Sasakian-Einstein structures on 9#(𝑆²×𝑆³)

Boyer, Charles; Galicki, Krzysztof; Nakamaye, Michael

doi:10.1090/S0002-9947-02-03015-5

Sasakian-Einstein structures on $9\#(S^2\times S^3)$
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by Charles P. Boyer, Krzysztof Galicki and Michael Nakamaye PDF

Trans. Amer. Math. Soc. 354 (2002), 2983-2996 Request permission

Abstract:

We show that $\scriptstyle {9\#(S^2\times S^3)}$ admits an 8-dimensional complex family of inequivalent non-regular Sasakian-Einstein structures. These are the first known Einstein metrics on this 5-manifold. In particular, the bound $\scriptstyle {b_2(M)\leq 8}$ which holds for any regular Sasakian-Einstein $\scriptstyle {M}$ does not apply to the non-regular case. We also discuss the failure of the Hitchin-Thorpe inequality in the case of 4-orbifolds and describe the orbifold version.

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