Intersection numbers and cup products.

Consider, once again, the torus. A basis for H ₁ is represented by a pair of embedded circles intersecting in a point The cup product of the dual basis elements is nonzero. Another example is the orientable surface of genus 2 with a basis for H ₁ represented in the picture. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429502408/295d3acc-dfa3-4421-a2d3-a9c22ffb681c/content/pg290_1.tif"/>