7
$\begingroup$

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

$\endgroup$
3
  • 1
    $\begingroup$ I haven't thought enough about this, but isn't a complex-orientable cohomology theory precisely one where compact complex manifolds are orientable, for some reasonable notion of orientability of a manifold ? $\endgroup$ Feb 11, 2021 at 9:19
  • 3
    $\begingroup$ @MaximeRamzi Not quite, it's one where complex vector bundles are 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 $E^*(BU_n)=E^*[[c_1,\dots,c_n]]$). $\endgroup$ Feb 11, 2021 at 20:19
  • $\begingroup$ @DenisNardin : aah right, I should have thought about it some more ! Thanks ! :) $\endgroup$ Feb 11, 2021 at 22:39

1 Answer 1

12
$\begingroup$

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.

$\endgroup$
4
  • $\begingroup$ Okay, thanks a lot! So (if I understand well) the definition of complex orientable cohomology theory is motivated (actually equivalent) by the fact that the AHSS for $\mathbb{C}P^{\infty}$ degenerates at the second page? $\endgroup$ Feb 11, 2021 at 9:59
  • $\begingroup$ So one could proceed as follows: since the singular cohomology (resp. K-theory) of $\mathbb{C}P^{\infty}$ is a power series with coefficient ring $\mathbb{Z}$ (resp. $K(pt)$) and this follows from the degeneracy of the AHSS, I ask myself: what conditions should I put on a generalized cohomology in order to make the AHSS of $\mathbb{C}P^{\infty}$ converge? Answer: I need precisely the conditions expressed in the definition of complex orientable cohomology. Am I right? $\endgroup$ Feb 11, 2021 at 10:08
  • $\begingroup$ Yes, exactly! (assuming "converge" in your last message was a typo and you meant "degenerate", it always converges, but might have differentials) $\endgroup$ Feb 11, 2021 at 11:55
  • $\begingroup$ Yes, of course I meant degenerates, thank you! $\endgroup$ Feb 11, 2021 at 12:39

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.