When two cycles have a transversal intersection 
on a smooth manifold 
, then 
is a cycle. Moreover, the homology class that 
represents depends only on the homology class of 
and 
. The sign of 
is determined by the orientations on 
, 
, and 
.
For example, two curves can intersect in one point on a surface transversally, since
 |
The curves can be deformed so that they intersect three times, but two of those intersections sum to zero since two intersect positively and one intersects negatively, i.e., with the manifold orientation of the curves being the reverse orientation of the ambient space.
On the torus illustrated above, the cycles intersect in one point.
The binary operation of intersection makes homology on a manifold into a ring. That is, it plays the role of multiplication, which respects the grading. When 
and 
, then 
. In fact, intersection is the dual to the cup product in Poincaré duality. That is, if 
is the Poincaré dual to 
and 
is the dual to 
then 
is the dual to 
.
Without the notion of transversal intersection, intersections are not well-defined in homology. On a more general space, even a manifold with singularities, the homology does not have a natural ring structure.
SEE ALSO:Codimension, Cup Product, Homology, Manifold, Manifold Orientation, Poincaré Duality, Transversal Intersection, Vector Space Orientation
This entry contributed by Todd Rowland
