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Nov 18, 2011 at 0:11 comment added William This idea of a "twisted double" is exactly what I was thinking of. I didn't want to get my hopes up by asking if you could split $M$ using two copies of the SAME manifold $W$, but I guess in the odd-dimensional case you always can. :) I think the appropriate course of action now would be to check out some of your books!
Nov 18, 2011 at 0:00 vote accept William
Nov 16, 2011 at 7:15 history answered Andrew Ranicki CC BY-SA 3.0