摘要
I. INTRODUCTIONLet φ∈（X）, where X is a compact topological space. We.denote by Ω<sub>n</sub>（X, φ）the nth normal bordism group of X with a coefficient φ. If X is path-connected, then by the natural hornomorphism I（ φ ）: π<sub>3</sub><sup>s</sup>≌ Ω<sub>3</sub> （*, 0 ）→Ω<sub>3</sub> （X, φ） and the classical stable J- homomorphism, we can define a unique generalized J-homomorphism J （φ）: π<sub>3</sub> （SO）→ Ω<sub>3</sub> （ X, φ）. The order of the finite cyclic group Im J（φ） is denoted by j （φ）. We refer readers to [1] for the details.
