**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](http://people.math.harvard.edu/%7Elurie/252x.html)


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



Motivation for the definition of complex orientable cohomology theory
As you wrote, complex orientability can be characterized by the cohomology of $\mathbb{C}P^\infty$: $E$ is complex orientable if $E^\*(\mathbb{C}P^\infty)$ splits according to the cell structure of $\mathbb{C}P^\infty$, i.e. $E$ doesn't "see" the attaching maps of $\mathbb{C}P^\infty$.
For example, this happens whenever $E\_\*$ is concentrated in even degrees, for degree reasons. This is the case for singular cohomology and complex K-theory. To contrast this with a negative example: For real K-theory, we don't get such a splitting: The attaching map of the $4$-cell of $\mathbb{C}P^\infty$ is the Hopf map $\eta$, which acts nontrivially on $KO$.


This kind of splitting can be expressed in terms of the Atiyah-Hirzebruch spectral sequence, and using the multiplicative structure you have on there, and the fact that singular cohomology of $\mathbb{C} P^\infty$ is polynomial, the question of whether the Atiyah-Hirzebruch spectral sequence splits boils down to the existence of a single element in $E^2(\mathbb{C}P^\infty)$ (which corresponds to the $2$-cell). The choice of such an element is precisely the usual definition of complex orientation.


