Proof of Theorem 1.4 . [01ZT]

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Proof of Theorem 1.4.

Let $M^{4}$ satisfy $|{\rm Ric}_{M^{4}}|\leq 3$ , ${\rm Vol}(M)>{\rm v}>0$ and ${\rm diam}(M^{4})\leq D$ . Then using Theorem 8.11, we can write

\displaystyle M^{4}\equiv\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}}\,.

(8.64)

First we will analyze each body region $\mathcal{B}^{k}_{j}$ . Indeed, by (1) and theorem 8.2, it follows that there are at most $C({\rm v},D)$ -diffeomorphism types for each $\mathcal{B}^{k}_{j}$ . By (4), there are at most $C({\rm v},D)$ such body regions, and by (2) and (3), there are at most $C({\rm v},D)$ diffeomorphism types that can arise by gluing them together, which proves the theorem. ∎