# Poincaré's cube manifolds

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The simplest cube manifold is probably the 3-torus. It was described before Poincaré by W. Killing in the context of his work on Clifford-Klein space-forms ([Killing1893, 288 - 289]). The faces of a cube are identified by three translations. Since these translations generate a subgroup of the isometries of ordinary $3$$3$-space which acts discontinuously and without fixed points, the resulting $3$$3$-torus - the name is neither used by Killing nor by Poincaré - is a locally flat 3D space-form.

In his work on Kleinean functions, which are special types of automorphic functions ([Poincaré1883]), Poincaré had met the situation that the faces of a polyhedron are identified in pairs ([Poincaré1953, 274-279]). The difference here is that those polyhedra were hyperbolic polyhedra whereas in his toplogical work the geometric situation is not taken into account. It should be mentioned that Poincaré announced his cube manifolds in the short Comptes rendus note of 1893 which opened his series of papers on topology ([Poincaré1893]). This is a hint to the importance which Poincaré attributed to those manifolds.

Here is Poincaré's description of the 3-torus: Cf. Figure 1. The coordinates of the points $A,B,C,D,A',B',C',D'$$A,B,C,D,A',B',C',D'$ are (0,0,0), (0,1,0), (1,0,0), (1,1,0), (0,0,1), (0,1,1), (1,0,1), (1,1,1) (actually this is only important in the sixth example).

Figure 1. Poincaré's description of the $3$$3$-torus ([Poincaré1953, 231]

In Poicaré's paper there is no picture - pictures are rather rare in his work! Here is one: see Figure 2.

Figure 2. Identifications of the cube

With the help of the relations defined by the identification scheme - the generators of the fundamental group of the torus are the three translations - Poincaré could calculate the fundamental group. In modern notation he arrived at the result that it is isomorphic to $\Z \times \Z \times \Z$$\Z \times \Z \times \Z$. Because this fundamental group is abelian, it was easy to compute the first Betti number as $4$$4$ and by duality also the second (equal to 4 also). It should be noted that Poincaré's Betti numbers are modern Betti numbers plus 1.

The second example given by Poincaré is one in which the identifications don't provide a manifold - so it is negative in a sense. The third example is more interesting: Cf. Figure 3.

Figure 3. Poincaré's third example

Here the translations are combined with rotations of $90^\circ$$90^\circ$. There are two cycles of vertices: $A\equiv{B'}\equiv{C'}\equiv{D}$$A\equiv{B'}\equiv{C'}\equiv{D}$ and $B\equiv{D'}\equiv{A'}\equiv{C}$$B\equiv{D'}\equiv{A'}\equiv{C}$ and four cycles of edges: Cf. Figure 4.

Figure 4. Edge cycles in Poincaré's third example

The calculation of the fundamental group shows that this is isomorphic to the group of units of the Hamiltonians. Therefore the 3-manifold of example 3 was later called (by Seifert and Threlfall ([Seifert&Threlfall1934, 198]; cf. also [Threlfall&Seifert1931, 60]) the quaternionian space´´. Poincaré himself related the group he had calculated to the symetries of the hypercube in 4-space. Calculating the Betti-numbers Poincaré found 1 and (by duality) 1. But there is a torsion coefficient in dimension one which wasn't noticed by Poincaré in his paper of 1895; this mistake was corrected in the first complement dating from 1900 ([Poincaré1900], [Poincaré1953, 292]).

The identification scheme of Poincaré's fourth example is the following: cf. Figure 5:

Figure 5. Poincaré's fourth example

There is only one cycle of vertices, that is, all vertices are identified but there are three cycles of edges: Cf. Figure 6:

Figure 6. Edge cycles in Poincaré's fourth example

Poincaré did not arrive at an explicit representation of the fundamental group of example four. He calculated only the Betti numbers as 2 not taking into account torsion.

The fifth example in Poincaré's paper is derived from an octahedron, it yields the projective space.

The sixth example is a whole family of cube manifolds; its role in Poincaré's paper is central because he shows with this construction that there are 3-manifolds with same Betti numbers but different fundamental groups. So the fundamental group is a stronger invariant then the Betti numbers.

The family of 3-manifolds which Poincaré called the sixth example is defined by three "substitutions" (that is mappings).

$\displaystyle f_1: {(x,y,z)}\mapsto{(x+1,y,z)}$
$\displaystyle f_2: {(x,y,z)}\mapsto{(x,y+1,z)}$
$\displaystyle f_3: {(x,y,z)}\mapsto{(\alpha\cdot{x}+\beta\cdot{y},\gamma\cdot{x}+\delta\cdot{y},z+1)}$

The numbers $\alpha$$\alpha$, $\beta$$\beta$, $\gamma$$\gamma$ and $\delta$$\delta$ are integers such that $\alpha \cdot \delta$$\alpha \cdot \delta$ - $\beta \cdot \gamma$$\beta \cdot \gamma$ equals 1. In modern language this means that the matrix $\left( \begin{array}{ll} \alpha & \beta \\ \gamma & \delta \end{array} \right)$$\left( \begin{array}{ll} \alpha & \beta \\ \gamma & \delta \end{array} \right)$ is an element of $SL(2,\Zz)$$SL(2,\Zz)$. We denote this matrix by $M(\alpha, \beta, \gamma, \delta)$$M(\alpha, \beta, \gamma, \delta)$. In contrast to the manner in which the other examples were constructed - labeled by Poincaré as "geometric" - the sixth example is defined by the action of a group that acts discontinuously and without fixpoints - Poincaré called this "Representation by a discontinuous group" ([Poincaré1953, 236]). In modern language the sixth example is an orbit space. From a geometric point of view there are two translations ($f_1$$f_1$ and $f_2$$f_2$) as in the case of the $3$$3$-torus (identiying the left face with the right face, the face in front with that in behind) plus a combination of a translation and a a composition of transvections which determines the identifiaction of the bottom with the top.

Poincaré gave some concrete examples. If we choose the matrix as the identity matrix we get the torus, if we choose it to describe a rotation about $90^\circ$$90^\circ$ we get the fourth example. Poincaré himself added the following example: If $\left( \begin{array}{ll} \alpha & \beta\\ \gamma & \delta \end{array} \right)$$\left( \begin{array}{ll} \alpha & \beta\\ \gamma & \delta \end{array} \right)$ equals $\left(\begin{array}{ll} 1 & 1 \\ 0 & 1 \end{array} \right)$$\left(\begin{array}{ll} 1 & 1 \\ 0 & 1 \end{array} \right)$ then the identification scheme is as follows: Cf. Figure 7.

Figure 7. The identification scheme for the special case of Poincaré's sixth example

The fundamental group of the manifold one gets - we call it $V(\alpha,\beta,\gamma,\delta)$$V(\alpha,\beta,\gamma,\delta)$ - is isomorphic to the group which is generated by the three defining transformations. By a long and complicated calculation Poincaré was able to calculate the first Betti number of the sixth example. His results was the following:

If $\left( \begin{array}{ll} \alpha & \beta\\ \gamma & \delta \end{array} \right)$$\left( \begin{array}{ll} \alpha & \beta\\ \gamma & \delta \end{array} \right)$ is $\left( \begin{array}{ll} 1 & 0\\ 0 & 1 \end{array} \right)$$\left( \begin{array}{ll} 1 & 0\\ 0 & 1 \end{array} \right)$, then $\beta_1(V(\alpha,\beta,\gamma,\delta))=4$$\beta_1(V(\alpha,\beta,\gamma,\delta))=4$; if $\alpha+\delta=2$$\alpha+\delta=2$ but the matrix $M(\alpha,\beta,\gamma,\delta)$$M(\alpha,\beta,\gamma,\delta)$ is not the identity matrix, then $\beta_1(M(\alpha,\beta,\gamma,\delta))=3$$\beta_1(M(\alpha,\beta,\gamma,\delta))=3$. In all other cases $\beta_1(M(\alpha,\beta,\gamma,\delta))=2$$\beta_1(M(\alpha,\beta,\gamma,\delta))=2$.

In order to get the result he was looking for - two manifolds with the same first and second Bettti numbers (always by duality) but different fundamental groups - Poincaré had to study the fundamental groups of the manifolds $V(\alpha,\beta,\gamma,\delta)$$V(\alpha,\beta,\gamma,\delta)$. For two such matrices, $M$$M$ and $M'$$M'$, the fundamental groups, $G$$G$ and $G'$$G'$, of the corresponding manifolds are isomorphic to the groups of deck transformations generated by the three substitutions´´ $f_1$$f_1$, $f_2$$f_2$ and $f_3$$f_3$: this is because the covering is simply connected. Poincaré arrived at the following result: The groups $G$$G$ and $G'$$G'$ are isomorphic if and only if the matrices $M$$M$ and $M'$$M'$ are in the same conjugacy class in $SL(2,\Zz)$$SL(2,\Zz)$. So Poincaré's problem is solved: An example is given by the matrices

$\displaystyle \left( \begin{array}{ll} 1&h \\ 0&1 \end{array} \right) \quad \text{and} \quad \left( \begin{array}{ll} 1&h'\\ 0&1 \end{array} \right),$

if the two numbers $h$$h$ and $h'$$h'$ have different absolute value. As a concrete example we may choose the matrices

$\displaystyle \left( \begin{array}{ll} 1&1\\ 0&1 \end{array} \right) \quad \text{and} \quad \left( \begin{array}{ll} 1&2 \\ 0&1 \end{array} \right)$

and so on.

Remark 0.1.[Sarkaria1996] corrected Poincaré's result: The conjugacy classes must be taken in $GL(2,Z)$$GL(2,Z)$ and the matrices and their inverses have to be taken into account. But the concrete conclusion of Poincaré is not harmed by this correction.

In his dissertation (1931) Seifert came back to Poincaré's sixth example in a different context. Imagine the two identifications by $f_1$$f_1$ and $f_2$$f_2$ are made. Then the result is the part of 3-space between two concentric tori. The identification by $f_3$$f_3$ has to glue the inner torus to the outer. Such a situation was called a Schalenraum´´ (shell space) by Seifert ([Seifert1931, 29]). For more details see [Volkert2002] (historical information), [Evans&Moser1972] (group theory), [Hempel1976, 127] (Poincaré's examples in another context) and [Sarkaria1996] (more on Poincaré's sixth example).