Let $M^{4}$ satisfy $|{\rm Ric}_{M^{4}}|\leq 3$, $Vol(M)\geq{\rm v}>0$ and ${\rm diam}(M)\leq D$. Then there exists a decomposition of $M^{4}$

$\displaystyle M^{4}=\mathcal{B}^{1}\cup\bigcup_{j_{2}=1}^{N_{2}}\mathcal{N}^{2}_{j_{2}}\cup\bigcup_{j_{2}=1}^{N_{2}}\mathcal{B}^{2}_{j_{2}}\cup\cdots\cup\bigcup_{j_{k}=1}^{N_{k}}\mathcal{N}^{k}_{j_{k}}\cup\bigcup_{j_{k}=1}^{N_{k}}\mathcal{B}^{k}_{j_{k}}\,,$

(8.51)

into open sets which satisfy the following:

1. (1)

   If $x\in\mathcal{B}^{\ell}_{j}$ then $r_{h}(x)>r_{0}(n,{\rm v},D)\cdot{\rm diam}(\mathcal{B}^{\ell}_{j})$.

2. (2)

   Each neck $\mathcal{N}^{\ell}_{j}$ is diffeomorphic to $\mathds{R}\times S^{3}/\Gamma^{\ell}_{j}$ for some $\Gamma^{\ell}_{j}<O(4)$.

3. (3)

   $\mathcal{N}^{\ell}_{j}\cap\mathcal{B}^{\ell}_{j}$ is diffeomorphic to $\mathds{R}\times S^{3}/\Gamma^{\ell}_{j}$.

4. (4)

   $\mathcal{B}^{\ell-1}_{j^{\prime}}\cap\mathcal{N}^{\ell}_{j}$ are either empty or diffeomorphic to $\mathds{R}\times S^{3}/\Gamma^{\ell}_{j}$.

5. (5)

   $N_{\ell}\leq N({\rm v},D)$ and $k\leq k({\rm v},D)$.