Why do we study complex orientable cohomology theories

Question

It seems that much of the literature in stable homotopy theory seems to study complex orientable cohomology theories. What is the reason of restricting to this class of multiplicative cohomology theories? Is it simply that they are more computable? Is there a good a priori reason that this is an important class of cohomology theories to study?

Answer 1

There is a sort of a priori reason why one would consider the cohomology theory $MU$, without first knowing of its connection to manifold geometry, to formal groups, … . Since complex-oriented cohomology theories are those cohomology theories with a ring map from $MU$, perhaps having a sufficiently strong interest in $MU$ would suffice to then motivate them.

$\newcommand{\co}{\colon\thinspace}\newcommand{\F}{\mathbb F}\newcommand{\Z}{\mathbb Z}\newcommand{\A}{\mathcal A}\newcommand{\from}{\leftarrow}$ There are a variety of constructions in homotopy theory that allow one to delete a class, where "class" can have different meanings and "delete" different levels of finesse. For instance, if you have a map $f\co S^n \to X$ which on homology sends the fundamental class $\iota_n \in H_n S^n$ to some class $f_* \iota_n \in H_n X$, then the mapping cone $C(f)$ has the property that $f_* \iota_n$ pushes forward to $0$ in $H_n C(f)$. In fact, if $f_* \iota_n$ is not torsion, then $H_* C(f)$ will be exactly $(H_* X) / f_* \iota_n$, as one can see by writing out the homology long exact sequence. By the same token, if $f_* \iota_n$ is $m$–torsion, the same long exact sequence will produce a fresh class in $H_{n+1} C(f)$, due to the death of $m \iota_n$ under $f_*$. This is a general truism in homotopy theory: if a class has been deleted "twice", then you get a fresh class one degree higher.

For a slight twist on this same idea, consider the mod–$p$ cohomology of a space, which is related to its homology by the universal coefficient sequence $$ 0 \to \operatorname{Ext}(H_{n-1}(X; \Z), \F_p) \to H^n(X; \F_p) \to \operatorname{Hom}(H_n(X; \Z), \F_p) \to 0. $$ A torsion-free class in $H_n(X; \mathbb Z)$ contributes a class only to $H^n(X; \F_p)$, but a $p$–power–torsion class contributes classes both to $H^n(X; \F_p)$ and $H^{n+1}(X; \F_p)$. There is actually a process by which one can marry these pairs of classes and reconstruct integral cohomology: the Bockstein spectral sequence has signature $$H^*(X; \F_p) \otimes \F_p[b] \Rightarrow H^*(X; \Z_p),$$ it converges for connected $X$ of finite type, and its differentials perform exactly these marriages. Its construction relies on the facts that $\Z_p$ is $p$–complete and that it participates in the short exact sequence $$0 \to \Z_p \xrightarrow{p \cdot -} \Z_p \to \F_p \to 0,$$ and for us it suffices to leave its explanation at that.

The first differential in the Bockstein spectral sequence takes the form of a stable, additive homomorphism $$\beta\co H^n(X; \F_p) \to H^{n+1}(X; \F_p).$$ In fact, this map is natural in $X$, and such natural transformations in general are called stable cohomology operations. The Steenrod algebra is the collection of all stable mod–$p$ cohomology operations, given by $$\A^* := [H\F_p, H\F_p]_* = H^*(H\F_p; \F_p).$$ This associative algebra is calculable from first principles [1]: $$\A^* = \F_p\langle \beta, P^n \mid j \ge 1 \rangle \, / \, (\text{various relations}),$$ where the angle brackets indicate that these generators do not commute; where $P^n$ is the "$p$th power operation", which sends a cohomology class of degree $2n$ to its $p$th power [2]; and where $\beta$ is the same Bockstein operation as above.

Taking this calculation for granted, one can then make a further calculation of a cousin of these operations: $$H^*(H\Z_p; \F_p) = [H\Z_p, H\F_p]_*.$$ Starting with same the quotient sequence $$H\Z_p \xrightarrow{p} H\Z_p \to H\F_p,$$ and applying $[-, H\F_p]_*$, the multiplication-by-$p$ map induces zero on cohomology, and hence the going-around map participates in a short-exact sequence of $\A$–modules $$0 \from [H\Z_p, H\F_p]_* \from \A^* \from [H\Z_p, H\F_p]_{*+1} \from 0.$$ The first map is onto, hence our $\A^*$–module of interest is cyclic; the second map presents it as a submodule of $\A^*$ generated by a class in degree $1$; and, since $\beta$ is the only operation in degree $1$, we learn $$[H\Z_p, H\F_p]_* = \A^* / (\beta \cdot \A^*).$$ In this sense, $\beta$ witnesses the double-quotient of $H\Z_p$ by $p$, in that the quotient appears on the left- and on the right-hand sides of $[H\F_p, H\F_p] = [H\Z_p / p, H\Z_p / p]$. By removing the quotient from the left and instead studying $[H\Z_p, H\Z_p/p] = [H\Z_p, H\F_p]$, we avoid killing $p$ twice, and $\beta$ disappears.

However, longer monomials in which $\beta$ appears in the middle still survive, and they contribute other odd-degree cohomology operations. One might wonder whether every such odd-degree cohomology operation belongs to some spectrum $X$ with a non-nilpotent endomorphism $v\co \Sigma^n X \to X$ whose quotient recovers $H\F_p$ and which admits an associated Bockstein spectral sequence.

It turns out that a version of this is true: $[MU, H\F_p]_*$ is (almost [3]) the quotient of $\A^*$ with $\beta$ fully deleted. In this sense, $MU$ is the "maximally unquotiented" version of $H\F_p$ within even-concentrated ring spectra. Even more than this, there are multiple theorems along these lines, coming at the same problem from different angles.

• The study of the mod–$p$ cohomology of $MU$ (and its various features, including its relation to that of $H\F_p$) is originally due to Milnor. The homotopy of $MU$ is given by $$\pi_* MU = \Z[x_1, x_2, \ldots],$$ and the generators $x_n$ (again, almost [3]) iteratively play the role of $v$ in the above fantasy resolution of $H\F_p$.
• Starting with the $p$–local sphere, one can iteratively remove its odd-degree homotopy while retaining even-concentrated homology. Priddy showed that this ultimately leads to a spectrum called $BP$, which is an indivisible $p$–local summand of $MU$.
• Starting with the $p$–local sphere, one can iteratively remove its odd-degree homotopy by $A_\infty$–algebra maps. This leads to a sequence of spectra $X(n)_{(p)}$ with $X(\infty)_{(p)} = MU_{(p)}$. The original study of this sequence of spectra is due to Ravenel, then taken up by Hopkins, Devinatz, and Smith, and this perspective in terms of iterated quotients is due to Beardsley.

Each of these objects begins with some desirable properties: the sphere spectrum has pleasant homology (but very knotty homotopy), and the Eilenberg–Mac Lane spectrum has pleasant homotopy (but knotty co/homology). By trying to correct the unpleasant part, one keeps ending up at (a chunk of) $MU$, which has even-concentrated homotopy, even-concentrated homology, even-concentrated co/operations, … . All I mean to point out by this is that $MU$ is an extremely natural object to bump into, especially if one has a preference for even-concentration, whether due to a preference for commutativity over graded-commutativity or due to an aversion toward unnecessarily killing classes twice.

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[1] - Here I'm restricting to odd primes, but you can say all these same words with slightly different formulas at $p = 2$.

[2] - $P^n$ does other, more mysterious things in degrees other than $2n$.

[3] - The precise statement is that the $p$–localization $MU_{(p)}$ splits as a sum of shifts of copies of a ring spectrum $BP$. This new spectrum has homotopy given by $$\pi_* BP = \Z_{(p)}[v_1, v_2, \ldots],$$ $[BP, H\F_p]$ is the submodule of $\A^*$ where $\beta$ is totally deleted, and these $v_j$s are the desired self-maps $v$. (In terms of $\pi_* MU$, $v_j$ is equivalent to $x_{p^j}$ modulo decomposables.) Although $BP$ has all these nice properties, it depends on the prime $p$, and $MU$ is to be thought of as the best integral object capturing all of them at once.

Answer 2

A surface-level motivation for complex oriented cohomology theories is that they are precisely those that admit a theory of generalized chern classes.

More surprisingly, the universal complex oriented theory $MU$ tends to see quite a lot of information about the stable homotopy groups. I am a novice, but there is a ton of history here-- you should really read some of the literature. Novikov used $MU$ in his version of the Adams spectral sequence. Quillen discovered the connection between $MU$ and formal group laws. Landweber exploited this to construct other interesting cohomology theories. Ravenel made a series of conjectures, which were mostly proved by Devinatz, Hopkins and Smith. Broadly these imply that $MU$ sees all non-nilpotent phenomena in stable homotopy theory. Thus, knowledge of $MU$ allows one to compute the Balmer spectrum of the stable homotopy category (analogous to the prime numbers for abelian groups). Because of the connection between formal group laws and $MU$, this spectrum can be interpreted in terms of formal group laws. To each point of the spectrum, there are cohomology theories, originally constructed by Morava $K(n,p)$ and $E(n,p)$. Various properties of these cohomology theories can be interpreted in terms of the algebraic geometry of the space (stack) of formal group laws. This is just a superficial glimpse of the picture of chromatic homotopy theory that we have today.