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\begin{document}

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\section*{A Survey of Elliptic Cohomology - cohomology theories}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology}

[[!include cohomology - contents]]

\hypertarget{higher_geometry}{}\paragraph*{{Higher geometry}}\label{higher_geometry}

[[!include higher geometry - contents]]

\hypertarget{higher_algebra}{}\paragraph*{{Higher algebra}}\label{higher_algebra}

[[!include higher algebra - contents]]

This is a sub-entry of

\begin{itemize}%
\item [[A Survey of Elliptic Cohomology]]

\end{itemize}
see there for background and context.

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

Next:

\begin{itemize}%
\item [[A Survey of Elliptic Cohomology - formal groups and cohomology]]

\end{itemize}
\hypertarget{rough_notes_from_a_talk}{}\subsection*{{rough notes from a talk}}\label{rough_notes_from_a_talk}

\begin{quote}%
the following are rough unpolished notes taken more or less verbatim from some seminar talk -- needs attention


\end{quote}
A complex oriented [[cohomology]] theory (meant is here and in all of the following a [[generalized (Eilenberg-Steenrod) cohomology]]) is one with a \emph{good notion of [[Thom class]]es, equivalently first [[Chern class]] for complex [[vector bundle]]}

\begin{quote}%
(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 $\mathbb{C}P^\infty \simeq \mathcal{B}U(1)$ is a power series ring. The formal variable of that is then identified with the universal first Chern class as seen by that theory).


\end{quote}
ordinary [[Chern class]] lives in [[integral cohomology]] $H^*(-,\mathbb{Z})$

or in [[K-theory]] $K^*(-)$ where for a [[vector bundle]] $V$ we would set $c_1(V) := ([V]-1)\beta$ where $\beta$ is the [[Bott generator]].

In the first case we have that under [[tensor product]] of [[vector bundle]]s the class behaves as

\begin{displaymath}
c_1(V\otimes W) = c_1(V) + c_1(W)
\end{displaymath}
whereas in the second case we get

\begin{displaymath}
c_1(V \otimes W) = c_1(V)c_1(W)\beta^{-1} + c_1(V) + c_1(W)
  \,.
\end{displaymath}
In general we will get that the [[Chern class]] of a tensor product is given by a certain [[power series]] $E^*(pt)$

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

\textbf{definition} of complex-orientation

there is an

\begin{displaymath}
x \in \tilde E^2(\mathbb{C}P^\infty)
\end{displaymath}
such that under the map

\begin{displaymath}
\tilde E^2(\mathbb{C}P^\infty)
  \to 
  \tilde E^2(\mathbb{C}P^1)
  \simeq
  \tilde E^2(S^1)
  \simeq E^0({*})
\end{displaymath}
induced by

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

we have $x \mapsto 1$

\textbf{remark} this also gives [[Thom class]]es since $\mathbb{C}P^\infty \to (\mathbb{C}P^\infty)^\gamma$ is a [[homotopy equivalence]]

\begin{displaymath}
\tilde E^2((\mathbb{C}P^\infty)^\gamma)
  \simeq
  \tilde E^2((\mathbb{C}P^\infty))
  \ni X
\end{displaymath}
Thom iso $\tilde H^{*+2}(X^\gamma) \simeq H^*(X)$

\ldots{}

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

\textbf{definition (Bott element and even periodic cohomology theory)}

\begin{itemize}%
\item An [[even cohomology theory]] is one whose odd cohomology rings vanish: $E^{2k+1}(X) = 0$.


\item A [[periodic cohomology theory]] is one with a [[Bott element]] $\beta \in E^2({*})$ which is invertible (under multiplication in the [[cohomology ring]] of the point)

so that gives an isomorphism $(-)\cdot \beta : E^*({*}) \simeq E^{*+2}({*})$



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

\begin{displaymath}
H^p(X, E^q({*})) \Rightarrow E^{p+q}(X)
\end{displaymath}
notice that since $\mathbb{C}P^\infty$ is [[homotopy equivalence|homotopy equivalent]] to the [[classifying space]] $\mathcal{B}U(1)$ (which is a topological group) it has a product on it

\begin{displaymath}
\mathbb{C}P^\infty \times  \mathbb{C}P^\infty
  \to 
  \mathbb{C}P^\infty
\end{displaymath}
which is the one that induces the [[tensor product]] of [[line bundle]]s classified by maps into $\mathbb{C}P^\infty$.

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

\begin{displaymath}
\itexarray{
    \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
  }
\end{displaymath}
this $f$ is called a [[formal group law]] if the following conditions are satisfied

\begin{enumerate}%
\item \textbf{commutativity} $f(x,y) = f(y,x)$


\item \textbf{identity} $f(x,0) = x$


\item \textbf{associtivity} f(x,f(y,z)) = f(f(x,y),z)



\end{enumerate}
\textbf{remark} the second condition implies that the constant term in the power series $f$ 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 $f$ always start as

\begin{displaymath}
f(x,y) = x + y + \cdots
\end{displaymath}
The [[Lazard ring]] is the ``universal formal group law''. it can be presented as by generators $a_{i j}$ with $i,j \in \mathbb{N}$

\begin{displaymath}
L = \mathbb{Z}[a_{i j}] / (relations 1-3 below)
\end{displaymath}
and relatins as follows

\begin{enumerate}%
\item $a_{i j} = a_{j i}$


\item $a_{10} = a_{01} = 1$; $\forall i \neq 0: a_{i 0} = 0$


\item the obvious associativity relation



\end{enumerate}
the universal formal group law we get from this is the power series in $x,y$ with coefficients in the [[Lazard ring]]

\begin{displaymath}
\ell(x,y) = \sum_{i,j} a_{i j} x^j y^j \in L[[x,y]]
  \,.
\end{displaymath}
\textbf{remark} the formal group law is not canonically associated to the cohomology theory, only up to a choice of rescaling of the elements $x$. But the underlying [[formal group]] is independent of this choice and well defined.

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

\textbf{remark} Quillen's theorem says that the Lazard ring is the ring of complex cobordisms

\textbf{some universal cohomology theories} $M U$ is the [[spectrum]] for [[complex cobordism cohomology theory]]. The corresponding [[spectrum]] is in degree $2 n$ given by

\begin{displaymath}
M U(2n) = Thom
  \left(
    standard associated bundle to universal bundle
    \itexarray{
      E U(n) \\ \downarrow \\ B U(n)
    }
  \right)
\end{displaymath}
periodic [[complex cobordism cohomology theory]] is given by

\begin{displaymath}
M P = \vee_{n \in \mathbb{Z}} \Sigma^{2 n} M U
\end{displaymath}
we get a canonical [[oriented cohomology theory|orientation]] from

\begin{displaymath}
\omega : \mathbb{C}P^\infty \stackrel{\simeq}{\to}
  M U(1)
  \;\;\;\;
  M U(\mathbb{C}P^\infty)
\end{displaymath}
this is the universal even periodic cohomology theory with orientation

\textbf{Theorem (Quillen)} the [[cohomology ring]] $M P(*)$ 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]] $(L,\ell)$

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

use the classifying map $M P({*}) \to R$ to build the [[tensor product]]

\begin{displaymath}
E^n(X) := M P^n(X) \otimes_{M P({*})} R
\end{displaymath}
this construction could however break the left exactness condition. However, $E$ 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).

[[!redirects A Survey of Eilliptic Cohomology - cohomology theories]]



\end{document}