On the symmetric squares of complex and quaternionic projective space

TL;DR

This paper computes the integral cohomology rings of symmetric squares of complex and quaternionic projective spaces using geometric, algebraic, and topological methods, extending classical results with new explicit descriptions.

Contribution

It provides the first explicit generators and relations for the integral cohomology rings of symmetric squares of complex and quaternionic projective spaces, utilizing geometric and combinatorial techniques.

Findings

Cohomology rings described via generators and relations.

Compatibility with existing mod 2 and homological results.

Use of generalized Fibonacci polynomials for expression.

Abstract

The problem of computing the integral cohomology ring of the symmetric square of a topological space has been of interest since the 1930s, but limited progress has been made on the general case until recently. In this work we offer a solution for the complex and quaternionic projective spaces $KP^n$, by taking advantage of their rich geometrical structure. Our description is in terms of generators and relations, and our methods entail ideas that have appeared in the literature of quantum chemistry, theoretical physics, and combinatorics. We deal first with the case $KP^\infty$, and proceed by identifying the truncation required for passage to finite n. The calculations rely upon a ladder of long exact cohomology sequences, which arises by comparing cofibrations associated to the diagonals of the symmetric square and the corresponding Borel construction. The cofibrations involve classic configuration spaces of unordered pairs of 1-dimensional subspaces of $K^{n+1}$, and their one-point compactifications; the latter are identified as Thom spaces by combining Lowdin's symmetric orthogonalisation process (and its quaternionic extension) with a dash of Pin geometry. The ensuing integral cohomology rings may be conveniently expressed using generalised Fibonacci polynomials. By way of validation, we note that our conclusions are compatible with mod 2 computations of Nakaoka and with homological results of Milgram.