A Survey of Elliptic Cohomology - cohomology theories

**cohomology** * cocycle, coboundary, coefficient * homology * chain, cycle, boundary * characteristic class * universal characteristic class * secondary characteristic class * differential characteristic class * fiber sequence/long exact sequence in cohomology * fiber ∞-bundle, principal ∞-bundle, associated ∞-bundle, twisted ∞-bundle * ∞-group extension * obstruction ### Special and general types ### * cochain cohomology * ordinary cohomology, singular cohomology * group cohomology, nonabelian group cohomology, Lie group cohomology * Galois cohomology * groupoid cohomology, nonabelian groupoid cohomology * generalized (Eilenberg-Steenrod) cohomology * cobordism cohomology theory * integral cohomology * K-theory * elliptic cohomology, tmf * taf * abelian sheaf cohomology * Deligne cohomology * de Rham cohomology * Dolbeault cohomology * etale cohomology * group of units, Picard group, Brauer group * crystalline cohomology * syntomic cohomology * motivic cohomology * cohomology of operads * Hochschild cohomology, cyclic cohomology * string topology * nonabelian cohomology * principal ∞-bundle * universal principal ∞-bundle, groupal model for universal principal ∞-bundles * principal bundle, Atiyah Lie groupoid * principal 2-bundle/gerbe * covering ∞-bundle/local system * (∞,1)-vector bundle / (∞,n)-vector bundle * quantum anomaly * orientation, Spin structure, Spin^c structure, String structure, Fivebrane structure * cohomology with constant coefficients / with a local system of coefficients * ∞-Lie algebra cohomology * Lie algebra cohomology, nonabelian Lie algebra cohomology, Lie algebra extensions, Gelfand-Fuks cohomology, * bialgebra cohomology ### Special notions * Čech cohomology * hypercohomology ### Variants ### * equivariant cohomology * equivariant homotopy theory * Bredon cohomology * twisted cohomology * twisted bundle * twisted K-theory, twisted spin structure, twisted spin^c structure * twisted differential c-structures * twisted differential string structure, twisted differential fivebrane structure * differential cohomology * differential generalized (Eilenberg-Steenrod) cohomology * differential cobordism cohomology * Deligne cohomology * differential K-theory * differential elliptic cohomology * differential cohomology in a cohesive topos * Chern-Weil theory * ∞-Chern-Weil theory * relative cohomology ### Extra structure * Hodge structure * orientation, in generalized cohomology ### Operations ### * cohomology operations * cup product * connecting homomorphism, Bockstein homomorphism * fiber integration, transgression * cohomology localization ### Theorems * universal coefficient theorem * Künneth theorem * de Rham theorem, Poincare lemma, Stokes theorem * Hodge theory, Hodge theorem nonabelian Hodge theory, noncommutative Hodge theory * Brown representability theorem * hypercovering theorem * Eckmann-Hilton-Fuks duality

Edit this sidebar

*** **higher geometry** / **derived geometry** ## Ingredients ## * higher topos theory * higher algebra ## Concepts ## * **geometric little (∞,1)-toposes** * structured (∞,1)-topos * geometry (for structured (∞,1)-toposes) * generalized scheme * **geometric big (∞,1)-toposes** * cohesive (∞,1)-topos * function algebras on ∞-stacks * geometric ∞-stacks ## Constructions * loop space object, free loop space object * fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos / of a locally ∞-connected (∞,1)-topos ## Examples * derived algebraic geometry * étale (∞,1)-site, Hochschild cohomology of dg-algebras * dg-geometry * dg-scheme * schematic homotopy type * derived noncommutative geometry * noncommutative geometry * derived smooth geometry * differential geometry, differential topology * derived smooth manifold, dg-manifold * smooth ∞-groupoid, ∞-Lie algebroid * higher symplectic geometry * higher Klein geometry * higher Cartan geometry ## Theorems * Isbell duality * Jones' theorem, Deligne-Kontsevich conjecture * Tannaka duality for geometric stacks

Edit this sidebar

*** **higher algebra** universal algebra ## Algebraic theories * algebraic theory / 2-algebraic theory / (∞,1)-algebraic theory * monad / (∞,1)-monad * operad / (∞,1)-operad ## Algebras and modules * algebra over a monad ∞-algebra over an (∞,1)-monad * algebra over an algebraic theory ∞-algebra over an (∞,1)-algebraic theory * algebra over an operad ∞-algebra over an (∞,1)-operad * action, ∞-action * representation, ∞-representation * module, ∞-module * associated bundle, associated ∞-bundle ## Higher algebras * monoidal (∞,1)-category * symmetric monoidal (∞,1)-category * monoid in an (∞,1)-category * commutative monoid in an (∞,1)-category * symmetric monoidal (∞,1)-category of spectra * smash product of spectra * symmetric monoidal smash product of spectra * ring spectrum, module spectrum, algebra spectrum * A-∞ algebra * A-∞ ring, A-∞ space * C-∞ algebra * E-∞ ring, E-∞ algebra * ∞-module, (∞,1)-module bundle * multiplicative cohomology theory * L-∞ algebra * deformation theory ## Model category presentations * model structure on simplicial T-algebras / homotopy T-algebra * model structure on operads model structure on algebras over an operad ## Geometry on formal duals of algebras * Isbell duality * derived geometry ## Theorems * Deligne conjecture * delooping hypothesis * monoidal Dold-Kan correspondence

Edit this sidebar

This is a sub-entry of

see there for background and context.

This entry reviews basics of periodic multiplicative cohomology theories and their relation to formal group laws.


rough notes from a talk

the following are rough unpolished notes taken more or less verbatim from some seminar talk – needs attention

A complex oriented cohomology theory (meant is here and in all of the following a generalized (Eilenberg-Steenrod) cohomology) is one with a good notion of Thom classes, equivalently first Chern class for complex vector bundle

(this “good notion” will boil down to certain extra assumptions such as multiplicativity and periodicity etc. What one needs is that the cohomology ring assigned by the cohomology theory to P U(1)\mathbb{C}P^\infty \simeq \mathcal{B}U(1) is a power series ring. The formal variable of that is then identified with the universal first Chern class as seen by that theory).

ordinary Chern class lives in integral cohomology H *(,) H^*(-,\mathbb{Z})

or in K-theory K *()K^*(-) where for a vector bundle VV we would set c 1(V):=([V]1)βc_1(V) := ([V]-1)\beta where β\beta is the Bott generator.

In the first case we have that under tensor product of vector bundles the class behaves as

c 1(VW)=c 1(V)+c 1(W) c_1(V\otimes W) = c_1(V) + c_1(W)

whereas in the second case we get

c 1(VW)=c 1(V)c 1(W)β 1+c 1(V)+c 1(W). c_1(V \otimes W) = c_1(V)c_1(W)\beta^{-1} + c_1(V) + c_1(W) \,.

In general we will get that the Chern class of a tensor product is given by a certain power series E *(pt)E^*(pt)

not all formal group laws arises this way. the Landweber criterion gives a condition under which there is a cohomology theory

definition of complex-orientation

there is an

xE˜ 2(P ) x \in \tilde E^2(\mathbb{C}P^\infty)

such that under the map

E˜ 2(P )E˜ 2(P 1)E˜ 2(S 1)E 0(*) \tilde E^2(\mathbb{C}P^\infty) \to \tilde E^2(\mathbb{C}P^1) \simeq \tilde E^2(S^1) \simeq E^0({*})

induced by

P 1P \mathbb{C}P^1 \to \mathbb{C}P^\infty

we have x1x \mapsto 1

remark this also gives Thom classes since P (P ) γ\mathbb{C}P^\infty \to (\mathbb{C}P^\infty)^\gamma is a homotopy equivalence

E˜ 2((P ) γ)E˜ 2((P ))X \tilde E^2((\mathbb{C}P^\infty)^\gamma) \simeq \tilde E^2((\mathbb{C}P^\infty)) \ni X

Thom iso H˜ *+2(X γ)H *(X)\tilde H^{*+2}(X^\gamma) \simeq H^*(X)

(here and everywhere the tilde sign is for reduced cohomology)

definition (Bott element and even periodic cohomology theory)

Periodic cohomology theories are complex-orientable. E *(P )E^*(\mathbb{C}P^\infty) can be calculated using the Atiyah-Hirzebruch spectral sequence

H p(X,E q(*))E p+q(X) H^p(X, E^q({*})) \Rightarrow E^{p+q}(X)

notice that since P \mathbb{C}P^\infty is homotopy equivalent to the classifying space U(1)\mathcal{B}U(1) (which is a topological group) it has a product on it

P ×P P \mathbb{C}P^\infty \times \mathbb{C}P^\infty \to \mathbb{C}P^\infty

which is the one that induces the tensor product of line bundles classified by maps into P \mathbb{C}P^\infty.

on (at least on even periodic cohomology theories) this induces a map of the form

P ×P P E(*)[[x,y]] E(*)[[t]] f(x,y) t \array{ \mathbb{C}P^\infty \times \mathbb{C}P^\infty &\to& \mathbb{C}P^\infty \\ E({*})[[x,y]] &\leftarrow& E(*)[[t]] \\ f(x,y) &\leftarrow |& t }

this ff is called a formal group law if the following conditions are satisfied

  1. commutativity f(x,y)=f(y,x)f(x,y) = f(y,x)

  2. identity f(x,0)=xf(x,0) = x

  3. associtivity f(x,f(y,z)) = f(f(x,y),z)

remark the second condition implies that the constant term in the power series ff is 0, so therefore all these power series are automatically invertible and hence there is no further need to state the existence of inverses in the formal group. So these ff always start as

f(x,y)=x+y+ f(x,y) = x + y + \cdots

The Lazard ring is the “universal formal group law”. it can be presented as by generators a ija_{i j} with i,ji,j \in \mathbb{N}

L=[a ij]/(relations13below) L = \mathbb{Z}[a_{i j}] / (relations 1-3 below)

and relatins as follows

  1. a ij=a jia_{i j} = a_{j i}

  2. a 10=a 01=1a_{10} = a_{01} = 1; i0:a i0=0\forall i \neq 0: a_{i 0} = 0

  3. the obvious associativity relation

the universal formal group law we get from this is the power series in x,yx,y with coefficients in the Lazard ring

(x,y)= i,ja ijx jy jL[[x,y]]. \ell(x,y) = \sum_{i,j} a_{i j} x^j y^j \in L[[x,y]] \,.

remark the formal group law is not canonically associated to the cohomology theory, only up to a choice of rescaling of the elements xx. But the underlying formal group is independent of this choice and well defined.

For any ring SS with formal group law g(x,y)powerseriesinx,ywithcoefficientsinSg(x,y) \in power series in x,y with coefficients in S there is a unique morphism LSL \to S that sends \ell to gg.

remark Quillen’s theorem says that the Lazard ring is the ring of complex cobordisms

some universal cohomology theories MUM U is the spectrum for complex cobordism cohomology theory. The corresponding spectrum is in degree 2n2 n given by

MU(2n)=Thom(standardassociatedbundletouniversalbundleEU(n) BU(n)) M U(2n) = Thom \left( standard associated bundle to universal bundle \array{ E U(n) \\ \downarrow \\ B U(n) } \right)

periodic complex cobordism cohomology theory is given by

MP= nΣ 2nMU M P = \vee_{n \in \mathbb{Z}} \Sigma^{2 n} M U

we get a canonical orientation? from

ω:P MU(1)MU(P ) \omega : \mathbb{C}P^\infty \stackrel{\simeq}{\to} M U(1) \;\;\;\; M U(\mathbb{C}P^\infty)

this is the universal even periodic cohomology theory with orientation

Theorem (Quillen) the cohomology ring MP(*)M P(*) of periodic complex cobordism cohomology theory over the point together with its formal group law is naturally isomorphic to the universal Lazard ring with its formal group law (L,)(L,\ell)

how one might make a formal group law (R,f(x,y))(R,f(x,y)) into a cohomology theory

use the classifying map MP(*)RM P({*}) \to R to build the tensor product

E n(X):=MP n(X) MP(*)R E^n(X) := M P^n(X) \otimes_{M P({*})} R

this construction could however break the left exactness condition. However, EE built this way will be left exact of the ring morphism M P{{*}) \to R is a flat morphism. This is the Landweber exactness condition (or maybe slightly stronger).

Revised on March 19, 2010 18:39:44 by Anonymous Coward (