A fake compact contractible 4-manifold

A Fake Compact Contractible 4-manifold

Here we construct a fake smooth structure on a compact contractible 4-manifold W , where W is a well-known Mazur manifold obtained by attaching in two-handle to S x B along its boundary as in Figure 1. * Here we use the conventions of [2]. The results of this paper imply: Theorem 1. There is a smooth contractible 4-manifold V with dV = d W, such that V is homeomorphic but not diffeomorphic to W...

Constructing a Fake 4-manifold by Gluck Construction to a Standard 4-manifold

LET S3 G S’ denote the twisted S3 bundle over S’. In this paper we will demonstrate an imbedding of a 2-sphere f:S* 4 S3 G S’ #S* x S* such that twisting S3 g S’ #S’ x S* along f(S*) (Gluck construction) produces a fake manifold M4. In fact M4 coincides with the fake S3 2 S’ #S* x S*‘s of [l] and 123. More specifically, if IV;= M-int(B3 g S’) (a fake B3 z S’ # S* x S*) and Q4 is the Cappell-Sha...

On images of continuous functions from a compact manifold to Euclidean space

A Hyperbolic 4-Manifold

There is a regular 4-dimensional polyhedron with 120 dodecahedra as 3-dimensional faces. (Coxeter calls it the "120-cell".) The group of symmetries of this polyhedron is the Coxeter group with diagram: For each pair of opposite 3-dimensional faces of this polyhedron there is a unique reflection in its symmetry group which interchanges them. The result of identifying opposite faces by these refl...

Decomposing a 4-manifold

A splitting of a manifold in Kirby’s problem list (no.4.97) is given Gauge theory provides many fake copies of oriented closed smooth manifolds X with b2 (X) odd. Unfortunately, there is no such example with b + 2 (X) even. In problem 4.97 of Kirby’s problem list [4] the author suggested possible a candidate for such an example. Here we show that our example fails to be fake. Along the way we s...