Let us recall that an affine structure on manifold $Y$ (smooth, of dimension $n$) is given by a torsion-free flat connection $\nabla$ on the tangent bundle $TY$.

We will give below three equivalent definitions of the notion of an integral affine structure.

It is easy to see that if $Y$ carries a ${\bf Z}$-affine structure then for any point $y\in Y$ there exist small neighborhood $U$, local coordinate system $(x_{1},...,x_{n})$ in $U$ such that $\nabla=d$ in coordinates $(x_{1},...,x_{n})$, and the lattice $(T_{x}Y)^{{\bf Z}},x\in U$ is a free abelian group generated by the tangent vectors $\partial/\partial x_{i}\in T_{x}Y,1\leq i\leq n$. Let us call ${\bf Z}$-affine such a coordinate system in $U$ (sometimes we will call such $U$ a ${\bf Z}$-affine chart). For a covering of $Y$ by ${\bf Z}$-affine charts the transition functions belong (locally) to $GL(n,{\bf Z})\ltimes{\bf R}^{n}$. Explicitly, a change of coordinates is given by the formula

$$x_{i}^{\prime}=\sum_{1\leq j\leq n}a_{ij}x_{j}+b_{i}\,\,,$$

where $(a_{ij})\in GL(n,{{\bf Z}}),(b_{i})\in{{\bf R}}^{n}$.

Hence, Definition 1 is equivalent to the following

In the above definition $Y$ is just a topological manifold, $C^{\infty}$-structure on it can be reconstructed canonically from ${\bf Z}$-affine structure.

We can restate the notion of ${\bf Z}$-affine structure in the language of sheaves of affine functions.

We say that a real-valued function $f$ on ${{\bf R}}^{n}$ is ${\bf Z}$-affine if it has the form

$$f(x_{1},\dots,x_{n})=a_{1}x_{1}+\dots+a_{n}x_{n}+b\,\,,$$

where $a_{1},\dots,a_{n}\in{\bf Z}$ and $b\in{\bf R}$. We will denote by $Aff_{{\bf Z},{{\bf R}}^{n}}$ the sheaf of functions on ${{\bf R}}^{n}$ which are locally ${\bf Z}$-affine.

Equivalence of the last two definitions follows from the observation that a homeomorphism between two open domains in ${{\bf R}}^{n}$ preserving the sheaf $Aff_{{\bf Z},{{\bf R}}^{n}}$ is given by the same formula $x^{\prime}=A(x)+b,A\in GL(n,{{\bf Z}}),b\in{{\bf R}}^{n}$ as the change of coordinates between two ${\bf Z}$-affine coordinate systems.

With a given affine structure on $Y$ we can associate a flat affine connection $\nabla^{aff}$ (see [KN]). The corresponding parallel transport acts on tangent spaces by affine transformations. For a ${\bf Z}$-affine structure the monodromy of $\nabla^{aff}$ belongs to $GL(n,{\bf Z})\ltimes{\bf R}^{n}$, i.e. $\forall y\in Y$ we have a monodromy representation

$$\rho:\pi_{1}(Y,y)\to GL(n,{\bf Z})\ltimes{{\bf R}}^{n}\,\,.$$

Alternatively, we can define the monodromy representation by covering a loop in $Y$ by ${\bf Z}$-affine coordinate charts and composing the corresponding transition functions.

Notice that a ${\bf Z}$-affine structure on $Y$ gives rise to a class

$$[\rho]\in H^{1}(Y,T^{{\bf Z}}\otimes{{\bf R}})=H^{1}(Y,T_{Y}^{\nabla})\,\,\,,$$

where $T_{Y}^{\nabla}\subset T_{Y}$ is the subsheaf of $\nabla$-flat sections¹¹ 1 Here we slightly abuse notations because $Y$ is not necessarily connected.. De Rham representative of class $[\rho]$ is given by a differential $1$-form $\theta\in\Omega^{1}(Y,T_{Y})$ such that $\theta(v)=v$ for any tangent vector $v$. In affine coordinates one has $\theta=\sum_{i}\partial/\partial x_{i}\otimes dx_{i}$. Clearly $\nabla(\theta)=0$.

We will need later an explicit formula for the ${\bf R}$-valued pairing of $[\rho]$ with a closed singular 1-chain with coefficients in the local system $(T^{\ast})^{\bf Z}=(T^{*}Y)^{\bf Z}$, the dual covariant lattice in $T^{\ast}Y$. With any singular $1$-chain $c$ with values in $(T^{\ast})^{\bf Z}$ we associate a real number $j(c)$ in the following way. Suppose that $c$ is given by a continuous map $\gamma:[0,1]\to Y$ and a section $\alpha\in\Gamma([0,1],\gamma^{\ast}(T^{\ast})^{\bf Z})$. Parallel transport via the connection $\nabla^{aff}$ gives rise to a map $\overline{\gamma}:[0,1]\to T_{\gamma(0)}Y,\,\,\,\overline{\gamma}(0)=0$. Let $\alpha_{0}=\alpha(0)\in(T_{\gamma(0)}^{\ast})^{{\bf Z}}\subset T^{*}_{\gamma(0)}Y$. We define $j(c)=\langle\alpha_{0},\overline{\gamma}(1)\rangle$. We extend $j(c)$ to an arbitrary singular $1$-chain $c$ by additivity. Then the class $[\rho]$ can be calculated as $\langle[\rho],[c]\rangle=j(c)$ for any closed $1$-chain $c\in C_{1}(Y,(T^{\ast})^{{\bf Z}})$.