On a generalized Petrie's conjecture via index theorems (Q390399)

Petrie's Conjecture [\textit{T. Petrie}, Invent. Math. 20, 139--146 (1973; Zbl 0262.57021)] states that if a homotopy complex projective space admits a smooth effective circle action, then its Pontrjagin class is standard. The conjecture is known to be true only in special cases.NEWLINENEWLINEThe author proves related results for spaces that are homotopy equivalent to products of complex projective spaces, \(M \simeq \prod_{i=1}^{m} \mathbb CP^{n_i}\). The cohomology of such a space is \(H^*(M, \mathbb Z) = \mathbb Z[x_1, \dots, x_m]/(x_i^{n_i+1})_{i=1}^m\). The concept of a torus manifold was introduced in [\textit{A. Hattori} and \textit{M. Masuda}, Osaka J. Math. 40, No. 1, 1--68 (2003; Zbl 1034.57031)]. The author showsNEWLINENEWLINE{Theorem } Let \(M\) be a torus manifold which is homotopy equivalent to \(\prod_{i=1}^{m} \mathbb CP^{n_i}\). If \(m\geq 2\) assume furthermore that one of the following two conditions holds:NEWLINENEWLINEThe \(\widehat{\mathcal A}\)--genus \(\widehat{\mathcal A}(M)\) does not vanish.NEWLINENEWLINEThe Todd genus \(Td (M)\) of \(M\) does not vanish.NEWLINENEWLINEThen the Pontrjagin class of \(M\) is of the standard form: NEWLINE\[NEWLINE p(M) = \prod_{i=1}^m (1+x_i^2)^{n_i+1}. NEWLINE\]