The U(1)-topological elliptic genus lifts to connective topological Jacobi forms and is surjective in homotopy.
Totaro, Chern numbers for singular varieties and elliptic homology , Annals of Mathematics (2000) 757–791
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The U(1)-topological elliptic genus is surjective
The U(1)-topological elliptic genus lifts to connective topological Jacobi forms and is surjective in homotopy.