem to complex dimensions, prove that the index as a function of dimension is meromorphic with a unified pole structure;6. We extend Poincar´e duality, Morse theory, characteristic class theory, cobordism theory, knot theory, and spectral sequence theory to complex dimensions;7. We establish the rigorous mathematical foundations for dimensional extension in topological quantum field theory, revealing the dual roles of alternating sums and total sums in the cobordism category;8. We prove the emergence of fractional topological dimensions in topological insulators and establish the connection between total sums and topological entropy.Dimensional extension theory achieves the most perfect unified form of the Euler-Poincar´e theorem in topology and establishes a dual theoretical system of alternating sums and total sums, where the parameters in the binomial theorem (a, b) can be extended to arbitrary complex numbers, corresponding to different types of topological invariants. This provides new perspectives and tools for multiple branches of topology, with significant theoretical value and application prospects.
S. B. Liu (Wed,) studied this question.