# The Atiyah Patodi Singer index formula for measured foliations - Mathematics > Differential Geometry

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Abstract: Let $X 0$ be a compact Riemannian manifold with boundary endowed with aoriented, measured even dimensional foliation with purely transverse boundary.Let $X$ be the manifold with cylinder attached and extended foliation. We provethat the $L^2$-measured index of a Dirac type operator is well defined and thefollowing Atiyah Patodi Singer index formula is true $$ind {L^2,\Lambda}D^+ =<\widehat{A}X, ablaChE-S,C \Lambda> +1-2\eta \LambdaD^{\mathcal{F} \partial} - h^+ \Lambda + h^- \Lambda.$$ Here$\Lambda$ is a holonomy invariant transverse measure,$\eta {\Lambda}D^{\mathcal{F} {\partial}}$ is the Ramachandran eta invariant\cite{Rama} of the leafwise boundary operator and the $\Lambda$-dimensions$h^\pm \Lambda$ of the space of the limiting values of extended solutions issuitably defined using square integrable representations of the equivalencerelation of the foliation with values on weighted Sobolev spaces on the leaves.

Autor: Paolo Antonini

Fuente: https://arxiv.org/