We give lower bounds on the maximal injectivity radius for a closed hyperbolic 3-manifold with first Betti number 2 under some additional topological hypotheses. The theme of this paper is the connection between topological properties of a closed orientable hyperbolic 3-manifold M and the maximal injectivity radius of M . In [4] we showed that if the first Betti number of M is at least 3, then the maximal injectivity radius of M is at least log 3. By contrast, the best known lower bound for the

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... maximal injectivity radius of M with no topological restriction on M is the lower bound of arcsinh( 1 4 ) = 0.24746 . . . due to Przeworski [7]. One of the results of this paper, Corollary 4, gives a lower bound of 0.32798 for the case where the first Betti number of M is 2 and M does not contain a "fibroid" (see below). Our main result, Theorem 3, is somewhat stronger than this. The proofs of our results combine a result due to Andrew Przeworski [7] with results from [5] and [6] . The results of [5] and [6] were motivated by applications to the study of hyperbolic volume, and these applications were superseded by the results of [2] . The applications presented in the present paper do not seem to be accessible by other methods. As in [5], we define a book of I-bundles to be a compact, connected, orientable topological 3-manifold (with boundary) W which has the form W = P ∪ B, where