Odd dimensional analogue of the Euler characteristic

A bstract When compact manifolds X and Y are both even dimensional, their Euler characteristics obey the Künneth formula χ ( X × Y ) = χ ( X ) χ ( Y ). In terms of the Betti numbers b p ( X ), χ ( X ) = Σ 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 × Y ) = χ ( X ) ρ ( Y ) when Y is odd dimensional. The unique solution is ρ ( Y ) = − Σ p ( − 1) p pb p ( Y ). Physical applications include: (1) ρ → ( − 1) m ρ under a generalized mirror map in d = 2 m + 1 dimensions, in analogy with χ → ( − 1) m χ in d = 2 m ; (2) ρ appears naturally in compactifications of M-theory. For example, the 4-dimensional Weyl anomaly for M-theory on X 4 × Y 7 is given by χ ( X 4 ) ρ ( Y 7 ) = ρ ( X 4 × Y 7 ) and hence vanishes when Y 7 is self-mirror. Since, in particular, ρ ( Y × S 1 ) = χ ( Y ), this is consistent with the corresponding anomaly for Type IIA on X 4 × Y 6 , given by χ ( X 4 ) χ ( Y 6 ) = χ ( X 4 × 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.