The algebraic intersection number of two closed submanifolds 
and 
in a manifold 
, when 
, 
, and 
have compatible orientations
and the dimensions of 
and 
are complementary, is their signed intersection count. When
their intersection is transversal, the
number is
 |
where the local sign 
is 1 or 
according to whether the orientations
of 
and 
at 
agree with the manifold orientation of 
.
When the intersection is not transversal,
the number is defined after perturbing the representatives so their intersection
is transversal.
The algebraic intersection number depends only on the corresponding homology classes. It is the numerical form of the homology intersection product and is dual to the cup product under Poincaré duality (Guillemin and Pollack 1974).
