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When compact manifolds X and Y are both even dimensional, their Euler characteristics obey the Kunneth formula χ ( X × Y ) = χ ( X ) χ ( Y ) .In terms of the Betti numbers of b_p ( X ) , χ ( X ) =∑_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 ) = − ∑_p (-1)^p p b_p(Y ). Physical applications include: (1) ρ → (−1)^m ρ under a generalized mirror map in d = 2m+1 dimensions, in analogy with χ → (−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 × 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.
