Recently I was trying to understand what’s behind the mysterious condition

for elements () to define an element in the Bloch group. It appears that the condition naturally appears if one studies hyperbolic -manifolds.

Let be an abelian group. Let be a triangulated oriented -dimensional manifold. Let be an oriented -dimensional simplex of the triangulation.

**Definition**

An angle structure on is a collection of elements of , denoted , one for each edge satisfying relations:

We think of as the angle between the faces of meeting at .

We have

This, in particular, implies

Suppose each tetrahedron of the triangulation has an angle structure. The corresponding angles will be denoted , which means the angle between and , i.e. for a simplex we have elements listed below:

Let us define additionally

Therefore the notation is invariant with respect to even permutations of vertices of .

**Definition**

For each edge of let be the -simplices adjacent to with , and . Suppose the following condition is satisfied:

Then we call our manifold *angled*.

Suppose we have a finite set of tetrahedra with angle structure. Is it possible to glue them together and obtain an angled oriented manifold? We provide a necessary condition.

**Theorem**

In an angled oriented manifold the following condition is satisfied in the group :

the sum is over the -simplices which compose the fundamental class of the manifold.

Note that if is invertible in , then the terms and are .

**Remark**

Let and each tetrahedron is realized as an ideal tetrahedron in the hyperbolic -space with cross-ratio . Then its angles are , , . To make the product one can change angles to , , . Then we see that up to -torsion the sum of tensors is , which is well known. In other words, hyperbolic -manifolds provide elements in *the Bloch group*. However our approach seems to be more general.

The rest of this text provides a proof of the theorem.

Suppose is angled. Then we can construct elements with the property

for each oriented simplex . If is another such family then there is a family with

Fix a vertex . Let and be edges. Join and by a sequence of triangles for , , . Put

This does not depend on the choice of the sequence since for any oriented -simplex we have

Therefore there is a family with the property

If is another such family, there exists a family with

In particular for any -simplex we have

We see that we can replace with to make satisfying

If is another family with this condition then there is a family with the property

Put . Then for any oriented -simplex

Therefore there exists a family (this is different from the one used before) with property

This means that for any -simplex we have

Let us summarize the properties of :

Let be a -simplex. Then

Consider the following element in :

This element is invariant under cyclic permutations:

Moreover,

Since and , we can also write as

Let

For any oriented -simplex put

Then

We may rewrite

Therefore

Taking into account that

we obtain

Now we turn to .

Therefore

We see that depends only on the angles and its sum over the manifold is zero. If is invertible in then

This specializes to in the case of ideal hyperbolic tertrahedron.

## 4 comments

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