Homology intersection assigns a homology class to the transversal intersection 
of closed oriented submanifolds representing homology
cycles in an oriented smooth
manifold 
.
The homology class of 
depends only on the homology
classes 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., the ordered tangent directions of the two curves at the negative intersection induce the opposite manifold orientation to that of the surface.
On the torus illustrated above, the homology cycles intersect in one point.
For a closed oriented 
-dimensional smooth manifold, the binary
operation of intersection makes homology
into a graded 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.
