I'm trying to find generating manifolds for the cobordism group $\mathit{MO}_5(K(\mathbb Z/2, 2))\cong (\mathbb Z/2)^4$, which can be represented as the cobordism group of closed 5-manifolds $M$ together with a class $B\in H^2(M;\mathbb Z/2)$. I've found three of the four generators; the fourth should be a 5-manifold $M$ such that $w_1(M)w_2(M)\in H^3(M;\mathbb Z/2)$ is nonzero, and here I've gotten stuck.

Such an $M$ cannot be a product of lower-dimensional manifolds, nor can it be the total space of a fiber bundle over the circle.

I'm happy to hear general approaches or ideas as well, such as ways of modifying a manifold to change its cohomology or Stiefel-Whitney classes in useful ways.

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