Čech (co)homology

In this post we briefly recall the construction of Čech cohomology as well as compute a few examples. Let $X$ be a topological space with a cover $\mathcal U = \{U_i\}$, $\mathcal F$ a $C$-valued sheaf on $X$, and $\widehat{\mathcal F}$ a $C$-valued cosheaf on $X$, for some category $C$ (usually abelian groups).

Definition: The nerve $N$ of $\mathcal U$ is the simplicial complex that has an $r$-simplex $\rho$ for every non-empty intersection of $r+1$ opens of $\mathcal U$. The support $U_\rho$ of $\rho$ is this non-empty intersection. The $r$-skeleton $N_r$ of $N$ is the collection of all $r$-simplices.

Remark: The sheaf $\mathcal F$ and cosheaf $\widehat {\mathcal F}$ may be viewed as being defined either on the opens of $\mathcal U$ over $X$, or on the nerve $N$ of $\mathcal U$. Indeed, the inclusion map $V\hookrightarrow U$ on opens is given by the forgetful map $\partial$. That is, $\partial_i:N_r\to N_{r-1}$ forgets the $i$th open defining $\rho\in N_r$, so if $U_\rho = U_0\cap \cdots \cap U_r$, then $U_{\partial_0\rho} = U_1\cap\cdots \cap U_r$.

The Čech (co)homology will be defined as the (co)homology of a particular complex, whose boundary maps will be induced by, equivalently, the inclusion map on opens or $\partial_i$ on simplices.

Definition: In the context above:

• a $p$-chain is a finite formal sum of elements $a_{\sigma_i}\in \widehat{\mathcal F}(U_{\sigma_i})$, for every $\sigma_i$ a $p$-simplex,
• a $q$-cochain is a finite formal sum of elements $b_{\tau_j}\in \mathcal F(U_{\tau_j})$, for every $\tau_j$ a $q$-simplex,
• the $p$-differential is the map $d_p:\check C_p(\mathcal U,\mathcal F) \to \check C_{p-1}(\mathcal U,\mathcal F)$ given by

$$d_p(a_\sigma) = \sum_{i=0}^p (-1)^i \widehat{\mathcal F}(\partial_i)(a_\sigma),$$

• the $q$-codifferential is the map $\delta^q:\check C^q(\mathcal U,\mathcal F) \to \check C^{q+1}(\mathcal U,\mathcal F)$ given by

$$\delta^q(b_\tau) = \sum_{j=0}^{q+1} (-1)^j \mathcal F(\partial_j)(b_\tau).$$ The collection of $p$-chains form a group $\check C_p(\mathcal U,\mathcal F)$ and the collection of $q$-cochains also form a group $\check C^q(\mathcal U,\mathcal F)$, both under the respective group operation in each coordinate. The Čech homology $H_*(\mathcal U,\mathcal F)$ is the homology of the chain complex of $\check C_p$ groups, and the Čech cohomology $H^*(\mathcal U,\mathcal F)$ is the cohomology of the cochain complex of $\check C^q$ groups.

Example: Let $X=S^1$ with a cover $\mathcal U = \{U,V,W\}$ and associated nerve $N_{\mathcal U}$ as below.

The cover is chosen so that all intersections are contractible. Let $k$ be a field. Let $\widehat{\mathcal F}$ be a cosheaf over $N$ and $\mathcal F$ a sheaf over $N$, with $\widehat {\mathcal F}(\text{0-cell})=\mathcal F(\text{1-cell}) = (1,1)\in k^2$ and $\widehat{\mathcal F}(\text{1-cell})=\mathcal F(\text{0-cell})=1\in k$, so that the natural extension and restriction maps work. Then all the degree 0 and 1 chain and cochain groups are $k^3$. Giving a counter-clockwise orientation to $X$, we easily see that
$$\begin{align*} d_1\sigma_{U\cap V} & = \sigma_V-\sigma_U, & \delta^0\sigma_U & = \sigma_{U\cap V}-\sigma_{W\cap U}, \\ d_1\sigma_{V\cap W} & = \sigma_W-\sigma_V, & \delta^0\sigma_V & = \sigma_{V\cap W}-\sigma_{U\cap V}, \\ d_1\sigma_{W\cap U} & = \sigma_U-\sigma_W, & \delta^0\sigma_W & = \sigma_{W\cap U}-\sigma_{V\cap W}.\end{align*}$$ If we give an ordered basis of $(\sigma_{U\cap V},\sigma_{V\cap W},\sigma_{W\cap U})$ to $\check C_1(\mathcal U,\widehat{\mathcal F})$ and $\check C^1(\mathcal U,\mathcal F)$, and $(\sigma_U,\sigma_V,\sigma_W)$ to $\check C_0(\mathcal U,\widehat{\mathcal F})$ and $\check C^0(\mathcal U,\mathcal F)$, we find that
$$d_1 = \begin{bmatrix} -1 & 0 & 1 \\ 1 & -1 & 0 \\ 0 & 1 & -1 \end{bmatrix} \sim \begin{bmatrix} 1 & 0 & -1 \\ 0 & 1 & -1 \\ 0 & 0 & 0 \end{bmatrix}, \hspace{1cm} \delta^0 = \begin{bmatrix} -1 & 1 & 0 \\ 0 & -1 & 1 \\ 1 & 0 & -1 \end{bmatrix} \sim \begin{bmatrix} 1 & 0 & -1 \\ 0 & 1 & -1 \\ 0 & 0 & 0 \end{bmatrix}.$$
The Čech chain and cochain complexes are then
$$0 \to \check C_1(\mathcal U,\widehat{\mathcal F}) \tov{d_1} \check C_0(\mathcal U,\widehat{\mathcal F}) \to 0, \hspace{1cm} 0 \to \check C^0(\mathcal U,\mathcal F) \tov{\delta^0} \check C^1(\mathcal U,\mathcal F) \to 0,$$ for which
$$\begin{align*} H_1(\mathcal U,\widehat{\mathcal F}) & = \ker(d_1) = k, & H^0(\mathcal U,\mathcal F) & = \ker(\delta^0) = k, \\ H_0(\mathcal U,\widehat{\mathcal F}) & = k^3/\im(d_1) = k^3/k^2 = k, & H^1(\mathcal U,\mathcal F) & = k^3/\im(\delta^0) = k^3/k^2 = k.\end{align*}$$ By the Čech-de Rham theorem, we know that the (co)homology groups should agree with the usual groups for $S^1$, as $\mathcal U$ was a good cover, which they do. Next we compute another example with a view towards persistent homology.

Definition: Let $X$ be a topological space and $f:X\to Y$ a map with $\mathcal U$ covering $f(X)$. The Leray sheaf $L^i$ of degree $i$ over $N_{\mathcal U}$ is defined by $L^i(\sigma) = H^i(f^{-1}(U_\sigma))$ and $L^i(\sigma\hookrightarrow \tau) = H^i(f^{-1}(U_\tau)\hookrightarrow f^{-1}(U_\sigma))$, whenever $\sigma$ is a face of $\tau$.

Theorem (Curry, Theorem 8.2.21): In the context above, if $N_{\mathcal U}$ is at most 1-dimensional, then for any $t\in \R$,
$$H^i(f^{-1}(-\infty,t])\cong H^0((-\infty,t],L^i)\oplus H^1((-\infty,t],L^{i-1}).$$
The idea is to apply this theorem in a filtration, for different values of $t$, but in the example below we will have $t$ large enough so that $X\subset f^{-1}(-\infty,t]$.

Example: Let $f:S^1\to \R$ be a projection map, and let $X = f(S^1)$ with a cover $\mathcal U = \{U,V\}$ as below.

Note that although $f^{-1}(U)\cap f^{-1}(V)$ is not contractible, $U\cap V$ is, and the Čech cohomology will be over $\mathcal U\subset \R$, so we are fine in applying the Čech-de Rham theorem. It is immediate that the only non-zero Leray sheaves are $L^0$, for which
$$L^0(\sigma_U) = k,\hspace{1cm} L^0(\sigma_V) = k,\hspace{1cm} L^0(\sigma_{U\cap V}) = k^2,$$ hence $\check C^0(\mathcal U,L^0)=\check C^1(\mathcal U,L^0) = k^2$. Giving $\check C^0(\mathcal U,L^0)$ the ordered basis $(\sigma_U,\sigma_V)$ and noting the homology maps $H^0(f^{-1}(U)\hookrightarrow f^{-1}(U\cap V))$ and $H^0(f^{-1}(V)\hookrightarrow f^{-1}(U\cap V))$ are simply $1\mapsto (1,1)$, the \v Cech complex is
$$0 \to \check C^0(\mathcal U,L^0) \tov{\left[\begin{smallmatrix}-1 & -1 \\ 1 & 1 \end{smallmatrix}\right]} \check C^1(\mathcal U,L^0) \to 0.$$
Hence $H^0(\mathcal U,L^0)=\ker(\delta^0)=k$ and $H^1(\mathcal U,L^0)=k^2/\im(\delta^0)=k^2/k=k$, allowing us to conclude, using Curry's and the Čech--de Rham theorems, that
$$\begin{align*} H^0(S^1) & \cong H^0(\mathcal U,L^0) \oplus H^1(\mathcal U,L^{-1}) = k\oplus 0 = k, \\ H^1(S^1) & \cong H^0(\mathcal U,L^1) \oplus H^1(\mathcal U,L^0) = 0\oplus k = k, \\ H^2(S^1) & \cong H^0(\mathcal U,L^2) \oplus H^1(\mathcal U,L^1) = 0\oplus 0=0,\end{align*}$$ as expected.

References: Bott and Tu (Differential forms in algebraic topology, Section 10), Bredon (Sheaf theory, Section VI.4), Curry (Sheaves, cosheaves, and applications, Section 8)