Let's fix a $(B,f)$ structure with Thom spectra $MB$. I'd like to know the condition for a not-necessarily-multiplicative generalized cohomology theory $E^\*$ such that for a fibration $X\to Y$ with $(B,f)$ structure with $n$-dimensional fiber to have the pushforward map $E^\*(X)\to E^{\*-n}(Y)$.


When $E$ is multiplicative, the condition is that there is a homomorphism $MB\to E$. When $E$ is not multiplicative, is the condition that $E$ is a module of $MB$ enough? I think it is...



pushforward in non-multiplicative generalized cohomology theory
Yes, and is explained in detail in any textbook. See e.g. Rudyak "On Thom Spectra, Orientability, and Cobordism", Chapter 5, Section 2.


