Odd dimensional analogue of the Euler characteristic

When compact manifolds X and Y are both even dimensional, their Euler characteristics obey the Kfinneth formula chi(X x Y) = chi(X)chi(Y). In terms of the Betti numbers b(p)(X), chi(X) = Sigma(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 x Y) = chi(X)rho(Y) when Y is odd dimensional. The unique solution is rho(Y) = Sigma(p)(-1)(p)pb(p) (Y). Physical applications include: (1) rho -> (-1)(m)rho under a generalized mirror map in d = 2m+1 dimensions, in analogy with chi -> (-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 x Y-7 is given by chi(X-4)rho(Y-7) = rho(X-4 x Y-7) and hence vanishes when Y-7 is self-mirror. Since, in particular, rho(Y x S-1) = chi(Y), this is consistent with the corresponding anomaly for Type IIA on X-4 x Y-6, given by chi(X-4)chi(Y-6) = chi(X-4 x Y-6), which vanishes when Y-6 is self-mirror; (3) In the partition function of p-form gauge fields, p appears in odd dimensions as x does in even.