Donaldson Freedman exotic R4

Let K denote the simply connected closed 4- manifoldMathworldPlanetmath given by


Let E8 denote the unique rank 8 unimodular symmetric bilinear formMathworldPlanetmath over , which is positive definitePlanetmathPlanetmath and with respect to which, the norm of any vector is even. Let B denote the rank 2 bilinear formPlanetmathPlanetmath over which may be represented by the matrix


Then we may regard H2(K;) as a direct sum MN, where the cup product induces the form E8E8 on M and BBB on N and we have M orthogonalMathworldPlanetmath to N. (This does not contradict Donaldson’s theorem as B has 1 and -1 as eigenvaluesMathworldPlanetmathPlanetmathPlanetmathPlanetmath.)

We may choose a (topological) open ball, U, in #3S2×S2 which contains a (topological) closed ball, V, such that we have a smooth embedding, f:#3S2×S2-VK satisfying the following property:

The map f induces an isomorphism from H2(#3S2×S2-U;) into the summand N .

If we could smoothly embed S3 into U-V, enclosing V, then by replacing the outside of the embedded S3 with a copy of B4, and regarding U-V as lying in K, we obtain a smooth simply connected closed 4- manifold, with bilinear form E8E8 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 S3. Hence U is an exotic 4.

By considering the three copies of B one at a time, we could have obtained our exotic 4 as an open subset of S2×S2.

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