Self-intersection
In mathematics, intersection theory is a branch of algebraic geometry, where subvarieties are intersected on an algebraic variety, and of algebraic topology, where intersections are computed within the cohomology ring. The theory for varieties is older, with roots in Bézout's theorem on curves and elimination theory. On the other hand, the topological theory more quickly reached a definitive form.
For a connected oriented manifold M of dimension 2n the intersection form is defined on the n-th cohomology group (what is usually called the 'middle dimension') by the evaluation of the cup product on the fundamental class [M] in H2n(M, ∂M). Stated precisely, there is a bilinear form
given by
with
This is a symmetric form for n even (so 2n = 4k doubly even), in which case the signature of M is defined to be the signature of the form, and an alternating form for n odd (so 2n= 4k + 2 singly even). These can be referred to uniformly as ε-symmetric forms, where ε = (−1)n = ±1 respectively for symmetric and skew-symmetric forms. It is possible in some circumstances to refine this form to an ε-quadratic form, though this requires additional data such as a framing of the tangent bundle. It is possible to drop the orientability condition and work with Z/2Z coefficients instead.
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