The cup length of a topological space with coefficients in a field is the supremum of the numbers of positive-degree cohomology classes with nonzero cup product. Using rational coefficients, write
 |
The empty product gives value 0 when no positive-degree product is nonzero. The value is infinite
if such products exist with arbitrarily many factors; a finite supremum is a maximum. A unit-inclusive
convention instead adds 1 to this number. For a positive-dimensional
sphere the cup length is 1, whereas that of 
is 
. On a closed smooth manifold,
one plus the cup length is a lower bound for the critical point number.
