An Exotic 4-manifold

In [1] we have constructed a fake smooth structure on a contractible 4-manifold W relative to boundary. This is a smooth manifold V with d V = d W such that the identity map d V —• d W extends to a homeomorphism but not to a difFeomorphism V -+ W. This is a relative result in the sense that V itself is diffeomorphic to W, even though no such diffeomorphism can extend the identity map on the boundary. Here we strengthen this result by dropping the boundary hypothesis at the expense of slightly enlarging W: We construct two compact smooth 4-manifolds Qx, Q2 which are homeomorphic but not diffeomorphic to each other. In particular no diffeomorphism dQ{ —• dQ2 can extend to a diffeomorphism Q{ -• Q2. Let Q*, / = 1, 2, be the 4-manifolds obtained by attaching 2-handles to B along knots Kn i = 1, 2, with -hi-framing (see Figures 1 and 2). Clearly Q{ and Q2 are homotopy equivalent to CP0 2 = CP int(5 4 ), and it will be shown that dQx = dQ2.