Introduction to the Intersection Theory for Twisted Homology and Cohomology Groups

We give an introduction to the intersection theory for twisted homology and cohomology groups associated with Euler type integrals of hypergeometric functions. We introuduce twisted homology and cohomology groups motivated by Twisted Stokes’ Theorem, and give their dimension formulas. We define an intersection form between twisted homology groups and that between twisted cohomology groups, and explain how to compute them. These intersection forms are compatible with the natural pairing between the twisted homology and cohomology groups. This compatibility yields a twisted period relation, which relates intersection numbers and period integrals regarded as some kinds of hypergeometric functions. In Appendix, we show that Elliott’s identity can be obtained from the twisted period relation.