**PRELIMINARY DEFINITIONS:**

Let $E^*$ be a multiplicative generalized cohomology theory. By the suspension isomorphism we have: $$ \tilde{E^2}(S^2)\cong\tilde{E^0}(S^0)=E^0(pt) $$ So there is a special element $t\in\tilde{E^2}(S^2)$ which corresponds to $1_E\in E^0(pt)$ under the above isomorphism.

$E^*$ is said to be **complex orientable** if the inclusion map $i:S^2=\mathbb{C}P^1\to\mathbb{C}P^{\infty}$ induces a sujective morphism $i^*:\tilde{E^2}(\mathbb{C}P^{\infty})\to\tilde{E^2}(S^2)$. A pair $(E^*,x_E)$ of a complex orientable cohomology theory $E^*$ and a choice of an element $x_E\in\tilde{E^2}(\mathbb{C}P^{\infty})$ such that $i^*(x_E)=t$ is called an **oriented cohomology theory**.

It is well known that singular cohomology, complex K-theory and complex cobordism are examples of orientable cohomology theories.

**QUESTION**:

I would like to motivate the previous definition. In particular, I would like to motivate this definition through some classical result for singular cohomology/ K-theory. So, I am searching for an answer of this kind:*"since for singular cohomology/K-theory holds theorem X (and we want to generalize this stuff to generalized cohomology), then it is natural to define oriented cohomology theories as above"*

**My attempt:** according to various textbooks, for example [1, Theorem 3.10], we have the following classical result: for any complex oriented cohomology theory $(E^*,x_E)$, there is an isomorphism
$$
E^*(\mathbb{C}P^{\infty})\cong E^*(pt)[[x_E]]
$$

this allows you to define the first Chern class in $E^*$-cohomology and other good stuff follows, for example you can associate to each cohomology theory a formal group law (see [2]). So, the definition of complex orientable cohomology given above is the good one if you want to define Chern classes for generalized cohomologies.

Any other ideas? Thank you in advance.

**REFERENCES:**

[1] A.Kono, D.Tamaki-Generalized cohomology

[2] J.Lurie-Chromatic homotopy theory course http://people.math.harvard.edu/~lurie/252x.html

[3] M.Hopkins-Complex oriented chomology theories and the language of staks https://people.math.rochester.edu/faculty/doug/otherpapers/coctalos.pdf

vector bundlesare orientable :). That is, it is a cohomology theory with a good theory of Chern classes (this is implicit in the current accepted answer, but essentially it boils down to the computation $E^*(BU_n)=E^*[[c_1,\dots,c_n]]$). $\endgroup$