SingularkettenFunktional

The SingularkettenFunktional, or SFF, is a linear functional defined on the singular n‑chain group C_n(X; R) of a topological space X. It assigns to each singular n-simplex σ a real value through a chosen weight function w(σ) and extends linearly to all chains. The construction depends on a fixed collection of weights assigned to representative simplices, so the value on a chain c = ∑ a_i σ_i is ∑ a_i w(σ_i).

A central feature is the behavior with respect to boundaries. If the weight function satisfies w(∂c) = 0 for all chains c, then the SFF vanishes on boundaries and hence factors through the n‑th singular homology H_n(X; R). In this case the functional defines a homology‑level invariant, pairing naturally with homology classes.

Naturality and computation are convenient when the weights are chosen to respect symmetries or orientation. Under a continuous map f: X → Y, the pushforward f_* on chains preserves the defined weights, giving a natural commuting relation SFF_Y ∘ f_* = f_* ∘ SFF_X. This makes SFF adaptable to comparisons across spaces and to functorial constructions in homology.

Variants of the SingularkettenFunktional arise by altering the weighting scheme, yielding orientation‑sensitive or boundary‑detecting forms. It relates to standard pairings with cohomology and to classical constructions such as the Kronecker pairing, when specialized appropriately.

History and status: The term is used in speculative or illustrative discussions and is not part of established literature. It serves as a conceptual tool to explore chain‑level invariants and homology‑level consequences.