Return to Answer

Even though we may not be able to associate a Lie algebroid with a singular foliation, we can associate a Lie $\infty$-algebroid with a singular foliation (satisfying certain not so strange conditions). This result is due to Camille Laurent-Gengoux, Sylvian Lavau and Thomas Storbl published as "The universal Lie $\infty$-algebroid of a singular foliation", whose arXiv version is available at https://arxiv.org/abs/1806.00475

I will try to explain the basic idea that I understood.

A singular foliation $\mathcal{F}$ is a locally generated $C^\infty(M)$-submodule of $\mathfrak{X}(M)$. Just like any other $R$-module, this $C^\infty(M)$-module $\mathcal{F}$ will also have resolution, $$\cdots\rightarrow P_{-3}\rightarrow P_{-2}\rightarrow P_{-1}\rightarrow \mathcal{F}\rightarrow 0,$$ where $P_{-i}$ are projective $C^\infty(M)$-modules for every $i\geq 1$.

As the singular foliation $\mathcal{F}$ is coming from a geometric structure on the manifold $M$, it is natural to focus on resolutions that comes from some geometric structures on $M$. This is where Serra-Swan theorem comes for help. When ever we have a finitely generated projective $C^\infty(M)$-module $P$, then, it has to be of the form $\Gamma(M,E)$ for some vector bundles $E\rightarrow M$.

Even though $\mathcal{F}$ is a locally finitely generated $C^\infty(M)$-module, we may not assure that there would be a projective resolution $$\cdots\rightarrow P_{-3}\rightarrow P_{-2}\rightarrow P_{-1}\rightarrow \mathcal{F}\rightarrow 0$$ where all these $P_{-i}$ are locally finitely generated. So, the question of $P_{-i}$ being finitely generated may be too much to hope for. They were able to get rid of this locally finitely generated and finitely generated issue by asking that $M$ is a compact manifold.

Such a projective resolution $$\cdots\rightarrow P_{-3}\rightarrow P_{-2}\rightarrow P_{-1}\rightarrow \mathcal{F}\rightarrow 0$$ gives vector bundles $E_{-i}\rightarrow M$ with $\Gamma(M,E_{-i})=P_{-i}$ and a morphism of vector bundles $E_{-i}\rightarrow E_{-i+1}$ coming from map of sections $P_{-i}\rightarrow P_{-i+1}$ (because of $C^\infty(M)$-linearity).

Thus, we have an exact sequence of vector bundles $$\cdots \rightarrow E_{-3}\rightarrow E_{-2}\rightarrow E_{-1}\rightarrow TM$$ such that the corresponding map of sections is the resolution of $\mathcal{F}$ that we mentioned above. This complex of vector bundles is what they have called as geometric resolution.

Under some mild conditions on the singular foliations, they were able to prove the following theorem

Theorem $2.4$ : A locally real analytic singular foliation admits a geometric resolution of length at most $\dim(M)+1$ over any relatively compact open subset of $M$.

It does not end here. I am not very sure but, may be the above result can be proved for any random locally finitely generated $C^\infty(M)$-submodule of $\mathfrak{X}(M)$, with out having the condition of being closed under Lie bracket (in other words $\mathcal{F}$ having structure of a Lie bracket). For obvious reasons, this extra structure on $\mathcal{F}$ should reflect somewhere in the sequence of vector bundles $\cdots \rightarrow E_{-3}\rightarrow E_{-2}\rightarrow E_{-1}\rightarrow TM$. This should remind the notion of Lie $\infty$-algebroid and this is what they say in the next result.

Theorem $2.7$ : Let $\mathcal{F}$ be a singular foliation on $M$ which permits a geometric resolution. Then, there is a (universal) Lie algebroid structure on the resolution.

It also explains in what sense it is "universal".

In this sense, any singular foliation (with mild assumptions) comes from a Lie $\infty$-algebroid, which they call as "the Lie $\infty$-algebroid of a singular foliation".

I can add some more details if anyone want to see.

Even though we may not be able to associate a Lie algebroid with a singular foliation, we can associate a Lie $\infty$-algebroid with a singular foliation (satisfying certain not so strange conditions). This result is due to Camille Laurent-Gengoux, Sylvian Lavau and Thomas Storbl published as "The universal Lie $\infty$-algebroid of a singular foliation", whose arXiv version is available at https://arxiv.org/abs/1806.00475

I will try to explain the basic idea that I understood.

A singular foliation $\mathcal{F}$ is a locally generated $C^\infty(M)$-submodule of $\mathfrak{X}(M)$. Just like any other $R$-module, this $C^\infty(M)$-module $\mathcal{F}$ will also have resolution, $$\cdots\rightarrow P_{-3}\rightarrow P_{-2}\rightarrow P_{-1}\rightarrow \mathcal{F}\rightarrow 0,$$ where $P_{-i}$ are projective $C^\infty(M)$-modules for every $i\geq 1$.

As the singular foliation $\mathcal{F}$ is coming from a geometric structure on the manifold $M$, it is natural to focus on resolutions that come from some geometric structures on $M$. This is where the Serra-Swan theorem comes for help. Whenever we have a finitely generated projective $C^\infty(M)$-module $P$, then, it has to be of the form $\Gamma(M,E)$ for some vector bundles $E\rightarrow M$.

Even though $\mathcal{F}$ is a locally finitely generated $C^\infty(M)$-module, we may not assure that there would be a projective resolution $$\cdots\rightarrow P_{-3}\rightarrow P_{-2}\rightarrow P_{-1}\rightarrow \mathcal{F}\rightarrow 0$$ where all these $P_{-i}$ are locally finitely generated. So, the question of $P_{-i}$ being finitely generated may be too much to hope for. They were able to get rid of this locally finitely generated and finitely generated issue by asking that $M$ is a compact manifold.

Such a projective resolution $$\cdots\rightarrow P_{-3}\rightarrow P_{-2}\rightarrow P_{-1}\rightarrow \mathcal{F}\rightarrow 0$$ gives vector bundles $E_{-i}\rightarrow M$ with $\Gamma(M,E_{-i})=P_{-i}$ and a morphism of vector bundles $E_{-i}\rightarrow E_{-i+1}$ coming from map of sections $P_{-i}\rightarrow P_{-i+1}$ (because of $C^\infty(M)$-linearity).

Thus, we have an exact sequence of vector bundles $$\cdots \rightarrow E_{-3}\rightarrow E_{-2}\rightarrow E_{-1}\rightarrow TM$$ such that the corresponding map of sections is the resolution of $\mathcal{F}$ that we mentioned above. This complex of vector bundles is what they have called as geometric resolution.

Under some mild conditions on the singular foliations, they were able to prove the following theorem

Theorem $2.4$ : A locally real analytic singular foliation admits a geometric resolution of length at most $\dim(M)+1$ over any relatively compact open subset of $M$.

It does not end here. I am not very sure but, may be the above result can be proved for any random locally finitely generated $C^\infty(M)$-submodule of $\mathfrak{X}(M)$, without having the condition of being closed under Lie bracket (in other words $\mathcal{F}$ having structure of a Lie bracket). For obvious reasons, this extra structure on $\mathcal{F}$ should reflect somewhere in the sequence of vector bundles $\cdots \rightarrow E_{-3}\rightarrow E_{-2}\rightarrow E_{-1}\rightarrow TM$. This should remind the notion of Lie $\infty$-algebroid and this is what they say in the next result.

Theorem $2.7$ : Let $\mathcal{F}$ be a singular foliation on $M$ which permits a geometric resolution. Then, there is a (universal) Lie algebroid structure on the resolution.

It also explains in what sense it is "universal".

In this sense, any singular foliation (with mild assumptions) comes from a Lie $\infty$-algebroid, which they call as "the Lie $\infty$-algebroid of a singular foliation".

I can add some more details if anyone wants to see them.

Even though we may not be able to associate a Lie algebroid with a singular foliation, we can associate a Lie $\infty$-algebroid with a singular foliation (satisfying certain not so strange conditions). This result is due to Camille Laurent-Gengoux, Sylvian Lavau and Thomas Storbl published as "The universal Lie $\infty$-algebroid of a singular foliation", whose arXiv version is available at https://arxiv.org/abs/1806.00475

I will try to explain the basic idea that I understood.

A singular foliation $\mathcal{F}$ is a locally generated $C^\infty(M)$-submodule of $\mathfrak{X}(M)$. Just like any other $R$-module, this $C^\infty(M)$-module $\mathcal{F}$ will also have resolution, $$\cdots\rightarrow P_{-3}\rightarrow P_{-2}\rightarrow P_{-1}\rightarrow \mathcal{F}\rightarrow 0,$$ where $P_{-i}$ are projective $C^\infty(M)$-modules for every $i\geq 1$.

As the singular foliation $\mathcal{F}$ is coming from a geometric structure on the manifold $M$, it is natural to focus on resolutions that comes from some geometric structures on $M$. This is where Serra-Swan theorem comes for help. When ever we have a finitely generated projective $C^\infty(M)$-module $P$, then, it has to be of the form $\Gamma(M,E)$ for some vector bundles $E\rightarrow M$.

Even though $\mathcal{F}$ is a locally finitely generated $C^\infty(M)$-module, we may not assure that there would be a projective resolution $$\cdots\rightarrow P_{-3}\rightarrow P_{-2}\rightarrow P_{-1}\rightarrow \mathcal{F}\rightarrow 0$$ where all these $P_{-i}$ are locally finitely generated. So, the question of $P_{-i}$ being finitely generated may be too much to hope for. They were able to get rid of this locally finitely generated and finitely generated issue by asking that $M$ is a compact manifold.

Such a projective resolution $$\cdots\rightarrow P_{-3}\rightarrow P_{-2}\rightarrow P_{-1}\rightarrow \mathcal{F}\rightarrow 0$$ gives vector bundles $E_{-i}\rightarrow M$ with $\Gamma(M,E_{-i})=P_{-i}$ and a morphism of vector bundles $E_{-i}\rightarrow E_{-i+1}$ coming from map of sections $P_{-i}\rightarrow P_{-i+1}$ (because of $C^\infty(M)$-linearity).

Thus, we have an exact sequence of vector bundles $$\cdots \rightarrow E_{-3}\rightarrow E_{-2}\rightarrow E_{-1}\rightarrow TM$$ such that the corresponding map of sections is the resolution of $\mathcal{F}$ that we mentioned above. This exact sequence of vector bundles is what they have called as geometric resolution.

Under some mild conditions on the singular foliations, they were able to prove the following theorem

Theorem $2.4$ : A locally real analytic singular foliation admits a geometric resolution of length at most $\dim(M)+1$ over any relatively compact open subset of $M$.

It does not end here. I am not very sure but, may be the above result can be proved for any random locally finitely generated $C^\infty(M)$-submodule of $\mathfrak{X}(M)$, with out having the condition of being closed under Lie bracket (in other words $\mathcal{F}$ having structure of a Lie bracket). For obvious reasons, this extra structure on $\mathcal{F}$ should reflect somewhere in the sequence of vector bundles $\cdots \rightarrow E_{-3}\rightarrow E_{-2}\rightarrow E_{-1}\rightarrow TM$. This should remind the notion of Lie $\infty$-algebroid and this is what they say in the next result.

Theorem $2.7$ : Let $\mathcal{F}$ be a singular foliation on $M$ which permits a geometric resolution. Then, there is a (universal) Lie algebroid structure on the resolution.

It also explains in what sense it is "universal".

In this sense, any singular foliation (with mild assumptions) comes from a Lie $\infty$-algebroid, which they call as "the Lie $\infty$-algebroid of a singular foliation".

I can add some more details if anyone want to see.

Even though we may not be able to associate a Lie algebroid with a singular foliation, we can associate a Lie $\infty$-algebroid with a singular foliation (satisfying certain not so strange conditions). This result is due to Camille Laurent-Gengoux, Sylvian Lavau and Thomas Storbl published as "The universal Lie $\infty$-algebroid of a singular foliation", whose arXiv version is available at https://arxiv.org/abs/1806.00475

I will try to explain the basic idea that I understood.

A singular foliation $\mathcal{F}$ is a locally generated $C^\infty(M)$-submodule of $\mathfrak{X}(M)$. Just like any other $R$-module, this $C^\infty(M)$-module $\mathcal{F}$ will also have resolution, $$\cdots\rightarrow P_{-3}\rightarrow P_{-2}\rightarrow P_{-1}\rightarrow \mathcal{F}\rightarrow 0,$$ where $P_{-i}$ are projective $C^\infty(M)$-modules for every $i\geq 1$.

As the singular foliation $\mathcal{F}$ is coming from a geometric structure on the manifold $M$, it is natural to focus on resolutions that comes from some geometric structures on $M$. This is where Serra-Swan theorem comes for help. When ever we have a finitely generated projective $C^\infty(M)$-module $P$, then, it has to be of the form $\Gamma(M,E)$ for some vector bundles $E\rightarrow M$.

Even though $\mathcal{F}$ is a locally finitely generated $C^\infty(M)$-module, we may not assure that there would be a projective resolution $$\cdots\rightarrow P_{-3}\rightarrow P_{-2}\rightarrow P_{-1}\rightarrow \mathcal{F}\rightarrow 0$$ where all these $P_{-i}$ are locally finitely generated. So, the question of $P_{-i}$ being finitely generated may be too much to hope for. They were able to get rid of this locally finitely generated and finitely generated issue by asking that $M$ is a compact manifold.

Such a projective resolution $$\cdots\rightarrow P_{-3}\rightarrow P_{-2}\rightarrow P_{-1}\rightarrow \mathcal{F}\rightarrow 0$$ gives vector bundles $E_{-i}\rightarrow M$ with $\Gamma(M,E_{-i})=P_{-i}$ and a morphism of vector bundles $E_{-i}\rightarrow E_{-i+1}$ coming from map of sections $P_{-i}\rightarrow P_{-i+1}$ (because of $C^\infty(M)$-linearity).

Thus, we have an exact sequence of vector bundles $$\cdots \rightarrow E_{-3}\rightarrow E_{-2}\rightarrow E_{-1}\rightarrow TM$$ such that the corresponding map of sections is the resolution of $\mathcal{F}$ that we mentioned above. This complex of vector bundles is what they have called as geometric resolution.

Under some mild conditions on the singular foliations, they were able to prove the following theorem

Theorem $2.4$ : A locally real analytic singular foliation admits a geometric resolution of length at most $\dim(M)+1$ over any relatively compact open subset of $M$.

It does not end here. I am not very sure but, may be the above result can be proved for any random locally finitely generated $C^\infty(M)$-submodule of $\mathfrak{X}(M)$, with out having the condition of being closed under Lie bracket (in other words $\mathcal{F}$ having structure of a Lie bracket). For obvious reasons, this extra structure on $\mathcal{F}$ should reflect somewhere in the sequence of vector bundles $\cdots \rightarrow E_{-3}\rightarrow E_{-2}\rightarrow E_{-1}\rightarrow TM$. This should remind the notion of Lie $\infty$-algebroid and this is what they say in the next result.

Theorem $2.7$ : Let $\mathcal{F}$ be a singular foliation on $M$ which permits a geometric resolution. Then, there is a (universal) Lie algebroid structure on the resolution.

It also explains in what sense it is "universal".

In this sense, any singular foliation (with mild assumptions) comes from a Lie $\infty$-algebroid, which they call as "the Lie $\infty$-algebroid of a singular foliation".

I can add some more details if anyone want to see.

Even though we may not be able to associate a Lie algebroid with a singular foliation, we can associate a Lie $\infty$-algebroid with a singular foliation (satisfying certain not so strange conditions). This result is due to Camille Laurent-Gengoux, Sylvian Lavau and Thomas Storbl published as "The universal Lie $\infty$-algebroid of a singular foliation", whose arXiv version is available at https://arxiv.org/abs/1806.00475

I will try to explain the basic idea that I understood.

A singular foliation $\mathcal{F}$ is a locally generated $C^\infty(M)$-submodule of $\mathfrak{X}(M)$. Just like any other $R$-module, this $C^\infty(M)$-module $\mathcal{F}$ will also have resolution, $$\cdots\rightarrow P_{-3}\rightarrow P_{-2}\rightarrow P_{-1}\rightarrow \mathcal{F}\rightarrow 0,$$ where $P_{-i}$ are projective $C^\infty(M)$-modules for every $i\geq 1$.

As the singular foliation $\mathcal{F}$ is coming from a geometric structure on the manifold $M$, it is natural to focus on resolutions that comes from some geometric structures on $M$. This is where Serra-Swan theorem comes for help. When ever we have a finitely generated projective $C^\infty(M)$-module $P$, then, it has to be of the form $\Gamma(M,E)$ for some vector bundles $E\rightarrow M$.

Even though $\mathcal{F}$ is a locally finitely generated $C^\infty(M)$-module, we may not assure that there would be a projective resolution $$\cdots\rightarrow P_{-3}\rightarrow P_{-2}\rightarrow P_{-1}\rightarrow \mathcal{F}\rightarrow 0$$ where all these $P_{-i}$ are locally finitely generated. So, the question of $P_{-i}$ being finitely generated may be too much to hope for. They were able to get rid of this locally finitely generated and finitely generated issue by asking that $M$ is a compact manifold.

Such a projective resolution $$\cdots\rightarrow P_{-3}\rightarrow P_{-2}\rightarrow P_{-1}\rightarrow \mathcal{F}\rightarrow 0$$ gives vector bundles $E_{-i}\rightarrow M$ with $\Gamma(M,E_{-i})=P_{-i}$ and a morphism of vector bundles $E_{-i}\rightarrow E_{-i+1}$ coming from map of sections $P_{-i}\rightarrow P_{-i+1}$ (because of $C^\infty(M)$-linearity).

Thus, we have an exact sequence of vector bundles $$\cdots \rightarrow E_{-3}\rightarrow E_{-2}\rightarrow E_{-1}\rightarrow TM$$ such that the corresponding map of sections is the resolution of $\mathcal{F}$ that we mentioned above. This exact sequence of vector bundles is what they have called as geometric resolution.

Under some mild conditions on the singular foliations, they were able to prove the following theorem

Theorem $2.4$ : A locally real analytic singular foliation admits a geometric resolution of length at most $\dim(M)+1$ over any relatively compact open subset of $M$.

Such geometric resolution/short exact sequence of vector bundles should remind the notion of Lie $\infty$-algebroid and this is what they say in the next result.

Theorem $2.7$ : Let $\mathcal{F}$ be a singular foliation on $M$ which permits a geometric resolution. Then, there is a (universal) Lie algebroid structure on the resolution.

It also explains in what sense it is "universal".

In this sense, any singular foliation (with mild assumptions) comes from a Lie $\infty$-algebroid, which they call as "the Lie $\infty$-algebroid of a singular foliation".

I can add some more details if anyone want to see.

Even though we may not be able to associate a Lie algebroid with a singular foliation, we can associate a Lie $\infty$-algebroid with a singular foliation (satisfying certain not so strange conditions). This result is due to Camille Laurent-Gengoux, Sylvian Lavau and Thomas Storbl published as "The universal Lie $\infty$-algebroid of a singular foliation", whose arXiv version is available at https://arxiv.org/abs/1806.00475

I will try to explain the basic idea that I understood.

A singular foliation $\mathcal{F}$ is a locally generated $C^\infty(M)$-submodule of $\mathfrak{X}(M)$. Just like any other $R$-module, this $C^\infty(M)$-module $\mathcal{F}$ will also have resolution, $$\cdots\rightarrow P_{-3}\rightarrow P_{-2}\rightarrow P_{-1}\rightarrow \mathcal{F}\rightarrow 0,$$ where $P_{-i}$ are projective $C^\infty(M)$-modules for every $i\geq 1$.

As the singular foliation $\mathcal{F}$ is coming from a geometric structure on the manifold $M$, it is natural to focus on resolutions that comes from some geometric structures on $M$. This is where Serra-Swan theorem comes for help. When ever we have a finitely generated projective $C^\infty(M)$-module $P$, then, it has to be of the form $\Gamma(M,E)$ for some vector bundles $E\rightarrow M$.

Even though $\mathcal{F}$ is a locally finitely generated $C^\infty(M)$-module, we may not assure that there would be a projective resolution $$\cdots\rightarrow P_{-3}\rightarrow P_{-2}\rightarrow P_{-1}\rightarrow \mathcal{F}\rightarrow 0$$ where all these $P_{-i}$ are locally finitely generated. So, the question of $P_{-i}$ being finitely generated may be too much to hope for. They were able to get rid of this locally finitely generated and finitely generated issue by asking that $M$ is a compact manifold.

Such a projective resolution $$\cdots\rightarrow P_{-3}\rightarrow P_{-2}\rightarrow P_{-1}\rightarrow \mathcal{F}\rightarrow 0$$ gives vector bundles $E_{-i}\rightarrow M$ with $\Gamma(M,E_{-i})=P_{-i}$ and a morphism of vector bundles $E_{-i}\rightarrow E_{-i+1}$ coming from map of sections $P_{-i}\rightarrow P_{-i+1}$ (because of $C^\infty(M)$-linearity).

Thus, we have an exact sequence of vector bundles $$\cdots \rightarrow E_{-3}\rightarrow E_{-2}\rightarrow E_{-1}\rightarrow TM$$ such that the corresponding map of sections is the resolution of $\mathcal{F}$ that we mentioned above. This exact sequence of vector bundles is what they have called as geometric resolution.

Under some mild conditions on the singular foliations, they were able to prove the following theorem

Theorem $2.4$ : A locally real analytic singular foliation admits a geometric resolution of length at most $\dim(M)+1$ over any relatively compact open subset of $M$.

It does not end here. I am not very sure but, may be the above result can be proved for any random locally finitely generated $C^\infty(M)$-submodule of $\mathfrak{X}(M)$, with out having the condition of being closed under Lie bracket (in other words $\mathcal{F}$ having structure of a Lie bracket). For obvious reasons, this extra structure on $\mathcal{F}$ should reflect somewhere in the sequence of vector bundles $\cdots \rightarrow E_{-3}\rightarrow E_{-2}\rightarrow E_{-1}\rightarrow TM$. This should remind the notion of Lie $\infty$-algebroid and this is what they say in the next result.

Theorem $2.7$ : Let $\mathcal{F}$ be a singular foliation on $M$ which permits a geometric resolution. Then, there is a (universal) Lie algebroid structure on the resolution.

It also explains in what sense it is "universal".

In this sense, any singular foliation (with mild assumptions) comes from a Lie $\infty$-algebroid, which they call as "the Lie $\infty$-algebroid of a singular foliation".

I can add some more details if anyone want to see.