Odd dimensional analogue of the Euler characteristic

Abstract

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