Smooth integral singular chain complex

Let $M$ be a finite-dimensional Hausdorff second-countable smooth manifold without boundary. For $k\ge 0$, let $C_k^{\mathrm{\infty }}(M;\mathbb{Z})$ be the free abelian group generated by the smooth singular $k$-simplices of $M$, with finite support, and put $C_k^{\mathrm{\infty }}(M;\mathbb{Z})=0$ for $k<0$. For $k\ge 1$, define ${\mathrm{\partial }}_k:C_k^{\mathrm{\infty }}(M;\mathbb{Z})\to C_{k-1}^{\mathrm{\infty }}(M;\mathbb{Z})$ by the alternating sum of affine face restrictions ${\mathrm{\partial }}_k\sigma ={\sum }_{i=0}^k(-1{)}^i\sigma \circ {\delta }_i$, using the face maps of Smooth singular simplex. Set ${\mathrm{\partial }}_0=0$ (and ${\mathrm{\partial }}_k=0$ for $k<0$). The smoothness convention is the neighbourhood-extension convention for smooth singular simplices. Set $Z_k^{\mathrm{\infty }}(M;\mathbb{Z})=\ker {\mathrm{\partial }}_k$, $B_k^{\mathrm{\infty }}(M;\mathbb{Z})=\mathrm{im}{\mathrm{\partial }}_{k+1}$, and $H_k^{\mathrm{\infty }}(M;\mathbb{Z})=Z_k^{\mathrm{\infty }}\mathrm{/}{B}_k^{\mathrm{\infty }}$.

Smooth singular simplex

Singular simplices and singular chain groups with coefficients

Every face of a smooth singular simplex is smooth by composition with its affine face map. For $k\ge 2$, each affine codimension-two face in ${\mathrm{\partial }}_{k-1}{\mathrm{\partial }}_k\sigma$ occurs in the two orders of deleting its vertices, with opposite alternating signs; hence ${\mathrm{\partial }}_{k-1}{\mathrm{\partial }}_k=0$. In degree $1$ this follows from ${\mathrm{\partial }}_0=0$. Thus $B_k^{\mathrm{\infty }}\subseteq Z_k^{\mathrm{\infty }}$, the displayed quotient is defined, and the smooth chains form a subcomplex of the ordinary integral singular chain complex. Constant and degenerate simplices are retained.