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By combining the join construction from Sasakian geometry with theHamiltonian 2-form construction from K\"ahler geometry, we recoverSasaki-Einstein metrics discovered by physicists. Our geometrical approachallows us to give an algorithm for computing the topology of theseSasaki-Einstein manifolds. In particular, we explicitly compute the cohomologyrings for several cases of interest and give a formula for homotopy equivalencein one particular 7-dimensional case. We also show that our construction givesat least a two dimensional cone of both Sasaki-Ricci solitons and extremalSasaki metrics.

The Sasaki Join, Hamiltonian 2-forms, and Sasaki-Einstein Metrics