I'm a complementarian and Perriman is an egalitarian, but I still found this to be a very good book and worthwhile read. Perriman offers very good and thought-provoking exegesis. He is more gracious and evenhanded with traditionalist scholarship than most egalitarians I've read. He opens the book with a study of κεφαλη and argues against both standard readings. I think he demolishes pretty well the egalitarian argument that κεφαλη means "source". I didn't find his argument against κεφαλη as "authority" nearly so convincing. Perriman argues that κεφαλη has the sense of "pre-eminence" and he makes a good case in some instances, but not all. I'm happy to grant that he is correct in some instances, but it seems that in many cases he ends up sacrificing the authority inherent in Jesus' lordship. Perriman undercuts a common egalitarian argument, but actually gives complementarians ammunition they didn't realise they had. Perriman also examines at length the various New Testament connections between the early chapters of Genesis and the role of men and women in the Church. There's some great exegesis here, but it's not always convincing. What I appreciate is that Perriman shows that Paul meant what he said rather than trying to explain it all away. In the end he relies on a "cultural" argument to justify his egalitarianism, arguing that the limitations Paul places on women are not theological, but cultural: Paul insisted that the churches not offend Jews or Greco-Romans by allowing women to assume positions that would have caused a scandal. Interestingly, Perriman also makes some tentative comments noting that due to the baggage of radical feminism today we might be wise to impose limitations similar to those of Paul.
So Perriman does, in fact, argue that Paul restricted the role of women in the churches, but that he did so that cultural sensibilities not be offended. This is where the argument breaks down for me. I simply cannot buy the argument that Paul was willing to sacrifice principle for the sake of not offending secular or Jewish culture. Everything about the Gospel offends that culture and in every other respect Paul condemns compromise.