Even rank of middle cohomology in dimension four k plus two ($b_{2k+1}(M^{4k+2})\text{ is even}$)

For a closed oriented manifold, the rational middle-dimensional cup-product pairing is nondegenerate by Poincare duality. Its degree is odd, so graded commutativity makes it skew-symmetric. A nondegenerate skew-symmetric form has even dimension. In particular a closed oriented six-manifold cannot have a rank-one third cohomology group.

// Target: algebra.bigb