Some identities for hypergeometric functions have geometric meaning behind. Aomoto proposed a method to study hypergeometric functions as pairings of cycles and cocycles about 30 years ago . This ingenious point of view has enabled us to yield a lot of formulas for hypergeometric functions.
The identities we have presented above
are quadratic relations for
We will see that it also has a geometric meaning
based on a work of Cho and Matsumoto.
They proved an analog of Riemann's period relation
for intersection numbers of cycles and cocycles and associated period
integral, of which entries are nothing but hypergeometric functions
The period relation yields a quadratic relation of hypergeometric
Therefore, the problem of deriving quadratic relations is reduced
to evaluation of intersection numbers.
Cho, Kita, Matsumoto and Yoshida gave formulas to evaluate
for a class of hypergeometric functions
expressed by a definite integral
of which integrant has a normally crossing singular locus
, , , .
has a multiple integral representation
but the singular locus of the integrant is not normally
Hence, the generalized hypergeometric functions are out of the class
for general , because their method requires a construction of resolutions
of singularities and it is difficult in general.
We will introduce a different approach to study the GHF.
The GHF is expressed in terms of the single integral
The first author studied a method to evaluate intersection numbers for cocycles with coefficients in locally constant sheaves of which rank is more than one. He applied the method for evaluating intersection numbers for cocycles associated to the Selberg type integrals , .
In this paper, we first reexamine the definition of intersection number in view of the topological cup product, because discussions by Kita and Yoshida  are not satisfactory to apply for our problem of and those by the first author in  are not satisfactory to be a rigorous foundation. We will see that the method of the first author in ,  is not only useful for computation, but also is a consequence of a general theory of duality. Next, we apply the method to evaluate intersection numbers for . The twisted cohomology and homology groups associated to the single integral representation of are direct sums of primary parts and degenerate parts. Only the primary parts stand for . This degeneration makes the evaluation of intersection numbers more complicated than the evaluation problem for the Selberg type integrals.