# Quadratic functions in geometry, topology, and M-theory

@inproceedings{Hopkins2002QuadraticFI,
title={Quadratic functions in geometry, topology, and M-theory},
author={Michael J. Hopkins and Isadore Manuel Singer},
year={2002}
}
We describe an interpretation of the Kervaire invariant of a Riemannian manifold of dimension $4k+2$ in terms of a holomorphic line bundle on the abelian variety $H^{2k+1}(M)\otimes R/Z$. Our results are inspired by work of Witten on the fivebrane partition function in $M$-theory (hep-th/9610234, hep-th/9609122). Our construction requires a refinement of the algebraic topology of smooth manifolds better suited to the needs of mathematical physics, and is based on our theory of "differential… CONTINUE READING

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