Odd dimensional analogue of the Euler characteristic

Abstract: When compact manifolds $X$ and $Y$ are both even dimensional, their Euler characteristics obey the K\"unneth formula $\chi(X\times Y)=\chi(X) \chi(Y)$. In terms of the Betti numbers $b_p(X)$, $\chi(X)=\sum_{p}(-1)^p b_p(X)$, implying that $\chi(X)=0$ when $X$ is odd dimensional. We seek a linear combination of Betti numbers, called $\rho$, that obeys an analogous formula $\rho(X\times Y)=\chi(X) \rho(Y)$ when $Y$ is odd dimensional. The unique solution is $\rho(Y)=-\sum_{p}(-1)^p p b_p(Y)$. Physical applications include: (1) $\rho \rightarrow (-1)^m \rho $ under a generalized mirror map in $d=2m+1$ dimensions, in analogy with $\chi \rightarrow (-1)^m \chi $ in $d=2m$; (2) $\rho$ appears naturally in compactifications of M-theory. For example, the 4-dimensional Weyl anomaly for M-theory on $X^4 \times Y^7$ is given by $\chi(X^4)\rho(Y^7)=\rho(X^4 \times Y^7) $ and hence vanishes when $Y^7$ is self-mirror. Since, in particular, $\rho(Y\times S^1)=\chi(Y)$, this is consistent with the corresponding anomaly for Type IIA on $X^4 \times Y^6$, given by $\chi(X^4)\chi(Y^6)=\chi(X^4 \times Y^6)$, which vanishes when $Y^6$ is self-mirror; (3) In the partition function of $p$-form gauge fields, $\rho$ appears in odd dimensions as $\chi$ does in even.