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The current-compatible intersection number is well-defined
Statement
For the compact transverse complementary submanifolds in Current-compatible signed intersection number, the displayed sum is a finite integer and does not depend on the positive bases used to compute its local signs.
Statement references
Facts & Assumptions
Given: The oriented ambient manifold and the compact embedded submanifolds from the definition.
Current-compatible signed intersection number defines each local sign by the determinant map induced by , comparing the image of the tensor orientation ray with the ambient orientation ray, and multiplies the finite sum by .
Transverse complementary submanifolds meet in finitely many points proves that is finite under the stated compactness and transversality hypotheses.
Proof
Given: The orientation rays and the hypotheses of [F1].
By [F2], only finitely many points occur in the sum in [F1]. At each such point, transversality and complementary dimensions make the addition map on tangent spaces an isomorphism. Its determinant-line map sends the tensor of the two orientation rays to exactly one of the ambient orientation ray and its opposite, so is uniquely determined. When positive-dimensional bases are used to represent the rays, replacing either basis by another positive basis preserves its ray and therefore preserves this sign; zero-dimensional orientation signs are already carried by their rays.
By [F1], each local sign is or and the finite sum has the fixed factor . That factor is independent of all basis choices, so the current-compatible intersection number is well-defined. Empty intersection gives the empty sum .