 On the other hand, the topological theory more quickly reached a definitive form.For a connected oriented manifold of dimension the intersection form is defined on the -th cohomology group (what is usually called the 'middle dimension') by the evaluation of the cup product on the fundamental class in .Stated precisely, there is a bilinear form

given by

with

This is a symmetric form for even (so doubly even), in which case the signature of is defined to be the signature of the form, and an alternating form for odd (so singly even).