\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*{spin^c}

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

\hypertarget{higher_spin_geometry}{}\paragraph*{{Higher spin geometry}}\label{higher_spin_geometry}

[[!include higher spin geometry - contents]]

\hypertarget{group_theory}{}\paragraph*{{Group Theory}}\label{group_theory}

[[!include group theory - contents]]

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

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{group_extension}{Group extension}\dotfill \pageref*{group_extension} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{AsHomotopyFiberOfSmoothW3}{As the homotopy fiber of the smooth $\mathbf{W}_3$}\dotfill \pageref*{AsHomotopyFiberOfSmoothW3} \linebreak
\noindent\hyperlink{relation_to_metaplectic_group_}{Relation to metaplectic group $Mp^c$}\dotfill \pageref*{relation_to_metaplectic_group_} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\begin{remark}
\label{}\hypertarget{}{}
By definition of the [[spin group]] $Spin(n)$ there is a canonical inclusion

\begin{displaymath}
\mathbb{Z}/2\mathbb{Z}\hookrightarrow Spin(n)
\end{displaymath}
of the [[group of order 2]]. For $Spin(n)\hookrightarrow GL_1(Cl(\mathbb{R}^n))$ canonically realized by even [[Clifford algebra]] elements of unit norm, this is given by the inclusion of $\{+1,-1\}$.

\end{remark}
\begin{quote}%
We frequently write $\mathbb{Z}_2$ as shorthand for $\mathbb{Z}/2\mathbb{Z}$.


\end{quote}
\begin{defn}
\label{DirectDefinitionOfSpinC}\hypertarget{DirectDefinitionOfSpinC}{}
For $n \in \mathbb{N}$, the [[Lie group]] $Spin^c(n)$ is the [[quotient group]]

\begin{displaymath}
\begin{aligned}
    Spin^c & \coloneqq Spin \times_{\mathbb{Z}_2} U(1)
    \\
    & = (Spin \times U(1))/{\mathbb{Z}_2}
    \,,
  \end{aligned}
\end{displaymath}
of the [[product]] of the [[spin group]] with the [[circle group]] by the common [[subgroup|sub]]-[[group of order 2]] $\mathbb{Z}_2 \hookrightarrow \mathbb{Z}$ and $\mathbb{Z}_2 \hookrightarrow U(1)$.

\end{defn}
Some authors (e.g. \hyperlink{Gompf97}{Gompf 97, p. 2}) denote this as

\begin{displaymath}
\begin{aligned}
    Spin^c(n) 
     & \coloneqq 
    Spin(n)\cdot Spin(2) 
     \\
    & \simeq Spin(n) \cdot U(1)
  \end{aligned}
\end{displaymath}
following the notation [[Sp(n).Sp(1)]] (see \href{SpnSp1#SpinnSpin2IsSpinc}{there}).

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

\hypertarget{group_extension}{}\subsubsection*{{Group extension}}\label{group_extension}

\begin{prop}
\label{}\hypertarget{}{}
We have a [[short exact sequence]]

\begin{displaymath}
U(1) \to Spin^c \to SO
 \,,
\end{displaymath}
where $U(1) \to Spin^c$ is the canonical inclusion into the defining product $U(1) \to Spin \times U(1) \to Spin \times_{\mathbb{Z}_2} U(1)$.

\end{prop}
\hypertarget{general}{}\subsubsection*{{General}}\label{general}

\hypertarget{AsHomotopyFiberOfSmoothW3}{}\subsubsection*{{As the homotopy fiber of the smooth $\mathbf{W}_3$}}\label{AsHomotopyFiberOfSmoothW3}

We discuss in the following that

\begin{enumerate}%
\item the universal third [[integral Stiefel-Whitney class]] $W_3$ has an essentially unique lift from [[∞Grpd]] $\simeq$ [[Top]] to [[Smooth∞Grpd]];


\item the smooth [[delooping]] $\mathbf{B}Spin^c \in Smooth\infty Grpd$ is the [[homotopy fiber]] of $\mathbf{W}_3$, hence is the [[circle n-bundle|circle 2-bundle]] over $\mathbf{B} SO$ classified by $\mathbf{W}_3$.



\end{enumerate}
\begin{prop}
\label{SpinCAsHomotopyPullbackOfW2AndC1}\hypertarget{SpinCAsHomotopyPullbackOfW2AndC1}{}
We have a [[homotopy pullback]] diagram

\begin{displaymath}
\itexarray{
     \mathbf{B} Spin^c &\stackrel{\mathbf{B}det}{\to}& \mathbf{B}U(1)
     \\
     \downarrow && \downarrow^{\mathrlap{\mathbf{c}_1 mod 2}}
     \\
     \mathbf{B} SO &\stackrel{\mathbf{w}_2}{\to}&
     \mathbf{B}^2 \mathbb{Z}_2 
  }
\end{displaymath}
in [[Smooth∞Grpd]], where

\begin{itemize}%
\item the right morphism is the universal first [[Chern class]] modulo 2;


\item the bottom morphism is the universal second [[Stiefel-Whitney class]].



\end{itemize}
\end{prop}
\begin{proof}
We present the sitation as usual in the projective [[model structure on simplicial presheaves]] over [[CartSp]] by [[∞-anafunctors]].

The first [[Chern class]] is given by the [[∞-anafunctor]]

\begin{displaymath}
\itexarray{
     \mathbf{B}(\mathbb{Z} \to \mathbb{R})
     &\stackrel{\mathbf{c}_1}{\to}&
     \mathbf{B}(\mathbb{Z} \to 1)
     = 
     \mathbf{B}^2 \mathbb{Z}
     \\
     \downarrow^{\mathrlap{\simeq}}
     \\
     \mathbf{B} U(1)     
  }
  \,,
\end{displaymath}
where $(G_1 \to G_0)$ denotes a presentation of a [[strict 2-group]] by a [[crossed module]].

The second [[Stiefel-Whitney class]] is given by

\begin{displaymath}
\itexarray{
    \mathbf{B}(\mathbb{Z}_2 \to Spin)
    &\stackrel{\mathbf{w}_2}{\to}&
    \mathbf{B}(\mathbb{Z}_2 \to 1)
    = 
    \mathbf{B}^2 \mathbb{Z}_2
    \\
    \downarrow^{\mathrlap{\simeq}}
    \\
    \mathbf{B} SO
  }
  \,.
\end{displaymath}
Notice that the top horizontal morphism here is a [[fibration]].

Therefore the [[homotopy pullback]] in question is (as discussed there) given by the ordinary [[pullback]] $Q$ in

\begin{displaymath}
\itexarray{
    Q
    &\to& 
    \mathbf{B}(\mathbb{Z} \to \mathbb{R})
    \\
    \downarrow && \downarrow
    \\
    \mathbf{B}(\mathbb{Z}_2 \to Spin)
    &\to&
    \mathbf{B}^2 \mathbb{Z}_2
  }
  \,.
\end{displaymath}
This pullback is $\mathbf{B}(\mathbb{Z} \stackrel{\partial}{\to} Spin \times \mathbb{R})$, where

\begin{displaymath}
\partial\colon n \mapsto ( n \,mod\, 2 , n)
  \,.
\end{displaymath}
This is equivalent to

\begin{displaymath}
\begin{aligned}
   (\mathbb{Z} \stackrel{\partial}{\to} Spin \times \mathbb{R})
   & \simeq 
  (\mathbb{Z}_2 \stackrel{\partial'}{\to} Spin \times (\mathbb{R}/2\mathbb{Z}))
   \\
  & \simeq
  (\mathbb{Z}_2 \stackrel{\partial'}{\to} Spin \times U(1))
  \end{aligned}
  \,,
\end{displaymath}
(notice the non-standard identification $U(1) \simeq \mathbb{R}/(2\mathbb{Z})$ here, which is important below in prop. \ref{UniversalDeterminantLineBundleMap} for the identification of $det$) where now $\partial'$ is the [[diagonal]] embedding of the [[subgroup]]

\begin{displaymath}
\partial'\colon \sigma \mapsto (\sigma, \sigma)
  \,.
\end{displaymath}
This in turn is equivalent to

\begin{displaymath}
\mathbf{B} (Spin \times_{\mathbb{Z}_2} U(1))
  \,,
\end{displaymath}
which is def. \ref{DirectDefinitionOfSpinC}.

\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
Compare this with the similar but different [[homotopy pullback]] that defines the [[spin group]]

\begin{displaymath}
\itexarray{
    \mathbf{B}Spin &\to& * 
    \\
    \downarrow && \downarrow 
    \\
    \mathbf{B}SO &\stackrel{\mathbf{w}_2}{\to}& \mathbf{B}^2 \mathbb{Z}_2
  }
\end{displaymath}
\end{remark}
\begin{prop}
\label{UniversalDeterminantLineBundleMap}\hypertarget{UniversalDeterminantLineBundleMap}{}
Under the identificaton $Spin^c \simeq Spin \underset{\mathbb{Z}_2}{\times} U(1)$ the ``universal [[determinant line bundle]] map''

\begin{displaymath}
det \colon Spin^c \to U(1)
\end{displaymath}
is given in components by

\begin{displaymath}
(g,c) \mapsto 2 c
\end{displaymath}
(where on the right we write the group structure additively).

\end{prop}
\begin{proof}
By the proof of prop. \ref{SpinCAsHomotopyPullbackOfW2AndC1} the $U(1)$-factor in $Spin^c \simeq Spin \underset{\mathbb{Z}_2}{\times}U(1)$ arises from the identification $U(1) \simeq \mathbb{R}/2\mathbb{Z}$. But under the horizontal map as it appears in the homotopy pullback in that proof this corresponds to multiplication by 2.

\end{proof}
\begin{prop}
\label{SmoothRefinementOfBockstein}\hypertarget{SmoothRefinementOfBockstein}{}
The third \emph{[[integral Stiefel-Whitney class]]}

\begin{displaymath}
W_3
   \coloneqq 
  \beta_2 \circ w_2 
   \colon
  B SO \stackrel{w_2}{\to} B^2 \mathbb{Z}_2
  \stackrel{\beta_2}{\to}
  B^3 \mathbb{Z}
\end{displaymath}
has an essentially unique lift through [[geometric realization]] ${\vert-\vert}\colon$ [[Smooth∞Grpd]] $\stackrel{\Pi}{\to}$ [[∞Grpd]] $\stackrel{\simeq}{\to}$ [[Top]]

given by

\begin{displaymath}
\mathbf{W}_3 = \mathbf{\beta}_2 \circ \mathbf{w}_2
  \colon 
  \mathbf{B} SO(n)
   \stackrel{w_2}{\to}
  \mathbf{B}^2 \mathbb{Z}_2
    \stackrel{\mathbf{\beta}_2}{\to}
  \mathbf{B}^2 U(1)
  \,,
\end{displaymath}
where $\mathbf{\beta}_2$ is simply given by the canonical subgroup embedding.

\end{prop}
\begin{proof}
Once we establish that this is a lift at all, the essential uniqueness follows from the respective theorem at [[smooth ∞-groupoid -- structures]].

The ordinary [[Bockstein homomorphism]] $\beta_2$ is presented by the [[∞-anafunctor]]

\begin{displaymath}
\itexarray{
    \mathbf{B}^2(\mathbb{Z} \stackrel{\cdot 2}{\to} \mathbb{Z})
    &\to&
    \mathbf{B}^2 (\mathbb{Z} \to 1)
    =
    \mathbf{B}^3 \mathbb{Z}
    \\
    \downarrow^{\mathrlap{\simeq}}
    \\
    \mathbf{B}^2 \mathbb{Z}_2
  }
  \,.
\end{displaymath}
Accordingly we need to lift the canonical presentation of $\mathbf{\beta}_2$ to a comparable $\infty$-anafunctor. This is accomplished by

\begin{displaymath}
\itexarray{  
    \mathbf{B}^2(\mathbb{Z} \stackrel{\cdot 2}{\to} \mathbb{Z})
    &\stackrel{\hat \mathbf{\beta}_2}{\to}&
    \mathbf{B}^2 (\mathbb{Z} \stackrel{\cdot 2}{\to} \mathbb{R})
    \\
    \downarrow^{\mathrlap{\simeq}}
     &&
    \downarrow^{\mathrlap{\simeq}}
    \\
    \mathbf{B}^2 \mathbb{Z}_2
     &\stackrel{\mathbf{\beta}_2}{\to}&
    \mathbf{B}^2 U(1)
  }
  \,.
\end{displaymath}
Here the top horizontal morphism is induced from the morphism of [[crossed module]]s that is given by the commuting diagram

\begin{displaymath}
\itexarray{
    \mathbb{Z} &\stackrel{id}{\to}& \mathbb{Z}
    \\
    \downarrow^{\mathrlap{\cdot 2}} && \downarrow^{\mathrlap{\cdot 2}}
    \\
    \mathbb{Z}
    &\stackrel{}{\hookrightarrow}&
    \mathbb{R}
  }
  \,.
\end{displaymath}
Since $\mathbb{R}$ is contractible, we have indeed under [[geometric realization]] an equivalence

\begin{displaymath}
\itexarray{
    \vert\mathbf{B}^2(\mathbb{Z} \stackrel{\cdot 2}{\to} \mathbb{Z})\vert
    &\stackrel{\vert \hat {\mathbf{\beta}}_2\vert}{\to}&
    \vert
      \mathbf{B}^2 (\mathbb{Z} \stackrel{\cdot 2}{\to} \mathbb{R})
    \vert
    \\
    \downarrow^{\mathrlap{\simeq}}
    &&
    \downarrow^{\mathrlap{\simeq}}
    \\
    \vert\mathbf{B}^2(\mathbb{Z} \stackrel{\cdot 2}{\to} \mathbb{Z})\vert
    &\to&
    \vert\mathbf{B}^2(\mathbb{Z} \to 1)\vert
    \\
    \downarrow^{\mathrlap{\simeq}}
    &&
    \downarrow^{\mathrlap{\simeq}}
    \\
    \vert B^2 \mathbb{Z}_2\vert
     & \stackrel{\beta_2}{\to}&
    \vert B^3 \mathbb{Z}\vert
  }
  \,.
\end{displaymath}
\end{proof}
\begin{prop}
\label{HomotopyFiberOfSmoothBeta2}\hypertarget{HomotopyFiberOfSmoothBeta2}{}
The sequence

\begin{displaymath}
\mathbf{B} U(1)
    \stackrel{\mathbf{c}_1 mod 2}{\to}
  \mathbf{B}^2 \mathbb{Z}_2 
    \stackrel{\mathbf{\beta}_2}{\to}
  \mathbf{B}^2 U(1)
  \,,
\end{displaymath}
where $\mathbf{\beta}_2$ is the smoothly refined [[Bockstein homomorphism]] from prop. \ref{SmoothRefinementOfBockstein}, is a [[fiber sequence]].

\end{prop}
\begin{proof}
The homotopy fiber of $\mathbf{B} \mathbb{Z}_2 \to \mathbf{B}U(1)$ is $U(1)/\mathbb{Z}_2 \simeq U(1)$. Thinking of this is $(\mathbb{Z} \stackrel{\cdot 1/2}{\to} \mathbb{R})$ one sees that the inclusion of this fiber is indeed $\mathbf{c}_1 mod 2$.

\end{proof}
\begin{prop}
\label{}\hypertarget{}{}
The [[delooping]] $\mathbf{B}Spin^c$ of the Lie group $Spin^c$ in [[Smooth∞Grpd]] is the [[homotopy fiber]] of the universal third smooth [[integral Stiefel-Whitney class]] from \ref{SmoothRefinementOfBockstein}.

\begin{displaymath}
\mathbf{B}Spin^c 
   \to 
  \mathbf{B} SO
   \stackrel{\mathbf{W}_3}{\to}
  \mathbf{B}^2 U(1)
  \,,
\end{displaymath}
\end{prop}
\begin{proof}
Then consider the [[pasting diagram]] of [[homotopy pullbacks]]

\begin{displaymath}
\itexarray{
     \mathbf{B}Spin^c 
       &\to& 
     \mathbf{B} U(1)
     &\to&
     {*}
     \\
     \downarrow && \downarrow^{\mathrlap{\mathbf{c}_1 mod 2}} && \downarrow
     \\
     \mathbf{B} SO 
       &\stackrel{\mathbf{w}_2}{\to}&
     \mathbf{B}^2 \mathbb{Z}_2
       &\stackrel{\mathbf{\beta}_2}{\to}&
     \mathbf{B}^2 U(1)
  }
  \,.
\end{displaymath}
The right square is a homotopy pullback by prop. \ref{HomotopyFiberOfSmoothBeta2}. The left square is a homotopy pullback by prop. \ref{SpinCAsHomotopyPullbackOfW2AndC1}. The bottom composite is the smooth $\mathbf{W}_3$ by prop \ref{SmoothRefinementOfBockstein}.

This implies by claim by the [[pasting law]].

\end{proof}
\hypertarget{relation_to_metaplectic_group_}{}\subsubsection*{{Relation to metaplectic group $Mp^c$}}\label{relation_to_metaplectic_group_}

There is a direct analogy between [[Spin]], [[Spin{\tt \symbol{94}}c]] and the [[metaplectic groups]] [[Mp]] and [[Mp{\tt \symbol{94}}c]] (see there for more).

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

\begin{itemize}%
\item [[spin group]], \textbf{$Spin^c$-group}, [[spin structure]]


\item [[spin{\tt \symbol{94}}c structure]], [[twisted spin{\tt \symbol{94}}c structure]]


\item [[string 2-group]], [[string structure]]


\item [[string{\tt \symbol{94}}c 2-group]]



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

See also

\begin{itemize}%
\item [[Robert Gompf]], \emph{$Spin^c$ structures and homotopy equivalences}, Geom. Topol. 1 (1997) 41-50 (\href{https://arxiv.org/abs/math/9705218}{arXiv:math/9705218})

\end{itemize}
[[!redirects spin{\tt \symbol{94}}c group]]

[[!redirects Spin{\tt \symbol{94}}c]]



\end{document}