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Missing `\operatorname`; PDF -> abs; inlined links
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LSpice
LSpice
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I don't know if this is quite the answer you're looking for, but elements of $H_3(PSL_2(\mathbb{C}),\mathbb{Z})$$H_3(\operatorname{PSL}_2(\mathbb{C}),\mathbb{Z})$ can be identified as fundamental classes of finite-volume hyperbolic $3$-manifolds. Neumann [1] showed how to construct the Cheeger-Chern-SimonsCheeger–Chern–Simons class as a map $\mathfrak{c} : H_3(\operatorname{PSL}_2(\mathbb{C}),\mathbb{Z}) \to \mathbb{C}/4\pi^2\mathbb{Z}$. Evaluating $i\mathfrak{c}([M])$ on a manifold computes the complex volume, whose real part is the volume of the hyperbolic metric and whose imaginary part is the Chern-SimonsChern–Simons invariant.

To effectively apply this formula (you need some extra combinatorial data called a flattening) it is helpful to use a particular coordinate system on the space of $\operatorname{PSL}_2(\mathbb{C})$-bundles over the manifold, the Ptolemy coordinates [2]. I think that these were originally discovered when thinking about things from the perspective of group homology: see [2, Sections 2 and 3].

[1] Extended Bloch group and the Cheeger–Chern–Simons classExtended Bloch group and the Cheeger–Chern–Simons class, Walter Neumann, https://arxiv.org/pdf/math/0307092.pdf.

[2] The volume and Chern-Simons invariant of a representationThe volume and Chern–Simons invariant of a representation, Christian K. Zickert, https://arxiv.org/abs/0710.2049.

I don't know if this is quite the answer you're looking for, but elements of $H_3(PSL_2(\mathbb{C}),\mathbb{Z})$ can be identified as fundamental classes of finite-volume hyperbolic $3$-manifolds. Neumann [1] showed how to construct the Cheeger-Chern-Simons class as a map $\mathfrak{c} : H_3(\operatorname{PSL}_2(\mathbb{C}),\mathbb{Z}) \to \mathbb{C}/4\pi^2\mathbb{Z}$. Evaluating $i\mathfrak{c}([M])$ on a manifold computes the complex volume, whose real part is the volume of the hyperbolic metric and whose imaginary part is the Chern-Simons invariant.

To effectively apply this formula (you need some extra combinatorial data called a flattening) it is helpful to use a particular coordinate system on the space of $\operatorname{PSL}_2(\mathbb{C})$-bundles over the manifold, the Ptolemy coordinates [2]. I think that these were originally discovered when thinking about things from the perspective of group homology: see [2, Sections 2 and 3].

[1] Extended Bloch group and the Cheeger–Chern–Simons class, Walter Neumann, https://arxiv.org/pdf/math/0307092.pdf

[2] The volume and Chern-Simons invariant of a representation, Christian K. Zickert, https://arxiv.org/abs/0710.2049

I don't know if this is quite the answer you're looking for, but elements of $H_3(\operatorname{PSL}_2(\mathbb{C}),\mathbb{Z})$ can be identified as fundamental classes of finite-volume hyperbolic $3$-manifolds. Neumann [1] showed how to construct the Cheeger–Chern–Simons class as a map $\mathfrak{c} : H_3(\operatorname{PSL}_2(\mathbb{C}),\mathbb{Z}) \to \mathbb{C}/4\pi^2\mathbb{Z}$. Evaluating $i\mathfrak{c}([M])$ on a manifold computes the complex volume, whose real part is the volume of the hyperbolic metric and whose imaginary part is the Chern–Simons invariant.

To effectively apply this formula (you need some extra combinatorial data called a flattening) it is helpful to use a particular coordinate system on the space of $\operatorname{PSL}_2(\mathbb{C})$-bundles over the manifold, the Ptolemy coordinates [2]. I think that these were originally discovered when thinking about things from the perspective of group homology: see [2, Sections 2 and 3].

[1] Extended Bloch group and the Cheeger–Chern–Simons class, Walter Neumann.

[2] The volume and Chern–Simons invariant of a representation, Christian K. Zickert.

mention flattenings
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Calvin McPhail-Snyder
Calvin McPhail-Snyder
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I don't know if this is quite the answer you're looking for, but elements of $H_3(PSL_2(\mathbb{C}),\mathbb{Z})$ can be identified as fundamental classes of finite-volume hyperbolic $3$-manifolds. Neumann [1] showed how to construct the Cheeger-Chern-Simons class as a map $\mathfrak{c} : H_3(\operatorname{PSL}_2(\mathbb{C})),\mathbb{Z}) \to \mathbb{C}/4\pi^2\mathbb{Z}$$\mathfrak{c} : H_3(\operatorname{PSL}_2(\mathbb{C}),\mathbb{Z}) \to \mathbb{C}/4\pi^2\mathbb{Z}$. Evaluating $\mathfrak{c}([M])$$i\mathfrak{c}([M])$ on a manifold computes the complex volume, whose real part is the volume of the hyperbolic metric and whose imaginary part is the Chern-Simons invariant. 

To effectively computeapply this formula (you need some extra combinatorial data called a flattening) it is helpful to use a particular coordinate system on the space of $\operatorname{PSL}_2(\mathbb{C})$-bundles over the manifold, the Ptolemy coordinates [2]. I think that these were originally discovered when thinking about things from the perspective of group homology: see [2, Sections 2 and 3].

[1] Extended Bloch group and the Cheeger–Chern–Simons class, Walter Neumann, https://arxiv.org/pdf/math/0307092.pdf [2]

[2] The volume and Chern-Simons invariant of a representation, Christian K. Zickert, https://arxiv.org/abs/0710.2049

I don't know if this is quite the answer you're looking for, but elements of $H_3(PSL_2(\mathbb{C}),\mathbb{Z})$ can be identified as fundamental classes of finite-volume hyperbolic $3$-manifolds. Neumann [1] showed how to construct the Cheeger-Chern-Simons class as a map $\mathfrak{c} : H_3(\operatorname{PSL}_2(\mathbb{C})),\mathbb{Z}) \to \mathbb{C}/4\pi^2\mathbb{Z}$. Evaluating $\mathfrak{c}([M])$ on a manifold computes the complex volume, whose real part is the volume of the hyperbolic metric and whose imaginary part is the Chern-Simons invariant. To effectively compute this formula it is helpful to use a particular coordinate system on $\operatorname{PSL}_2(\mathbb{C})$-bundles over the manifold, the Ptolemy coordinates [2]. I think that these were originally discovered when thinking about things from the perspective of group homology: see [2, Sections 2 and 3].

[1] Extended Bloch group and the Cheeger–Chern–Simons class, Walter Neumann, https://arxiv.org/pdf/math/0307092.pdf [2] The volume and Chern-Simons invariant of a representation, Christian K. Zickert, https://arxiv.org/abs/0710.2049

I don't know if this is quite the answer you're looking for, but elements of $H_3(PSL_2(\mathbb{C}),\mathbb{Z})$ can be identified as fundamental classes of finite-volume hyperbolic $3$-manifolds. Neumann [1] showed how to construct the Cheeger-Chern-Simons class as a map $\mathfrak{c} : H_3(\operatorname{PSL}_2(\mathbb{C}),\mathbb{Z}) \to \mathbb{C}/4\pi^2\mathbb{Z}$. Evaluating $i\mathfrak{c}([M])$ on a manifold computes the complex volume, whose real part is the volume of the hyperbolic metric and whose imaginary part is the Chern-Simons invariant. 

To effectively apply this formula (you need some extra combinatorial data called a flattening) it is helpful to use a particular coordinate system on the space of $\operatorname{PSL}_2(\mathbb{C})$-bundles over the manifold, the Ptolemy coordinates [2]. I think that these were originally discovered when thinking about things from the perspective of group homology: see [2, Sections 2 and 3].

[1] Extended Bloch group and the Cheeger–Chern–Simons class, Walter Neumann, https://arxiv.org/pdf/math/0307092.pdf

[2] The volume and Chern-Simons invariant of a representation, Christian K. Zickert, https://arxiv.org/abs/0710.2049

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Calvin McPhail-Snyder
Calvin McPhail-Snyder
  • 2.5k
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  • 27

I don't know if this is quite the answer you're looking for, but elements of $H_3(PSL_2(\mathbb{C}),\mathbb{Z})$ can be identified as fundamental classes of finite-volume hyperbolic $3$-manifolds. Neumann [1] showed how to construct the Cheeger-Chern-Simons class as a map $\mathfrak{c} : H_3(\operatorname{PSL}_2(\mathbb{C})),\mathbb{Z}) \to \mathbb{C}/4\pi^2\mathbb{Z}$. Evaluating $\mathfrak{c}([M])$ on a manifold computes the complex volume, whose real part is the volume of the hyperbolic metric and whose imaginary part is the Chern-Simons invariant. To effectively compute this formula it is helpful to use a particular coordinate system on $\operatorname{PSL}_2(\mathbb{C})$-bundles over the manifold, the Ptolemy coordinates [2]. I think that these were originally discovered when thinking about things from the perspective of group homology: see [2, Sections 2 and 3].

[1] Extended Bloch group and the Cheeger–Chern–Simons class, Walter Neumann, https://arxiv.org/pdf/math/0307092.pdf [2] The volume and Chern-Simons invariant of a representation, Christian K. Zickert, https://arxiv.org/abs/0710.2049