Suppose a Hilbert manifold $M$ is covered by the union of $\infty$-balls (in the sense of Baire spaces), namely $M=\bigcup_{\alpha\in A}B^{\infty}$, without knowledge of intersections. The only requirement is that the union covers $M$. Let $\Sigma$ be a finite set of points in $\mathbb{R}^{\infty}$. For $r>0$, consider balls $B_r(x)=x+rB^{\infty}$ for each $x\in \Sigma$. The Čech complex is $\text{Čech}(r)=\left\{\sigma\subseteq\Sigma\big|\bigcap_{x\in\sigma}B_r(x)\ne\emptyset\right\}$. The Čech complex should look something like this:
That is, we are allowed to arrange the balls in a configuration of our choosing, so long as the configuration still covers $M$. (Indeed, we can "pull" the balls apart as much as possible so that they still cover $M$-an optimal configuration-with the least amount of balls used).
How do we compute homology groups $H_k(M;\mathbb{Z})$ from the Čech complex $\text{Čech}(r)$ for $B^{\infty}$ balls covering the Hilbert manifold $M$?
Any help would be much appreciated. Thanks in advance!