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\section*{Atiyah 2-framing}

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

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

[[!include cohomology - contents]]

\hypertarget{manifolds_and_cobordisms}{}\paragraph*{{Manifolds and cobordisms}}\label{manifolds_and_cobordisms}

[[!include manifolds and cobordisms - 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{RelationToBounding4Manifolds}{Relation to bounding 4-manifolds}\dotfill \pageref*{RelationToBounding4Manifolds} \linebreak
\noindent\hyperlink{RelationToStringStructures}{Relation to String-structures}\dotfill \pageref*{RelationToStringStructures} \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}

In the general terminology of \emph{$n$-[[framing]]} then a \emph{2-framing} of a [[manifold]] $\Sigma$ of [[dimension]] $d \leq 2$ is a trivialization of $T \Sigma \oplus \mathbb{R}^{2-d}$.

In (\hyperlink{Atiyah1990}{Atiyah 90}) the term ``2-framing'' is instead used for a trivialization of the double of the tangent bundle of a 3-manifold. So this is a different concept, but it turns out to be closely related to the \emph{3-framing} (in the previous sense) of surfaces.

For $X$ a [[compact space|compact]], [[connected]], [[orientation|oriented]] 3-[[dimension|dimensional]] [[manifold]], write

\begin{displaymath}
2 T X := T X \oplus T X
\end{displaymath}
for the [[direct sum of vector bundles|fiberwise direct sum]] of the [[tangent bundle]] with itself. Via the [[diagonal]] embedding

\begin{displaymath}
SO(3) \to SO(3) \times SO(3) \hookrightarrow SO(6)
\end{displaymath}
this naturally induces a [[special orthogonal group|SO(6)]]-[[principal bundle]].

\begin{prop}
\label{}\hypertarget{}{}
The underlying $SO(6)$-principal bundle of $2 T X$ always admits a [[lift of structure group|lift]] to a [[spin group|spin(6)]]-[[principal bundle]].

\end{prop}
\begin{proof}
By the sum-rule for [[Stiefel-Whitney classes]] (see at \href{Stiefel-Whitney+class#AxiomaticDefinition}{SW class -- Axiomatic definition}) we have that

\begin{displaymath}
w_2(2 T X) = 2 w_0(T X) \cup w_2(T X) + w_1(T X) w_1(T X)
  \,.
\end{displaymath}
Since $T X$ is assumed [[orientation|oriented]], $w_1(T X) = 0$ (since this is the [[obstruction]] to having an [[orientation]]). So $w_2(2 T X) = 0 \in H^2(X,\mathbb{Z}_2)$ and since this in turn is the further [[obstruction]] to having a [[spin structure]], this does exist.

\end{proof}
Therefore the following definition makes sense

\begin{defn}
\label{}\hypertarget{}{}
A \textbf{2-framing} in the sense of (\hyperlink{Atiyah1990}{Atiyah 90}) on a [[compact space|compact]], [[connected]], [[orientation|oriented]] 3-[[dimension|dimensional]] [[manifold]] $X$ is the [[homotopy class]] of a trivializations of the [[spin-group]]-[[principal bundle]] underlying [[Whitney sum|twice]] its [[tangent bundle]].

\end{defn}
More in detail, we may also remember the [[groupoid]] of 2-framings and the smooth structure on collections of them:

\begin{defn}
\label{}\hypertarget{}{}
The [[moduli stack]] $At2\mathbf{Frame}$ is the [[homotopy pullback]] in

\begin{displaymath}
\itexarray{
    At2\mathbf{Frame} &\to& *
    \\
    \downarrow && \downarrow
    \\
    \mathbf{B}SO(3) &\stackrel{}{\to}&
    \mathbf{B} Spin(6)
  }
\end{displaymath}
in [[Smooth∞Grpd]].

\end{defn}
In terms of this a 2-framing on $X$ with [[orientation]] $\mathbf{o} \colon X \to \mathbf{B}SO(3)$ is a lift $\hat {\mathbf{o}}$ in

\begin{displaymath}
\itexarray{
    && At 2 \mathbf{Frame}
    \\
    & {}^{\mathllap{\hat {\mathbf{o}}}}\nearrow & \downarrow
    \\
    X &\stackrel{\mathbf{o}}{\to}& \mathbf{B}SO(3)
  }
  \,.
\end{displaymath}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{RelationToBounding4Manifolds}{}\subsubsection*{{Relation to bounding 4-manifolds}}\label{RelationToBounding4Manifolds}

In (\hyperlink{Atiyah1990}{Atiyah}) it is shown how a framing on a compact connected oriented 3-manifold $X$ is induced by a 4-manifold $Z$ with [[boundary]] $\partial Z \simeq X$. In fact, a framing is equivalently a choice of [[cobordism]] class of bounding 4-manifolds (\hyperlink{Kerler}{Kerler}).

Discussion of 2-framing entirely in terms of bounding 4-manifolds is for instance in (\hyperlink{Sawin}{Sawin}).

\hypertarget{RelationToStringStructures}{}\subsubsection*{{Relation to String-structures}}\label{RelationToStringStructures}

By (\hyperlink{Atiyah1990}{Atiyah 2.1}) an Atiyah 2-framing of a 3-manifold $X$ is equivalently a\newline $p_1$-[[twisted differential c-structures|structure]], where $p_1$ is the first [[Pontryagin class]], hence is a homotopy class of a trivialization of

\begin{displaymath}
p_1(X) \colon X \to B SO(3) \stackrel{p_1}{\to} K(\mathbb{Z},4)
  \,.
\end{displaymath}
This perspective on Atiyah 2-framings is made explicit in (\hyperlink{BunkeNaumann}{Bunke-Naumann, section 2.3}). It is mentioned for instance also in (\hyperlink{Freed08}{Freed, page 6, slide 5}).

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

\begin{itemize}%
\item [[framed manifold]]


\item [[p1-structure]], [[string structure]].



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

The notion of ``2-framing'' in the sense of framing of the double of the tangent bundle is due to

\begin{itemize}%
\item [[Michael Atiyah]], \emph{On framings of 3-manifolds} , Topology, Vol. 29, No 1, pp. 1-7 (1990) (\href{http://www.maths.ed.ac.uk/~aar/papers/atiyahfr.pdf}{pdf})

\end{itemize}
making explicit a structure which slightly implicit in the discussion of the [[perturbation theory|perturbative]] [[path integral]] [[quantization of 3d Chern-Simons theory]] in

\begin{itemize}%
\item [[Edward Witten]], \emph{Quantum field theory and the Jones Polynomial} , Comm. Math. Phys. 121 (1989)

\end{itemize}
reviewed for instance in

\begin{itemize}%
\item M. B. Young, section 2 of \emph{Chern-Simons theory, knots and moduli spaces of connections} (\href{http://www.math.sunysb.edu/~myoung/CS.pdf}{pdf})

\end{itemize}
(see \hyperlink{Atiyah}{Atiyah, page 6}). For more on the role of 2-framings in [[Chern-Simons theory]] see also

\begin{itemize}%
\item [[Daniel Freed]], Robert Gompf, \emph{Computer calculation of Witten's 3-Manifold invariant}, Commun. Math. Phys. 141,79-117 (1991) (\href{http://www.maths.ed.ac.uk/~aar/papers/freedgompf.pdf}{pdf})


\item [[Gregor Masbaum]], section 2 of \emph{Spin TQFT and the Birman-Craggs Homomorphism}, Tr. J. of Mathematics 19 (1995) \href{http://journals.tubitak.gov.tr/math/issues/mat-95-19-2/pp-189-199.pdf}{pdf}


\item [[Daniel Freed]], \emph{Remarks on Chern-Simons theory} (\href{http://arxiv.org/abs/0808.2507}{arXiv:0808.2507}, \href{http://www.ma.utexas.edu/users/dafr/MSRI_25.pdf}{pdf slides})



\end{itemize}
and for discussion in the context of the [[M2-brane]] from p. 7 on in

\begin{itemize}%
\item [[Hisham Sati]], \emph{[[Geometric and topological structures related to M-branes]] II: Twisted $String$ and $String^c$-structures} (\href{http://arxiv.org/abs/1007.5419}{arXiv:1007.5419}).

\end{itemize}
The relation to $p_1$-structure is made explicit in

\begin{itemize}%
\item [[Ulrich Bunke]], [[Niko Naumann]], section 2.3 of \emph{Secondary Invariants for String Bordism and tmf}, Bull. Sci. Math. 138 (2014), no. 8, 912--970 (\href{http://arxiv.org/abs/0912.4875}{arXiv:0912.4875})


\item C. Blanchet, N. Habegger, [[Gregor Masbaum]], [[Pierre Vogel]], \emph{Topological quantum field theories derived from the Kauffman bracket}, Topology Vol 34, No. 4, pp. 883-927 (1995) (\href{http://www.maths.ed.ac.uk/~aar/papers/bhmv.pdf}{pdf})



\end{itemize}
More discussion in terms of bounding 4-manifolds is in

\begin{itemize}%
\item Thomas Kerler, \emph{Bridged links and tangle presentations of cobordism categories}. Adv. Math., 141(2):207--281, (1999) (\href{http://arxiv.org/abs/math/9806114}{arXiv:math/9806114})


\item Stephen F. Sawin, \emph{Three-dimensional 2-framed TQFTS and surgery} (2004) (\href{http://digitalcommons.fairfield.edu/cgi/viewcontent.cgi?article=1020&context=mathandcomputerscience-facultypubs}{pdf})



\end{itemize}
and page 9 of

\begin{itemize}%
\item Stephen Sawin, \emph{Invariants of Spin Three-Manifolds From Chern-Simons Theory and Finite-Dimensional Hopf Algebras} (\href{http://arxiv.org/abs/math/9910106}{arXiv:math/9910106}).

\end{itemize}
and more discussion for 3-manifolds with boundary includes

\begin{itemize}%
\item Thomas Kerler, [[Volodymyr Lyubashenko]], section 1.6.1 of \emph{Non-semisimple topological quantum field theories for 3-manifolds with corners}, Lecture notes in mathematics 2001

\end{itemize}
See also

\begin{itemize}%
\item [[Greg Kuperberg]], \emph{\href{http://mathoverflow.net/a/4389/381}{MO comment}}

\end{itemize}
[[!redirects 2-framing]] [[!redirects 2-framings]]



\end{document}