\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\newtheorem*{utheorem}{Theorem}
\newtheorem*{ulemma}{Lemma}
\newtheorem*{uprop}{Proposition}
\newtheorem*{ucor}{Corollary}
\theoremstyle{definition}
\newtheorem{defn}{Definition}
\newtheorem{example}{Example}
\newtheorem*{udefn}{Definition}
\newtheorem*{uexample}{Example}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
\newtheorem{note}{Note}
\newtheorem*{uremark}{Remark}
\newtheorem*{unote}{Note}

%-------------------------------------------------------------------

\begin{document}

%-------------------------------------------------------------------


\section*{genus}

\begin{quote}%
this entry is about the notion of \emph{genus} in [[algebraic topology]]/[[cohomology]]. For classification of [[surfaces]] see instead the (related) entry \emph{[[genus of a surface]]}, \emph{[[genus of a curve]]}. There is also [[genus of a lattice]].


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

\hypertarget{index_theory}{}\paragraph*{{Index theory}}\label{index_theory}

[[!include index theory - contents]]

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

[[!include cohomology - contents]]

\hypertarget{cobordism_theory}{}\paragraph*{{Cobordism theory}}\label{cobordism_theory}

[[!include cobordism theory -- contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{basic}{Basic}\dotfill \pageref*{basic} \linebreak
\noindent\hyperlink{InStableHomotopyTheory}{In stable homotopy theory and generalized cohomology theory}\dotfill \pageref*{InStableHomotopyTheory} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{Rationalization}{Rationalization}\dotfill \pageref*{Rationalization} \linebreak
\noindent\hyperlink{LogarithmAndCharacteristicSeries}{Logarithm and Characteristic series}\dotfill \pageref*{LogarithmAndCharacteristicSeries} \linebreak
\noindent\hyperlink{definition_in_components}{Definition in components}\dotfill \pageref*{definition_in_components} \linebreak
\noindent\hyperlink{HirzebruchSeriesViaOrientationsInGeneralizedCohomology}{Definition via orientations in generalized cohomology}\dotfill \pageref*{HirzebruchSeriesViaOrientationsInGeneralizedCohomology} \linebreak
\noindent\hyperlink{HirzebruchFormula}{The Hirzebruch formula}\dotfill \pageref*{HirzebruchFormula} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{todd_genus}{Todd genus}\dotfill \pageref*{todd_genus} \linebreak
\noindent\hyperlink{signature_genus}{Signature genus}\dotfill \pageref*{signature_genus} \linebreak
\noindent\hyperlink{AhatGenus}{$\hat A$-genus}\dotfill \pageref*{AhatGenus} \linebreak
\noindent\hyperlink{elliptic_genus}{Elliptic genus}\dotfill \pageref*{elliptic_genus} \linebreak
\noindent\hyperlink{witten_genus}{Witten genus}\dotfill \pageref*{witten_genus} \linebreak
\noindent\hyperlink{nonexamples}{Non-examples}\dotfill \pageref*{nonexamples} \linebreak
\noindent\hyperlink{euler_characteristic}{Euler characteristic}\dotfill \pageref*{euler_characteristic} \linebreak
\noindent\hyperlink{RelatedConcepts}{Related concepts}\dotfill \pageref*{RelatedConcepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

\hypertarget{basic}{}\subsubsection*{{Basic}}\label{basic}

For $R$ a ([[commutative ring|commutative]]) [[ring]], an $R$-valued \textbf{genus} is a ring [[homomorphism]] into $R$

\begin{displaymath}
\sigma : \Omega_\bullet  \to R
\end{displaymath}
from a [[cobordism ring]] for [[cobordisms]] with specified structure; typical choices being [[orientation]] or [[stable complex structure]]. Often the rationalization of such a morphism is meant, see below at \emph{\hyperlink{Rationalization}{Properties -- Rationalization}}.

To emphasize that this is indeed a ring homomorphism and hence in particular respects the multiplicative structure, a genus is sometimes (especially in older literature) synonymously called a \emph{multiplicative genus}.

\hypertarget{InStableHomotopyTheory}{}\subsubsection*{{In stable homotopy theory and generalized cohomology theory}}\label{InStableHomotopyTheory}

Since the [[cobordism ring]] is the ring of coefficients of the corresponding universal [[Thom spectrum]], e.g $M U$, $M SO$, so a genus may also be written as a ring homomorphism of the form

\begin{displaymath}
M SO_\ast \longrightarrow R
\end{displaymath}
or

\begin{displaymath}
M U_\ast \longrightarrow R
\end{displaymath}
respectively. Written this way it is immediate that genera arise naturally as the value on [[homotopy groups]] (the ``[[decategorification]]'' or ``de-homotopification'') of homomorphisms of [[E-∞ ring]] [[spectra]] from an actual universal [[Thom spectrum]] to some [[E-∞ ring]] $E$ with [[coefficient]] ring $R$

\begin{displaymath}
M SO \longrightarrow E
\end{displaymath}
or

\begin{displaymath}
M U \longrightarrow E
  \,.
\end{displaymath}
This in turn induces multiplicative morphisms of the [[cohomology theories]] [[Brown representability theorem|represented]] by these spectra (the domain being hence [[cobordism cohomology theory]]), and these multiplicative maps are the ``families version'' of the given genus/[[index]] (\hyperlink{Hopkins94}{Hopkins 94, section 3}).

Such homomorphisms in turn arise naturally from [[orientation in generalized cohomology|universal orientations in generalized E-cohomology]]. Namely such an orientation is a [[homotopy]] of the form

\begin{displaymath}
\itexarray{
     \ast && \longleftarrow && B SO
     \\
       & {}_{\mathllap{0}}\searrow & \swArrow_\sigma  & \swarrow_{\mathrlap{J}}
     \\
     && B GL_1(E)
     \\
     && \downarrow
     \\
     && E Mod
  }
\end{displaymath}
(a trivialization of the $E$-[[(∞,1)-module bundle]] [[associated ∞-bundle|associated]] to the [[spherical fibration]] given by the [[J-homomorphism]]) and under forming [[homotopy colimits]] in $E$[[(∞,1)Mod]] this becomes an $E$-linear map

\begin{displaymath}
M SO \wedge E  \longleftarrow E
\end{displaymath}
hence a map

\begin{displaymath}
M SO  \longleftarrow E
  \,.
\end{displaymath}
At least in some important cases, genera seem to be naturally understood as encoding [[sigma-model]] [[quantum field theories]]. For $G$ some structure, the [[Thom spectrum]] $M G$ is the classifying space of [[manifolds]] with [[G-structure]], and hence may be thought of as classifying [[target spaces]] for [[sigma-models]]. The codomain spectrum $R$ itself may then be thought of as a classifying space for a certain class of QFTs, and hence the genus $\sigma : M G \to R$ can be thought of as assigning to any target space the corresponding [[sigma-model]].

This is for instance the case at least over the point for the [[A-hat genus]] $M Spin \to K O$, which may be thought of as sending manifolds with [[spin structure]] to the corresponding [[(1,1)-dimensional Euclidean field theories and K-theory|(1,1)-supersymmetric EFT]] (``[[spinning particle]]''); and for the [[Witten genus]] $M String \to tmf$, which can be thought of as sending a manifold with [[string structure]] to the corresponding [[(2,1)-dimensional Euclidean field theories and tmf|(2,1)-supersymmetric EFT]] (``[[heterotic string]]'').

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{Rationalization}{}\subsubsection*{{Rationalization}}\label{Rationalization}

When the [[coefficient]] [[ring]] $R$ does not have additive [[torsion]], then any ring homomorphism

\begin{displaymath}
\phi \colon \Omega^{U/SO}_\bullet \longrightarrow R
\end{displaymath}
is determined already by its rationalization

\begin{displaymath}
\phi \colon \Omega^{U,SO}_\bullet\otimes \mathbb{Q} \longrightarrow R \otimes \mathbb{Q}
\end{displaymath}
which is traditionally denoted by the same symbol. The rational [[cobordism rings]] in turn are known to be [[polynomial rings]]

\begin{displaymath}
\Omega^U_\bullet\otimes \mathbb{Q} \simeq \mathbb{Q}[\mathbb{C}P^1,\mathbb{C}P^2, \cdots ]
\end{displaymath}
\begin{displaymath}
\Omega^{SO}_\bullet\otimes \mathbb{Q} \simeq \mathbb{Q}[\mathbb{C}P^2,\mathbb{C}P^4, \cdots ]
\end{displaymath}
whose generators are identified with the cobordism classes of the manifolds which are the complex [[projective spaces]], as indicated.

\hypertarget{LogarithmAndCharacteristicSeries}{}\subsubsection*{{Logarithm and Characteristic series}}\label{LogarithmAndCharacteristicSeries}

\hypertarget{definition_in_components}{}\paragraph*{{Definition in components}}\label{definition_in_components}

Given a (rational) genus $\phi \colon \Omega^{U,SU}_\bullet\otimes \mathbb{Q} \to R \otimes \mathbb{Q}$ one defines (we follow (\hyperlink{Hopkins94}{Hopkins 94}))

\begin{enumerate}%
\item its \emph{logarithm} to be the [[formal power series]] over $R \otimes \mathbb{Q}$ given by

\begin{displaymath}
log_\phi(z) \coloneqq \sum_{n \in \mathbb{N}} \phi(\mathbb{C}P^n) \frac{z^{n+1}}{n+1};
\end{displaymath}

\item its \emph{characteristic series} (or \emph{Hirzebruch series}) to be the formal power series over $R \otimes \mathbb{Q}$

\begin{displaymath}
K_\phi(z) \coloneqq \frac{z}{\exp_\phi(z)}
  \,,
\end{displaymath}
where $\exp_\phi$ is the inverse of the logarithm;


\item its \emph{characteristic class} as the [[universal characteristic class]] which via the [[splitting principle]] is fixed by its value on the universal line bundle as

\begin{displaymath}
K_\phi(c_1) \in H^\bullet(B U(1),R \otimes \mathbb{Q})
\end{displaymath}
where $c_1 \in H^2(B U(1), \mathbb{Z})$ denotes the universal [[first Chern class]]; hence its value on a [[direct sum]] $L_1 \oplus \cdots \oplus L_k$ of [[complex line bundles]] is

\begin{displaymath}
\prod_{i} K_\phi(c_1(L_i))
  \,.
\end{displaymath}


\end{enumerate}
\hypertarget{HirzebruchSeriesViaOrientationsInGeneralizedCohomology}{}\paragraph*{{Definition via orientations in generalized cohomology}}\label{HirzebruchSeriesViaOrientationsInGeneralizedCohomology}

Suppose that the given genus $\Omega_\bullet^{SO} \longrightarrow R$ indeed comes from an [[orientation in generalized cohomology]] (as discussed \hyperlink{InStableHomotopyTheory}{above}) hence from a [[homomorphism]] of [[E-∞ rings]]

\begin{displaymath}
\beta \;\colon\;  M SO \longrightarrow E
\end{displaymath}
for an [[E-∞ ring]] $E$ with [[homotopy groups]] $R\simeq \pi_\bullet(E)$. (And suppose that $E$ defines a [[complex oriented cohomology theory]].)

This defines (\hyperlink{AndoHopkinsRezk10}{Ando-Hopkins-Rezk 10, prop. 2.11}) a universal [[orientation in generalized cohomology|orientation]] of real vector bundles and hence of [[complex vector bundles]] and hence of [[complex line bundles]] in $E$-cohomology

\begin{displaymath}
\beta \;\colon\; M U \longrightarrow M SO \stackrel{\beta}{\longrightarrow} E
  \,.
\end{displaymath}
Now rationally, i.e. for $E\otimes \mathbb{Q}$, there is a canonical such orientation, given by the composite

\begin{displaymath}
\alpha \;\colon\; M U \longrightarrow M SO \longrightarrow H \mathbb{Q} \simeq \mathbb{S}\otimes \mathbb{Q} \stackrel{}{\longrightarrow} E \otimes \mathbb{Q}
  \,.
\end{displaymath}
Thus, given any orientation $\beta$, its rationalization may be compared to $\alpha$. Since these rational orientations are equivalently trivializations of maps to $B GL_1(E \otimes \mathbb{R})$, their difference is a class $\beta/\alpha$ with coefficients in $GL_1(E\otimes R)$, hence over any space $X$ the difference is a class in $H^0(X, \pi_\bullet E\otimes \mathbb{Q})^\times$.

Specifically consider the [[delooping]] $X= B U(1)$ of the [[circle group]]. For this the [[cohomology ring]] is the [[power series]] ring in a single [[variable]] (the universal [[first Chern class]] $c_1(L)$). Under the canonical inclusion $B U(1)\to B U$ both the above orientations $\beta$ and $\alpha$ pull back, so that we have a difference

\begin{displaymath}
K_\beta \coloneqq \beta/\alpha \in (E_\bullet \otimes \mathbb{Q})[ [ c_1(L) ] ]
  \,.
\end{displaymath}
This is the Hirzebruch series of $\beta$ (\hyperlink{AndoHopkinsRezk10}{Ando-Hopkins-Rezk 10, def. 3.10}).

If $F$ denotes the [[formal group law]] classified via $MU_\bullet \to M SO_\bullet \stackrel{\beta_\bullet}{\to} E_\bullet$ then

\begin{displaymath}
K_\beta(z) = \frac{z}{\exp_F(z)}
\end{displaymath}
\hypertarget{HirzebruchFormula}{}\subsubsection*{{The Hirzebruch formula}}\label{HirzebruchFormula}

The central theorem of (\hyperlink{Hirzebruch66}{Hirzebruch 66}) expresses the genus of an arbitrary ([[cobordism class]] of a) manifold $X$ of [[dimension]] $2n$ via the formula

\begin{displaymath}
\phi(X)
  =
  \langle \prod_{i = 1}^n K_\phi(x_i(T X)), [X] \rangle
\end{displaymath}
in terms of the Hirzebruch characteristic series $K_\phi$ discussed \hyperlink{LogarithmAndCharacteristicSeries}{above}, and via the [[splitting principle]]:

This means that $\prod_{i = 1}^n K_\phi(x_i(T X))$ is the function of [[Chern classes]] $c_k$ (i.e. [[Pontryagin classes]] $P_{2k}$ and [[Euler classes]] $\chi$) obtained by rewriting the [[polynomial]] in the $x_i$ (the ``[[Chern roots]]'') as a polynomial in [[elementary symmetric polynomials]] $\sigma_k(x_1,\cdots, x_n)$ and then substituting for each of these by $c_k(T X)$.

(see also e.g. \href{http://www.map.mpim-bonn.mpg.de/Formal_group_laws_and_genera#Construction}{ManifoldAtlas -- Genera -- 4.1 Construction}).

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{todd_genus}{}\subsubsection*{{Todd genus}}\label{todd_genus}

The [[Todd genus]] is the genus with logarithm

\begin{displaymath}
-log_{Todd}(1-x) = \sum_{n \in \mathbb{N}} \frac{x^n}{n}
\end{displaymath}
\hypertarget{signature_genus}{}\subsubsection*{{Signature genus}}\label{signature_genus}

The [[signature genus]];

\hypertarget{AhatGenus}{}\subsubsection*{{$\hat A$-genus}}\label{AhatGenus}

The [[A-hat genus]] is the [[index of a Dirac operator]] coming from a [[spin bundle]] in KO-theory. It is given by the characteristic series

The \href{genus#LogarithmAndCharacteristicSeries}{characteristic series} of the $\hat A$-genus is

\begin{displaymath}
\begin{aligned}
    K_{\hat A}(e)
     & =
    \frac{z}{e^{z/2} - e^{-z/2}}
    \\
     &= 
    \exp\left(
       - \sum_{k \geq 2} \frac{B_k}{k} \frac{z^k}{k!}
     \right)
  \end{aligned}
  \,,
\end{displaymath}
where $B_k$ is the $k$th [[Bernoulli number]] (\hyperlink{AndoHopkinsRezk10}{Ando-Hopkins-Rezk 10, prop. 10.2}).

The $\hat A$-genus is an [[integer]] on manifolds with [[spin structure]].

\hypertarget{elliptic_genus}{}\subsubsection*{{Elliptic genus}}\label{elliptic_genus}

An [[elliptic genus]] is one whose logarithm is given by

\begin{displaymath}
log_{ell}(z) = \int_0^z (1-2 \delta t^2 + \epsilon t^4)^{-1/2} d t
\end{displaymath}
for constants $\delta, \epsilon$ with non-degenerate values $\delta^2 \neq \epsilon$ and $\epsilon = 0$.

For degenerate choices this reproduces the [[signature genus]] and the [[A-hat genus]] above, see at \emph{[[elliptic genus]]} for more. For non-degenerate values one may regard $\epsilon$ and $\delta$ as values of [[modular forms]] of the same name and hence regard all elliptic genera together as one single genus with [[coefficients]] in $MF_\bullet(\Gamma_0(2))$. This ``universal'' elliptic genus is the \emph{[[Witten genus]]}.

\hypertarget{witten_genus}{}\subsubsection*{{Witten genus}}\label{witten_genus}

The [[Witten genus]]

\begin{displaymath}
w \colon \Omega^{SO}_\bullet \longrightarrow \mathbb{Q}[ [q] ]
\end{displaymath}
is the genus with [[coefficients]] in the [[power series]] ring $\mathbb{Q}[ [ q ] ]$ with characteristic series given by

\begin{displaymath}
\begin{aligned}
    K_w(z)
    & =
    \frac{z/2}{sinh(z/2)}
    \prod_{n \geq 1}
    \frac{(1-q^n)^2}{(1-q^n e^z)(1-q^n e^{-z})}
    \\
    & = 
    \exp\left(
      \sum_{k \geq 2} G_k \frac{z^k}{k!}
    \right)
  \end{aligned}
  \,,
\end{displaymath}
where $G_k$ are the [[Eisenstein series]] (\href{AndoHopkinsStrickland01}{Ando-Hopkins-Strickland 01}, \hyperlink{AndoHopkinsRezk10}{Ando-Hopkins-Rezk 10, prop. 10.9}). (Notice that the constant term in $G_k$ is proportional to the $k$th [[Bernoulli number]], so that indeed the exponential expression matches that for the \hyperlink{AhatGenus}{A-hat genus above}.)

On [[manifolds]] with [[spin structure]] the Witten genus takes values in $\mathbb{Z}[ [ q ] ]$

On manifolds with rational [[string structure]] it takes values in (the $q$-expansion of) [[modular forms]] for $SL_2(\mathbb{Z})$, meaning that setting $q = e^{2 \pi i \tau}$ then as a function $f$ of the parameter $\tau$ taking values in the [[upper half plane]] the Witten genus satisfies

\begin{displaymath}
f(-1/\tau) = (-\tau)^n f(\tau)
  \,.
\end{displaymath}
Finally on manifolds with actual [[string structure]] it takes values in [[topological modular forms]]. See at \emph{[[Witten genus]]} for more.

\hypertarget{nonexamples}{}\subsection*{{Non-examples}}\label{nonexamples}

\hypertarget{euler_characteristic}{}\subsubsection*{{Euler characteristic}}\label{euler_characteristic}

The [[Euler characteristic]] $X \mapsto \chi(X)$ is close to being a genus, but is \emph{not} [[cobordism]] invariant

(this is the [[index]] of the [[Dirac operator]] $D = d + d^\dagger$)

\hypertarget{RelatedConcepts}{}\subsection*{{Related concepts}}\label{RelatedConcepts}

[[!include genera and partition functions - table]]

\hypertarget{references}{}\subsection*{{References}}\label{references}

The abstract concept of genus is due to [[Friedrich Hirzebruch]]. It had evolved out of the older concept of (arithmetic) [[genus of a surface]] via the concept of [[Todd genus]] introduced in

\begin{itemize}%
\item [[John Arthur Todd]], \emph{The arithmetical invariants of algebraic loci}, Proc. London Math. Soc. (2), Ser. 43, 1937, 190--225.

\end{itemize}
An review of the history is at the beginning of (\hyperlink{HirzebruchKreck09}{Hirzebruch-Kreck 09})

The theory of multiplicative sequences and characteristic series of genera is due to

\begin{itemize}%
\item [[Friedrich Hirzebruch]], \emph{Neue topologische Methoden in der algebraischen Geometrie}, Ergebnisse der Mathematik und ihrer Grenzgebiete, 9, Springer-Verlag, Berlin-G\"o{}ttingen-Heidelberg, 1965.


\item [[Friedrich Hirzebruch]], \emph{Topological methods in algebraic geometry}, Springer-Verlag, New York, 1966. MR0202713 (34 \#2573) Zbl 0843.14009



\end{itemize}
Reviews include

\begin{itemize}%
\item [[Serge Ochanine]], \emph{What is\ldots{} an elliptic genus}?, Notices of the AMS, volume 56, number 6 (2009) (\href{http://www.ams.org/notices/200906/rtx090600720p.pdf}{pdf})


\item [[Friedrich Hirzebruch]], [[Matthias Kreck]], \emph{On the concept of genus in topology and complex analysis}, Notices of the AMS, volume 56, number 6 (2009) \href{http://www.ams.org/notices/200906/rtx090600713p.pdf}{pdf}


\item [[Michael Hopkins]], section 2 of \emph{Topological modular forms, the Witten Genus, and the theorem of the cube}, Proceedings of the International Congress of Mathematics, Z\"u{}rich 1994 (\href{http://www.mathunion.org/ICM/ICM1994.1/Main/icm1994.1.0554.0565.ocr.pdf}{pdf})


\item Manifold Atlas, \emph{\href{http://www.map.mpim-bonn.mpg.de/Formal_group_laws_and_genera}{Formal group laws and genera}}


\item Wikipedia, \emph{\href{http://en.wikipedia.org/wiki/Genus_of_a_multiplicative_sequence}{Genus of a multiplicative series}}


\item \href{wwwmath.uni-muenster.de/reine/inst/lueck/lehre/.../summary6.}{}



\end{itemize}
Discussion in terms of [[orientations in generalized cohomology]] and specifically for the [[A-hat genus]] and the [[Witten genus]] is in

\begin{itemize}%
\item [[Matthew Ando]], [[Michael Hopkins]], [[Neil Strickland]], \emph{Elliptic spectra, the Witten genus and the theorem of the cube}, Invent. Math. 146 (2001) 595--687 MR1869850


\item [[Matthew Ando]], [[Mike Hopkins]], [[Charles Rezk]], \emph{Multiplicative orientations of KO-theory and the spectrum of topological modular forms}, 2010 (\href{http://www.math.uiuc.edu/~mando/papers/koandtmf.pdf}{pdf})



\end{itemize}
[[!redirects genera]]

[[!redirects Hirzebruch series]]



\end{document}