# Donaldson Freedman exotic R4

Let $K$ denote the simply connected closed 4- manifold^{} given by

$$K=\{x:y:z:w\in \u2102{P}^{3}|{x}^{4}+{y}^{4}+{z}^{4}+{w}^{4}=0\}$$ |

Let ${E}_{8}$ denote the unique rank 8 unimodular symmetric bilinear form^{} over $\mathbb{Z}$, which is positive definite^{} and with respect to which, the norm of any vector is even. Let $B$ denote the rank 2 bilinear form^{} over $\mathbb{Z}$ which may be represented by the matrix

$$\left(\begin{array}{cc}\hfill 0\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 0\hfill \end{array}\right)$$ |

Then we may regard ${H}_{2}(K;\mathbb{Z})$ as a direct sum $M\oplus N$, where the cup product induces the form ${E}_{8}\oplus {E}_{8}$ on $M$ and $B\oplus B\oplus B$ on $N$ and we have $M$ orthogonal^{} to $N$. (This does not contradict Donaldson’s theorem as $B$ has 1 and -1 as eigenvalues^{}.)

We may choose a (topological) open ball, $U$, in ${\mathrm{\#}}_{3}{S}^{2}\times {S}^{2}$ which contains a (topological) closed ball, $V$, such that we have a smooth embedding, $f:{\mathrm{\#}}_{3}{S}^{2}\times {S}^{2}-V\to K$ satisfying the following property:

The map $f$ induces an isomorphism from ${H}_{2}({\mathrm{\#}}_{3}{S}^{2}\times {S}^{2}-U;\mathbb{Z})$ into the summand $N$ .

If we could smoothly embed ${S}^{3}$ into $U-V$, enclosing $V$, then by replacing the outside of the embedded ${S}^{3}$ with a copy of ${B}^{4}$, and regarding $U-V$ as lying in $K$, we obtain a smooth simply connected closed 4- manifold, with bilinear form ${E}_{8}\oplus {E}_{8}$ induced by the cup product. This contradicts Donaldson’s theorem.

Therefore, $U$ has the property of containing a compact set which is not enclosed by any smoothly embedded ${S}^{3}$. Hence $U$ is an exotic ${\mathbb{R}}^{4}$.

By considering the three copies of $B$ one at a time, we could have obtained our exotic ${\mathbb{R}}^{4}$ as an open subset of ${S}^{2}\times {S}^{2}$.

Title | Donaldson Freedman exotic R4 |
---|---|

Canonical name | DonaldsonFreedmanExoticR4 |

Date of creation | 2013-03-22 15:37:36 |

Last modified on | 2013-03-22 15:37:36 |

Owner | whm22 (2009) |

Last modified by | whm22 (2009) |

Numerical id | 13 |

Author | whm22 (2009) |

Entry type | Application |

Classification | msc 57R12 |

Classification | msc 14J80 |

Related topic | Donaldsonstheorem |

Related topic | exoticR4s |

Related topic | ExoticR4s |

Related topic | DonaldsonsTheorem |