# Donaldson’s theorem

Theorem Let $M$ be a smooth simply connected, closed 4- manifold. If the bilinear form induced on $H_{2}(M;\mbox{\boldmathZ})$ by the cup product is positive definite, then it may be represented by the identity matrix.

One application is the existence of exotic $\mbox{\boldmathR}^{4}$’s. See Donaldson Freedman exotic R4.

Title Donaldson’s theorem DonaldsonsTheorem 2013-03-22 15:37:30 2013-03-22 15:37:30 whm22 (2009) whm22 (2009) 9 whm22 (2009) Theorem msc 57R12 msc 14J80 exoticR4s DonaldsonFreedmanexoticR4 DonaldsonFreedmanExoticR4 ExoticR4s