Donaldson Freedman exotic R4
Let denote the simply connected closed 4- manifold given by
Let denote the unique rank 8 unimodular symmetric bilinear form over , which is positive definite and with respect to which, the norm of any vector is even. Let denote the rank 2 bilinear form over which may be represented by the matrix
We may choose a (topological) open ball, , in which contains a (topological) closed ball, , such that we have a smooth embedding, satisfying the following property:
The map induces an isomorphism from into the summand .
If we could smoothly embed into , enclosing , then by replacing the outside of the embedded with a copy of , and regarding as lying in , we obtain a smooth simply connected closed 4- manifold, with bilinear form induced by the cup product. This contradicts Donaldson’s theorem.
Therefore, has the property of containing a compact set which is not enclosed by any smoothly embedded . Hence is an exotic .
By considering the three copies of one at a time, we could have obtained our exotic as an open subset of .
|Title||Donaldson Freedman exotic R4|
|Date of creation||2013-03-22 15:37:36|
|Last modified on||2013-03-22 15:37:36|
|Last modified by||whm22 (2009)|