group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
geometry, complex numbers, complex line
$dim = 1$: Riemann surface, super Riemann surface
A complex oriented cohomology theory is a Whitehead-generalized cohomology theory which is oriented on all complex vector bundles. Examples include ordinary cohomology, complex topological K-theory, elliptic cohomology and cobordism cohomology.
The collection of all complex oriented cohomology theories turns out to be parameterized over the moduli stack of formal group laws. The stratification of this stack by the height of formal group leads to the stratification of complex oriented cohomology theory by “chromatic level”, a perspective also known as chromatic homotopy theory.
For more detailed introduction see at Introduction to Cobordism and Complex Oriented Cohomology.
Write $\mathbb{C}P^\infty \simeq B U(1) \simeq K(\mathbb{Z},2)$ for the infinite complex projective space, equivalently the classifying space for circle group-principal bundles (an Eilenberg-MacLane space); write $S^2$ for the 2-sphere and write
for a representative of $1 \in \mathbb{Z} \simeq \pi_2(B U(1))$, classifying the universal complex line bundle. Regard both $S^2$ and $B U(1)$ as pointed homotopy types and take $i$ to be a pointed morphism.
Let $E^\bullet$ be a multiplicative cohomology theory, i.e. a functor $X \mapsto \pi_\bullet[X,E]$ for $E$ a ring spectrum. Write $\tilde E^\bullet$ for the corresponding reduced cohomology on pointed topological spaces, such that for any pointed space $X$ there is a canonical direct sum decomposition (this prop.)
By the suspension isomorphism there is an identification
with the commutative ring underlying $E$. Write $1 \in \pi_0(E)$ for the multiplicative identity element in this ring.
(complex oriented cohomology theory)
A multiplicative cohomology theory $E$ is complex orientable if the following equivalent conditions hold:
The morphism
is surjective.
The morphism on reduced cohomology
is surjective.
The ring unit $1 \in \pi_0(E)$ is in the image of the morphism $\tilde i^\ast$.
A complex orientation on a multiplicative cohomology theory $E^\bullet$ is an element
(the “first generalized Chern class”) such that
Since $B U(1) \simeq K(\mathbb{Z},2)$ is the classifying space for complex line bundles, it follows that a complex orientation on $E^\bullet$ induces an $E$-generalization of the first Chern class which to a complex line bundle $\mathcal{L}$ on $X$, classified by $\phi \colon X \to B U(1)$ assigns the class $c_1(\mathcal{L}) \coloneqq \phi^\ast c_1^E$. This construction extends to a general construction of $E$-Chern classes (Conner-Floyd Chern classes).
(complex $E$-orientation by extensions and their obstructions)
In terms of classifying maps, Def. means that a complex orientation $c_1^E$ in $E$-cohomology theory is equivalently an extension (in the classical homotopy category) of the map $\Sigma^2 1 \,\colon\, \mathbb{C}P^1 \longrightarrow \Omega^{\infty-2} E$ (which classifies the suspended identity in the cohomology ring) along the canonical inclusion of complex projective spaces
Notice that the complex projective spaces form a cotower
where each inclusion stage is (by this Prop., see at cell structure of projective spaces) the coprojection of a pushout of topological spaces (or rather: of pointed topological spaces) of the form
(where $h^{2n+1}_{\mathbb{C}}$ is the complex Hopf fibration in dimension $2n+1$) hence of a homotopy pushout of underlying homotopy types (rather: of pointed homotopy types) of this form:
Therefore, a complex orientation by extension (1) is equivalently the homotopy colimiting map of a sequence
of finite-stage extensions
Moreover, by the defining universal property of the homotopy pushout, the extension $c_1^{E,n+1}$ of $c_1^{E,n}$ is equivalently a choice of homotopy which trivializes the pullback of $c_1^{E,n}$ to the 2n+1-sphere:
This means, first of all, that the non-triviality of the pullback class
is the obstruction to the existence of the extension/orientation at this stage.
It follows that if these obstructions all vanish, then a complex $E$-orientation does exist.
A sufficient condition for this is, evidently, that the reduced $E$-cohomology of all odd-dimensional spheres vanishes, hence, that the graded $E$-cohomology ring $E_\bullet$ is trivial in odd degrees (an even cohomology theory).
(see also Lurie, Lecture 6, Remark 4)
(complex $E$-orientations from first $E$-Chern class)
To see that a generalized Chern class $c^E_1$ in the above sense indeed has to do with complex orientations, namely of, first of all, fibers of complex line bundles, notice that any complex projective space is equivalently the Thom space of the dual tautological line bundle (see there) on the complex projective space of one complex dimension lower:
The morphism of E-cohomology Milnor sequences that this transformation induces shows (see the proof of Prop. for details) that pullback along the zero section identifies the $E$-cohomology of the Thom space of the universal complex line bundle with that of the base classifying space:
(In fact the zero-section of the Thom space of the universal complex line bundle is itself a weak homotopy equivalence, see this Lemma.)
But this means that the $E$-Chern class $c^E_1$ has a unique pre-image which is an $E$-Thom class $th^E_1$ on the universal complex line bundle, and hence pulls back to an $E$-Thom class on any complex line bundle.
In fact more is true: The choice of $c^E_1$ induces a sequence of universal Conner-Floyd Chern classes $c^E_n$ for all $n \in \mathbb{N}$, and by an elaboration of the previous argument these are pullbacks along the zero-section of universal $E$-Thom classes on the universal complex vector bundle of any rank $n$ (see this Lemma).
Complex orientation in the above sense is indeed universal MU-orientation in generalized cohomology:
For $E$ a homotopy commutative ring spectrum, there is a bijection between complex orientations of $E$-cohomology and homotopy commutative ring spectrum-homomorphisms $MU \longrightarrow E$ out of MU.
(Hopkins 99, section 4, Lurie, lecture 6, theorem 8)
See at universal complex orientation on MU.
One might hope that the above universal property can be refined to say that that if $E$ is an $E_\infty$-ring spectrum, then complex orientations of $E$ are in bijection with $E_\infty$ ring maps $MU \to E$. This is not the case; Ando classified $H_\infty$ ring maps $MU \to E$ in his thesis and in particular showed that not every complex orientation gives rise to an $H_\infty$ ring map out of $MU$. More recently, Hopkins and Lawson have classified the further structure constituting an $E_\infty$ ring map out of $MU$.
Examples of complex orientable cohomology theories:
For $E = H \mathbb{Z}$ the Eilenberg-MacLane spectrum, the ordinary first Chern class
defines a complex orientation of $H\mathbb{Z}$.
For $E = KU$ complex topological K-theory, then the class of the image of the universal complex line bundle $\mathcal{O}(1)$ in reduced K-theory is a complex orientation.
The induced formal group law (by prop. ) is the multiplicative formal group law.
For details see at topological K-theory the section Complex orientation and Formal group law.
(complex cobordism)
For $E = MU$ complex cobordism cohomology theory, the canonical map
defines a complex orientation.
Brown-Peterson cohomology$E = B P^\bullet$.
Given a complex oriented cohomology theory $(E^\bullet, c^E_1)$ according to def. , then there are isomorphisms of graded rings
$E^\bullet(B U(1)) \simeq E^\bullet(\ast)[ [ c_1^E ] ]$
(between the $E$-cohomology ring of $B U(1)$ and the formal power series (but see remark ) in one generator of even degree over the $E$-cohomology ring of the point);
$E^\bullet(B U(1) \times B U(1)) \simeq E^\bullet(\ast)[ [ c_1^E \otimes 1 , 1 \otimes c_1^E ] ]$.
We may realize the classifying space $B U(1)$ as the infinite complex projective space $\mathbb{C}P^\infty = \underset{\longleftarrow}{\lim}_n \mathbb{C}P^n$ (exmpl.). There is a standard CW-complex-structure on the classifying space $\mathbb{C}P^\infty$, given by inductively identifying $\mathbb{C}P^{n+1}$ with the result of attaching a single $2n$-cell to $\mathbb{C}P^n$ (prop.). With this structure, the unique 2-cell inclusion $i \;\colon\; S^2 \hookrightarrow \mathbb{C}P^\infty$ is identified with the canonical map $S^2 \to B U(1)$.
Then consider the Atiyah-Hirzebruch spectral sequence for the $E$-cohomology of $\mathbb{C}P^n$.
Since (prop.) the ordinary cohomology with integer coefficients of projective space is
where $c_1$ represents a unit in $H^2(S^2, \mathbb{Z})\simeq \mathbb{Z}$, and since similarly the ordinary homology of $\mathbb{C}P^n$ is a free abelian group (prop.), hence a projective object in abelian groups (prop.), the Ext-group vanishes in each degree ($Ext^1(H_n(\mathbb{C}P^n), E^\bullet(\ast)) = 0$) and so the universal coefficient theorem (prop.) gives that the second page of the spectral sequence is
By the standard construction of the Atiyah-Hirzebruch spectral sequence (here) in this identification the element $c_1$ is identified with a generator of the relative cohomology
(using, by the above, that this $S^2$ is the unique 2-cell of $\mathbb{C}P^n$ in the standard cell model).
This means that $c_1$ is a permanent cocycle of the spectral sequence (in the kernel of all differentials) precisely if it arises via restriction from an element in $E^2(\mathbb{C}P^n)$ and hence precisely if there exists a complex orientation $c_1^E$ on $E$. Since this is the case by assumption on $E$, $c_1$ is a permanent cocycle. (For the fully detailed argument, see (Pedrotti 16)).
The same argument applied to all elements in $E^\bullet(\ast)[c]$, or else the $E^\bullet(\ast)$-linearity of the differentials (prop.), implies that all these elements are permanent cocycles.
Since the AHSS of a multiplicative cohomology theory is a multiplicative spectral sequence (prop.) this implies that the differentials in fact vanish on all elements of $E^\bullet(\ast) [c_1] / (c_1^{n+1})$, hence that the given AHSS collapses on the second page to give
or in more detail:
Moreover, since therefore all $\mathcal{E}_\infty^{p,\bullet}$ are free modules over $E^\bullet(\ast)$, and since the filter stage inclusions $F^{p+1} E^\bullet(X) \hookrightarrow F^{p}E^\bullet(X)$ are $E^\bullet(\ast)$-module homomorphisms (prop.) the extension problem trivializes, in that all the short exact sequences
split (since the Ext-group $Ext^1_{E^\bullet(\ast)}(\mathcal{E}_\infty^{p,\bullet},-) = 0$ vanishes on the free module, hence projective module $\mathcal{E}_\infty^{p,\bullet}$).
In conclusion, this gives an isomorphism of graded rings
A first consequence is that the projection maps
are all epimorphisms. Therefore this sequence satisfies the Mittag-Leffler condition (def., exmpl.) and therefore the Milnor exact sequence for generalized cohomology (prop.) finally implies the claim:
where the last step is this prop..
There is in general a choice to be made in interpreting the cohomology groups of a multiplicative cohomology theory $E$ as a ring:
a priori $E^\bullet(X)$ is a sequence
of abelian groups, together with a system of group homomorphisms
one for each pair $(n_1,n_2) \in \mathbb{Z}\times\mathbb{Z}$.
In turning this into a single ring by forming formal sums of elements in the groups $E^n(X)$, there is in general the choice of whether allowing formal sums of only finitely many elements, or allowing arbitrary formal sums.
In the former case the ring obtained is the direct sum
while in the latter case it is the Cartesian product
These differ in general. For instance if $E$ is ordinary cohomology with integer coefficients and $X$ is infinite complex projective space $\mathbb{C}P^\infty$, then (prop.)
and the product operation is given by
for all $n_1, n_2$ (and zero in odd degrees, necessarily). Now taking the direct sum of these, this is the polynomial ring on one generator (in degree 2)
But taking the Cartesian product, then this is the formal power series ring
A priori both of these are sensible choices. The former is the usual choice in traditional algebraic topology. However, from the point of view of regarding ordinary cohomology theory as a multiplicative cohomology theory right away, then the second perspective tends to be more natural;
The cohomology of $\mathbb{C}P^\infty$ is naturally computed as the inverse limit of the cohomolgies of the $\mathbb{C}P^n$, each of which unambiguously has the ring structure $\mathbb{Z}[c_1]/((c_1)^{n+1})$. So we may naturally take the limit in the category of commutative rings right away, instead of first taking it in $\mathbb{Z}$-indexed sequences of abelian groups, and then looking for ring structure on the result. But the limit taken in the category of rings gives the formal power series ring (see here).
See also for instance remark 1.1. in Jacob Lurie: A Survey of Elliptic Cohomology.
(topological K-theory of classifying space of circle group) For the case of topological K-theory $E =$ KU, the cohomology ring $KU^\bullet(B U(1)) \;\simeq\; \mathbb{Z}[ [ c_1^{KU}] ]$ (Prop. ) may also be computed via the Atiyah-Segal completion theorem, see the Example there.
Let again $B U(1)$ be the classifying space for complex line bundles, modeled, in particular, by infinite complex projective space $\mathbb{C}P^\infty)$.
There is a continuous function
which represents the tensor product of line bundles in that under the defining equivalence, and for $X$ any paracompact Hausdorff space (notably a CW-complex, since all CW-complexes are paracompact Hausdorff spaces), then
where $[-,-]$ denotes the hom-sets in the (Serre-Quillen-)classical homotopy category and $\mathbb{C}LineBund(X)_{/\sim}$ denotes the set of isomorphism classes of complex line bundles on $X$.
Together with the canonical point inclusion $\ast \to \mathbb{C}P^\infty$, this makes $\mathbb{C}P^\infty$ an abelian group object in the classical homotopy category (an abelian H-group).
By the Yoneda lemma (the fully faithfulness of the Yoneda embedding) there exists such a morphism $\mathbb{C}P^\infty \times \mathbb{C}P^\infty \longrightarrow \mathbb{C}P^\infty$ in the classical homotopy category. But since $\mathbb{C}P^\infty$ admits the structure of a CW-complex (prop.) it is cofibrant in the standard model structure on topological spaces, as is its Cartesian product with itself (prop.). Since moreover all spaces are fibrant in the classical model structure on topological spaces, it follows (by this lemma) that there is an actual continuous function representing that morphism in the homotopy category.
That this gives the structure of an abelian group object now follows via the Yoneda lemma from the fact that each $\mathbb{C}LineBund(X)_{/\sim}$ has the structure of an abelian group under tensor product of line bundles, with the trivial line bundle (wich is classified by maps factoring through $\ast \to \mathbb{C}P^\infty$) being the neutral element, and that this group structure is natural in $X$.
The space $B U(1) \simeq \mathbb{C}P^\infty$ has in fact more structure than that of an H-group from lemma . As an object of the homotopy theory represented by the classical model structure on topological spaces, it is a 2-group, a 1-truncated infinity-group.
Let $(E, c_1^E)$ be a complex oriented cohomology theory. Under the identification
of pullback in $E$-cohomology along the maps from lemma constitutes a 1-dimensional graded-commutative formal group law (exmpl.) over the graded commutative ring $\pi_\bullet(E)$ (prop.). If we consider $c_1^E$ to be in degree 2, then this formal group law is compatibly graded.
The associativity and commutativity conditions follow directly from the respective properties of the map $\mu$ in lemma . The grading follows from the nature of the identifications in prop. .
That the grading of $c_1^E$ in prop. is in negative degree is because by definition
(rmk.).
Under different choices of orientation, one obtains different but isomorphic formal group laws.
The formal group law of complex cobordism cohomology theory, example is universal in that for every commutative ring $R$ there is a natural bijection
$MU^\bullet$ is the Lazard ring.
This is Milnor-Quillen's theorem on MU (involving Lazard's theorem).
The formal group law of Brown-Peterson cohomology theory, example is universal for $p$-local cohomology theories in that $\mathbb{G}_{B P}$ is universal among $p$-local, p-typical formal group laws.
For $E$ a complex oriented cohomology theory and $n \in \mathbb{N}$, restriction along the canonical map
induces an isomorphism
of $E^\bullet(B U(n))$ with the cyclic group-invariants in $E^\bullet((B U(1))^n)$, hence with the power series ring in the elementary symmetric polynomials $c_i^E$ (the generalized Chern classes) in the $c_1^E$-s (the generalized first Chern classes of prop. ).
Use this proposition to reduce to the situation for ordinary Chern classes. (e.g. Lurie 10, lecture 4)
The follows says that complex oriented cohomology theories in the sense of def. , indeed canonically have an orientation in generalized cohomology for the (spherical fibration of) any complex vector bundle.
For more details see at universal complex orientation on MU.
For $E$ any cohomology theory and $n \in \mathbb{N}$, $n \geq 1$, there is a canonical isomorphism of relative cohomology
where $\zeta_n \coloneqq E U(n) \underset{U(n)}{\times} \mathbb{R}^{2n}$ is the universal complex vector bundle.
Observe that the sphere bundle $S(\zeta_n) \to B U(n)$ of the universal complex vector bundle is equivalently the canonical map $B U(n-1) \to B U(n)$.
This follows form the fact that $S^{2n-1} \simeq U(n)/U(n-1)$ and that hence the unit sphere bundle is equivalently the quotient of the $U(n)$-universal principal bundle by $U(n-1)$
The unit ball bundle $B(\zeta_n)$ is weakly equivalent to $B U(n)$, and under this identification the map $S(\zeta_n) \to B(\zeta_n)$ is equivalent to $B U(n-1) \to B U(n)$.
For $E$ a complex oriented cohomology theory, its $n$th generalized Chern class $c^E_n$, prop. , identified as an element of $E^\bullet(B(\zeta_n), S(\zeta_n))$ via prop. , is a Thom class.
(e.g. Lurie 10, lecture 5, prop. 6)
chromatic level | complex oriented cohomology theory | E-∞ ring/A-∞ ring | real oriented cohomology theory |
---|---|---|---|
0 | ordinary cohomology | Eilenberg-MacLane spectrum $H \mathbb{Z}$ | HZR-theory |
0th Morava K-theory | $K(0)$ | ||
1 | complex K-theory | complex K-theory spectrum $KU$ | KR-theory |
first Morava K-theory | $K(1)$ | ||
first Morava E-theory | $E(1)$ | ||
2 | elliptic cohomology | elliptic spectrum $Ell_E$ | |
second Morava K-theory | $K(2)$ | ||
second Morava E-theory | $E(2)$ | ||
algebraic K-theory of KU | $K(KU)$ | ||
3 …10 | K3 cohomology | K3 spectrum | |
$n$ | $n$th Morava K-theory | $K(n)$ | |
$n$th Morava E-theory | $E(n)$ | BPR-theory | |
$n+1$ | algebraic K-theory applied to chrom. level $n$ | $K(E_n)$ (red-shift conjecture) | |
$\infty$ | complex cobordism cohomology | MU | MR-theory |
Textbook accounts:
Pierre Conner, Edwin Floyd, around Theorem 7.6 of: The Relation of Cobordism to K-Theories, Lecture Notes in Mathematics 28 Springer 1966 (doi:10.1007/BFb0071091, MR216511)
Frank Adams, part II, section 2 of: Stable homotopy and generalised homology, 1974
John Michael Boardman, Section 5 of: Stable Operations in Generalized Cohomology [pdf, pdf] in: Ioan Mackenzie James (ed.) Handbook of Algebraic Topology Oxford 1995 (doi:10.1016/B978-0-444-81779-2.X5000-7)
Stanley Kochmann, section 4.3 of: Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996
Doug Ravenel, Section 4.1 of: Complex cobordism and stable homotopy groups of spheres
Dai Tamaki, Akira Kono, Section 3.2 in: Generalized Cohomology, Translations of Mathematical Monographs, American Mathematical Society, 2006 (ISBN: 978-0-8218-3514-2)
Introduction includes
Riccardo Pedrotti, Complex oriented cohomology, generalized orientation and Thom isomorphism, 2016, 2018 (pdf)
The perspective of chromatic homotopy theory originates in
and is further developed in
Jacob Lurie, Chromatic Homotopy Theory, Lecture notes, 2010 (web)
Lecture 4 Complex-oriented cohomology theories (pdf)
Lecture 6 MU and complex orientations (pdf)
See also the references at equivariant cohomology – References – Complex oriented cohomology-.
A comparison between complex orientations and $H_\infty$ ring maps out of $MU$ was given in
More recent developments include
Refinement to equivariant complex oriented cohomology theory:
John Greenlees, Equivariant formal group laws and complex oriented cohomology theories, Homology Homotopy Appl. Volume 3, Number 2 (2001), 225-263 (euclid:hha/1139840255)
Michael Cole, John Greenlees, Igor Kriz, The universality of equivariant complex bordism, Math Z 239, 455–475 (2002) (doi:10.1007/s002090100315)
Discussion of complex orientation (in Whitehead generalized cohomology) on (only) those complex vector bundles which are pulled back from base spaces of bounded cell-dimension (Hopkins 84, 1.2, Ravenel 86, 6.5.2) – or rather, for the most part, of Ravenel's Thom spectra $X(n)$ and $T(m)$ (Ravenel 84, Sec. 3) which co-represent these:
Douglas Ravenel, section 3 of: Localization with Respect to Certain Periodic Homology Theories, American Journal of Mathematics Vol. 106, No. 2 (Apr., 1984), pp. 351-414 (doi:10.2307/2374308, jstor:2374308)
Michael Hopkins, Stable decompositions of certain loop spaces, Northwestern 1984 (pdf)
Douglas Ravenel, section 6.5 of: Complex cobordism and stable homotopy groups of spheres, 1986
Ethan Devinatz, Michael Hopkins, Jeffrey Smith, Theorem 3 of: Nilpotence and Stable Homotopy Theory I, Annals of Mathematics Second Series, Vol. 128, No. 2 (Sep., 1988), pp. 207-241 (jstor:1971440)
Doug Ravenel, The first Adams-Novikov differential for the spectrum $T(m)$, 2000 (pdf, pdf)
Ippei Ichigi, Katsumi Shimomura, The Modulo Two Homotopy Groups of the $L_2$-Localization of the Ravenel Spectrum, CUBO A Mathematical Journal, Vol. 10, No 03, (43–55). October 2008 (cubo:1498)
Gabe Angelini-Knoll, J. D. Quigley, The Segal Conjecture for topological Hochschild homology of the Ravenel spectra, Journal of Topology 4.3 (2011): 591-622 (arXiv:1705.03343, doi:10.1112/jtopol/jtr015)
Jonathan Beardsley, A Theorem on Multiplicative Cell Attachments with an Application to Ravenel’s $X(n)$ Spectra, Journal of Homotopy and Related Structures volume 14, pages 611–624 (2019) (arXiv:1708.03042, doi:10.1007/s40062-018-0222-6)
Jonathan Beardsley, Topological Hochschild homology of $X(n)$ (arXiv:1708.09486)
Xiangjun Wang, Zihong Yuan, The homotopy groups of $L_2 T(m)/\big(p^{[\tfrac{m}{2}]+2}, v_1 \big)$ for $m \gt 1$, New York J. Math.24 (2018) 1123–1146 (pdf)
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