The graph of a foliation $\mathcal{F}$ of a manifold $M$ is the space of all triple $(x,y,[\gamma])$ where $x,y$ lies on the same leaf $F$ and $[\gamma]$ is the equivalent class of a path in $F$ joining $x$ to $y$. The equivalent class is defined as follows: Two paths are equivalents if they generate the same local holonomy.

The graph of foliations generates some special $C^*$ algebras associated to the foliation and this $C^*$ algebra contains useful information about the foliation.

In this post we try to consider two kinds of generalization of the graph of a foliation. Our generalization is only based on a generalization of the equivalent class of paths $\gamma$ joining $x,y$ on the same leaf. We do not change other axioms of the definition of the graph of a foliation. Then we ask whether these generalizations introduce some new objects associated to a foliation?

  1. Let we have a foliation of $\mathbb{R}^n$. We consider affine transversal sections for our foliation. Two path $\gamma_1, \gamma_2$ are equivalent if $\gamma_1 \circ \gamma_2^{-1}$ is an affine map.

2.Let we have a foliation of a manifold. Two path $\gamma_1, \gamma_2$ are equivalent if they have the same linear part.


Your Answer

By clicking "Post Your Answer", you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.