This is not an answer to your question.  I just wanted to point out that there are plenty of  examples of complete Einstein manifolds of infinite topological type.  I am aware of the following two papers at least:

1. [Complete Ricci-flat Kåhler manifolds of infinite topological type][2], by Anderson, Kronheimer and LeBrun

2. [Continued fractions and Einstein manifolds of infinite topological type][1], by Calderbank and Singer




  [1]: http://arxiv.org/abs/math/0508614v1
  [2]: http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.cmp/1104179632