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Let $p$ be a natural number, and for $i\in \{0, ..., p-1\}$, denote the irreducible rank one complex representation of $\mathbb{Z}/p$. by $\rho_{i}$. Let $a=(a_{1},\ldots a_{d}) $ be a sequence of natural numbers between 1 and p-1, and consider $ L^{2d-1}(a)$, the Lens space defined as the quotient of the unitary sphere in the rank $d$ complex representation $\rho_{a_{1}}\oplus \ldots \oplus \rho_{a_{d}}$. Is the $\hat{A}$- genus

$$ \hat{A}(L^{2d-1}(a) )$$ documented somewhere in the literature? in particular, is there a closed formula in terms of the $a_{i}$ for each dimension $2d-1$?

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    $\begingroup$ Isn't the A-hat genus zero for manifolds whose dimension is not divisible by 4, more or less by definition? $\endgroup$
    – Oscar Randal-Williams
    Commented May 28 at 6:59
  • $\begingroup$ As Oscar says, the $\hat{A}$-genus isn't defined in odd dimensions. But you can find the Pontrjagin classes by the same method as for projective spaces. see Ewing, John; Moolgavkar, Suresh; Smith, Larry; Stong, R. E. Stable parallelizability of lens spaces. J. Pure Appl. Algebra10 (1977/78), no.2, 177–191. At that point, you can work out the A-polynomial, which is perhaps what you mean. $\endgroup$
    – Danny Ruberman
    Commented May 28 at 16:14

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