Motivation for the definition of complex orientable cohomology theory

Question

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

Answer 1

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.