Some New Four-Manifolds

Abstract

The quotient space Q = X'/Z, of the involution is a smooth manifold of the (simple) homotopy type of real projective 4-space P', but not diffeomorphic or even piecewise linear (PL) homeomorphic to P'. As a corollary of the theorem and the vanishing of L,(Z2, -), there are precisely two s-cobordism classes of homotopy 4-dimensional real projective spaces. The manifold Q is obtained from Pi by removing the complement of a tube around P2 c P', a non-orientable 3-disk bundle over S', and replacing it by a certain bundle over S' with fibre the complement of an open cell in the 3-torus T3. In [6] we gave a classification theory of 4-manifolds, modulo connected sum with copies of S2 x S2, extending the results of Wall for the simplyconnected case. In particular, the connected sum of Q with many copies of S2 x SI is diffeomorphic to the manifold constructed in [6, 2.4], and the connected sum of Q with arbitrarily many copies of S2 x S2 is not PL homeomorphic or even h-cobordant to the connected sum of Pi with arbitrarily many copies of S2 x S2. Let V be a manifold with dim V > 1, and dim V > 2 if Vhas non-empty boundary. Then one can show, using topological surgery and the open h-cobordism theorem, that Pi x V and Q x V are topologically homeomorphic. It is not known if any of the manifolds Q constructed below are actually homeomorphic to P'. It is also not known if their universal covering spaces are diffeomorphic, PL homeomorphic, or even just homeomorphic to the 4-sphere S. 1 Partially supported by NSF.