Unoriented analogue of elliptic genera (Q1343893)

Let \(I\) be the ideal of the unoriented bordism ring \({\mathcal N}_*\) consisting of all bordism classes \([\mathbb{R} P(\xi^{2m})]\) of the total spaces of odd dimensional real projective space bundles over closed manifolds. The author proves that a multiplicative genus \(\phi : {\mathcal N}_* \to \mathbb{Z}/2\mathbb{Z}\) annihilates \(I\) if and only if its logarithm \(g(u)\) is given by \[ g(u) = \int^u_0 \;{du\over 1 + \phi([\mathbb{R} P_2])u^2} . \] This is the unoriented analogue of a well known result of Ochanine in oriented bordism.