Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Invariants of hyperbolic 3-manifolds in relative group homology   Let $M$ be a complete oriented hyperbolic $3$--manifold of finite volume.
Using classifying spaces for families of subgroups we construct a class
$\beta_P(M)$ in the Adamson relative homology group
$H_3([PSL_2(\mathbb{C}):\bar{P}];\mathbb{Z})$, where $\bar{P}$ is the subgroup
of parabolic transformations which fix $\infty$ in the Riemann sphere. We also
prove that the classes $F(M)$ in the Takasu relative homology groups
$H_3(PSL_2(\mathbb{C}),\bar{P};\mathbb{Z})$ constructed by Zickert, which are
not well-defined and depend of a choice of decorations by horospheres, are all
mapped to $\beta_P(M)$ via a canonical comparison homomorphism
$H_3(PSL_2(\mathbb{C}),\bar{P};\mathbb{Z})\to
H_3([PSL_2(\mathbb{C}):\bar{P}];\mathbb{Z})$. To do this, we simplify the
construction of the classes $F(M)$ using a simpler complex which computes
$H_3(PSL_2(\mathbb{C}),\bar{P};\mathbb{Z})$, getting a simple simplicial
formula for $F(M)$, which in turn gives a simpler and more efficient formula to
compute the volume and Chern--Simons invariant than the one given by Zickert.
The constructions can be extended for any boundary-parabolic
$PSL_2(\mathbb{C})$-representation.

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The categories of this paper are: physics
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The categories of this paper are: q-alg
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The categories of this paper are: cs
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The categories of this paper are: chao-dyn
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The categories of this paper are: plasm-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: q-fin
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The categories of this paper are: q-bio
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The categories of this paper are: cond-mat
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The categories of this paper are: hep-th
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The categories of this paper are: acc-phys
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The categories of this paper are: nlin
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The categories of this paper are: ao-sci
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The categories of this paper are: econ
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The categories of this paper are: hep-lat
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The categories of this paper are: gr-qc