Even rank of middle cohomology in dimension four k plus two
= Even rank of middle cohomology in dimension four k plus two
{title2=$b_{2k+1}(M^{4k+2})\text{ is even}$}
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