Cobordism of manifolds with \(\mathbb{Z}_2\times\mathbb{Z}_2\)-action having isolated stationary points (Q1964670)

This paper considers a smooth fibring \(p: E\to M\) of manifolds with \(\mathbb{Z}_2\times \mathbb{Z}_2\) action having only isolated fixed points. If \(p^{-1}(x)\) bounds for every fixed point \(x\in M\), then the action on \(E\) bounds equivariantly. The result is much easier than the author's proof indicates. For an action of \(\mathbb{Z}_2\times \mathbb{Z}_2\) with isolated fixed points the action bounds if and only if the fixed set has even Euler characteristic and equivalently if and only if the manifold has even Euler characteristic. Since each \(p^{-1}(x)\) bounds, its fixed set has even Euler characteristic, so the fixed set of \(E\) also has even Euler characteristic.