The integral homology of the symmetric square of a CW complex has been known since 1960’s, although the general answer is very complicated. However, the situation for the integral cohomology ring remains an open problem, except for a few special cases. One of the main difficulties is the computation of its ring structure.  In this talk I shall outline the solution for quaternionic projective spaces, including the infinite version. Rather than give the details, I shall focus on the overall structure of the results, which are being extended to the complex case in forthcoming joint work. The analogous methods also work for the octonionic projective plane.

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