Fixed point sets of orientation reversing involutions on 3-manifolds (Q919323)

Let \(\tau: M\to M\) be an orientation-reversing involution \((\tau^ 2\) is the identity) of a closed orientable 3-manifold, with fixed point set Fix \(\tau\). The author proves that \[ \dim H_ 1(Fix \tau;{\mathbb{Z}}_ 2)\quad \leq \quad \dim H_ 1(M;{\mathbb{Z}}_ 2)+\beta_ 1(M) \] where the dimension is over \({\mathbb{Z}}_ 2\) and \(\beta_ 1\) denotes the first Betti number. He also observes that this inequality is best possible.