Chapters authored
Smooth Structures on Spin Manifolds in Four Dimensions
The estimate of the coefficient of the magnitude of the signature, which are defined by the number of positive and negative eigenvalues in the inequality representing smooth, oriented, simply connected, compact, spin four-manifolds with indefinite intersection forms can be increased until it is equal to the conjectured value. Therefore, if the intersection form is mE8⊕n0110, the oriented, simply connected, compact, spin four-manifold will admit a smooth structure if and only if n≥32m. The inequality is changed to n≥32m−1−12i, there is a 2i-fold spin covering of a non-spin manifold M given the demonstration of n≥32m for oriented, compact, spin manifolds. A closer examination of the proof reveals that the lower bound for b+ can be increased to 3|k| + 1, where k=315σ for a spin manifold, yielding b2≥118σ+2. The projection of a spin covering to a non-spin manifold yields the lower bound b2≥118σ, which establishes the prediction for the coefficients of intersection forms for this class of smooth, oriented, simply connected, compact four-manifolds.
Part of the book: Manifolds III