| Summary: | A smooth manifold which is homeomorphic but not diffeomorphic to R4 is called an exotic R4 . In this paper we construct several examples of such spaces, and define a notion of complexity for smooth manifolds homeomorphic to R4 . In particular, for R homeomorphic to R4 , we define the complexity, m(R), to be the supremum, taken over all compact codimension zero submanifolds K of R, of the minimal first betti number of any smooth 3-manifold Sigma3 which separates K from infinity, i.e., mR=sup K&parl0;min S&parl0;b1&parl0;S&parr0; &parr0;&parr0;∈0,1,2,&ldots;,infinity . Examples are given, and various upper and lower bounds for complexity are computed. We also extend a result of Bizaca and Etnyre by showing how end summing with exotic R4 's of increasing complexity can be used to construct infinitely many smooth structures on any open 4-manifold with at least one topologically collarable end, i.e., an end homeomorphic to Sigma3 x R for some closed 3-manifold Sigma3.
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