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Obstruction theory and the level n elliptic genus

Part of: Operations and obstructions Homology and cohomology theories Homotopy theory

Published online by Cambridge University Press:  03 August 2023

Andrew Senger
Andrew Senger*
Affiliation:
Department of Mathematics, Harvard University, 1 Oxford Street, Cambridge, MA 02138, USA senger@math.harvard.edu
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Abstract

Given a height at most two Landweber exact $\mathbb {E}_\infty$-ring $E$ whose homotopy is concentrated in even degrees, we show that any complex orientation of $E$ which satisfies the Ando criterion admits a unique lift to an $\mathbb {E}_\infty$-complex orientation $\mathrm {MU} \to E$. As a consequence, we give a short proof that the level $n$ elliptic genus lifts uniquely to an $\mathbb {E}_\infty$-complex orientation $\mathrm {MU} \to \mathrm {tmf}_1 (n)$ for all $n\, {\geq}\, 2$.

Keywords

topological modular formscomplex orientationobstruction theory

MSC classification

Primary: 55N34: Elliptic cohomology 55P43: Spectra with additional structure ($E_infty$, $A_infty$, ring spectra, etc.) 55S35: Obstruction theory
Type
Research Article
Information
Compositio Mathematica , Volume 159 , Issue 9 , September 2023 , pp. 2000 - 2021
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Copyright
© 2023 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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Footnotes

During the course of this work, the author was supported by NSF Grants DGE-1745302 and DMS-2103236.

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