Donaldson Freedman exotic R4

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

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

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

$\left(\begin{array}[]{cc}0&1\\ 1&0\end{array}\right)$

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

The map $f$ induces an isomorphism from $H_{2}(\#_{3}S^{2}\times S^{2}\,\,-\,\,U;\mathbb{Z})$ 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