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\begin{document}

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\section*{framed manifold}

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

\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{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{stable_homotopy_elements}{Stable homotopy elements}\dotfill \pageref*{stable_homotopy_elements} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{IntersectionPairing}{Relation to intersection pairing and Kervaire invariant}\dotfill \pageref*{IntersectionPairing} \linebreak
\noindent\hyperlink{properties_2}{Properties}\dotfill \pageref*{properties_2} \linebreak
\noindent\hyperlink{moduli_of_framings}{Moduli of framings}\dotfill \pageref*{moduli_of_framings} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{for_2manifolds}{For 2-manifolds}\dotfill \pageref*{for_2manifolds} \linebreak
\noindent\hyperlink{for_3manifolds}{For 3-manifolds}\dotfill \pageref*{for_3manifolds} \linebreak


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

In one sense of the term, a \emph{framing} of a [[manifold]] is a choice of trivialization of its [[tangent bundle]], hence a choice of [[section]] of the corresponding [[frame bundle]].

A manifold that admits a framing is also called a \textbf{parallelizable manifold}. A manifold equipped with a framing is also called a \textbf{parallelized manifold}.

More generally, one means by a \emph{framing} not a trivialization of the tangent bundle itself, but

\begin{itemize}%
\item of the [[normal bundle]] if the manifold is understood [[embedding|embedded]] in some [[Cartesian space]] $\mathbb{R}^d$


\item of the \emph{[[stable tangent bundle]]}.



\end{itemize}
Accordingly, a \emph{framed cobordism} is a [[cobordism]] equipped with a framing on the underlying manifold.

For $dim(X)$ the [[dimension]] of the manifold and $n \geq dim(X)$, then one also speaks of an \emph{$n$-framing} to mean a trivialization of the ``$n$-stabilized tangent bundle'' $T X \oplus \mathbb{R}^{n-dim(X)}$ (where the right [[direct sum|direct summand]] denotes the trivial real [[vector bundle]] of [[rank]] $n - dim(X)$). These $n$-framed manifolds appear in particular in the construction of the [[cobordism category]] of framed $n$-dimensional cobordisms. The [[cobordism hypothesis]] asserts essentially that the [[(∞,n)-category of cobordisms]] with $n$-framing is the [[free construction|free]] [[symmetric monoidal (∞,n)-category with duals]].

Beware that there is also the term ``[[2-framing]]'' due to (\hyperlink{Atiyah}{Atiyah}), which is related but different.

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

\begin{prop}
\label{}\hypertarget{}{}
Every [[Lie group]] is a parallelizable manifold.

\end{prop}
\begin{proof}
Every non-zero [[invariant vector field]] on the Lie group provides an everywhere non-vanishing section of the tangent bundle.

\end{proof}
The following is obvious:

\begin{prop}
\label{SpinManifoldAdmitsFraming}\hypertarget{SpinManifoldAdmitsFraming}{}
Every 3-[[dimension|dimensional]] [[manifold]] with [[spin structure]] admits a framing.

\end{prop}
\begin{proof}
That a 3-manifold $X$ has [[spin structure]] means that we have a [[reduction of the structure group]] of the tangent bundle to the [[spin group]], and hence the tangent bundle is classified by a map $X \to B Spin(3)$. But $Spin(3)$ has vanishing [[homotopy groups]] in degree $0 \leq k \leq 2$. Therefore its [[delooping]] [[classifying space]] $B Spin(3)$ has vanishing homotopy groups below degree 4 and hence every morphism out of a 3-dimensional manifold into it is homotopically constant.

\end{proof}
But in fact the following stronger statement is also true.

\begin{prop}
\label{EveryOrientable3ManifoldsIsParallelizable}\hypertarget{EveryOrientable3ManifoldsIsParallelizable}{}
Every [[orientation|orientable]] 3-[[dimension|dimensional]] [[manifold]] admits a framing.

\end{prop}
\begin{proof}
By the argument in the proof of prop. \ref{SpinManifoldAdmitsFraming}, the only possible obstruction is the [[second Stiefel-Whitney class]] $w_2$. By the discussion at \emph{[[Wu class]]}, this vanishes on an oriented manifold precisely if the second [[Wu class]] vanishes. This in turn is by definition defined to represent the [[Steenrod square]] under [[cup product]], and this vanishes on a 3-manifold by degree reasons.

\end{proof}
\begin{remark}
\label{In4dFirstOrderGravity}\hypertarget{In4dFirstOrderGravity}{}
Prop. \ref{EveryOrientable3ManifoldsIsParallelizable} has some impact in the context of the [[first order formulation of gravity]], where one is interested in [[vielbein]] fields on 4-dimensional [[spacetime manifolds]], and in particular in [[globally hyperbolic spacetimes]], which, as [[topological spaces]], are the [[Cartesian product]] of the [[real line]] (time) with a [[3-manifold]] (space). Prop. \ref{EveryOrientable3ManifoldsIsParallelizable} implies that already when that spatial 3-manifold is orientable, then the whole globally hyperbolic spacetime admits a framing. (See also at \emph{[[teleparallel gravity]]}.)

\end{remark}
\begin{theorem}
\label{}\hypertarget{}{}
The $n$-[[spheres]] that admit a framing are precisely only

\begin{itemize}%
\item the 0-sphere $S^0 = \ast \coprod \ast$, the unit [[real numbers]]


\item the 1-sphere $S^1$, the [[circle]] underlying the [[circle group]] (the unit [[complex numbers]]);


\item the 3-sphere $S^3$, underlying the [[special unitary group]] $SU(2)$, is isomorphic to the unit [[quaternions]];


\item the 7-sphere $S^7$, which underlies a [[Moufang loop]] internal to [[Diff]], namely the unit [[octonions]],



\end{itemize}
where the algebras appearing are precisely the four [[normed division algebras]].

\end{theorem}
This is due to (\hyperlink{Adams58}{Adams 58}), proven with the [[Adams spectral sequence]].

\hypertarget{stable_homotopy_elements}{}\subsubsection*{{Stable homotopy elements}}\label{stable_homotopy_elements}

Left invariant framings $\mathcal{L}$ on compact connected Lie groups $G$ with $dim(G)=k$ give rise to elements $[G,\mathcal{L}]$ in the stable homotopy group $\pi_k^s$ of spheres. One can restrict attention to semisimple Lie groups $G$ since this construction behaves well with respect to products, $G \to T\times G/T$ gives a framed diffeomorphism and $[S^1,\mathcal{L}] \in \pi_1^s$ is the generator. The following facts are assembled from (\href{https://doi.org/10.1016/0040-9383(82%2990013-1}{Ossa 1982}) and (\href{http://projecteuclid.org/euclid.jmsj/1468956166}{Minami 2016})

\begin{itemize}%
\item For any \emph{semisimple} compact connected Lie group $G$, Ossa proved that $72[G,\mathcal{L}]=0$. In particular the only possible torsion is at the primes 2 and 3.


\item The left invariant framings on $SO(n),Spin(n)$ and $SU(n)$ give the zero element for $n\geq 7$, those on $Sp(n)$ for $n\geq 4$, as do those on $F_4,E_6,E_7,E_8, SO(4),SO(6),SU(5),SU(6)$. (this means that the undetermined cases of rank 4 in Ossa's Table 1 are in fact all 0)


\item The left invariant framings on the remaining groups ($SU(2)=Spin(3)=Sp(1),SU(3),SU(4)=Spin(6),Sp(2)=Spin(5),Sp(3),SO(3),SO(5)$ and $G_2$) represent known nonzero classes in $\pi_\ast^s$.



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

\hypertarget{IntersectionPairing}{}\subsubsection*{{Relation to intersection pairing and Kervaire invariant}}\label{IntersectionPairing}

On a framed manifold there is a canonical [[quadratic refinement]] of the [[intersection pairing]]. The associated invariant is the \emph{[[Kervaire invariant]]}.

\hypertarget{properties_2}{}\subsection*{{Properties}}\label{properties_2}

\hypertarget{moduli_of_framings}{}\subsubsection*{{Moduli of framings}}\label{moduli_of_framings}

The [[homotopy type]] of the [[moduli space of framed manifolds|moduli space of framings]] on a fixed manifold is a disjoint union of subgroups of the oriented [[mapping class group]] which fix a given isotopy type of framings.

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

\begin{itemize}%
\item [[normal framing]]


\item [[2-framing]]


\item [[framed elliptic curve]]


\item [[moduli space of framed manifolds]]


\item [[teleparallel gravity]]



\end{itemize}
Formalization in [[differential cohesion]] is discussed \href{http://ncatlab.org/nlab/show/differential+cohesive+%28infinity%2C1%29-topos#GLnTangentBundles}{there}.

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

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

Rieview in the context of the [[Kervaire invariant]] includes

\begin{itemize}%
\item [[Akhil Mathew]], \emph{The Kervaire invariant I} (\href{https://amathew.wordpress.com/2012/10/01/the-kervaire-invariant-i/#more-3888}{web})

\end{itemize}
The theorem about the parallizablitiy of spheres is due to

\begin{itemize}%
\item [[John Adams]], \emph{On the Non-Existence of Elements of Hopf Invariant One} Bull. Amer. Math. Soc. 64, 279-282, 1958, Ann. Math. 72, 20-104, 1960.

\end{itemize}
Relation to existence of [[flat connections]] on the [[tangent bundle]] is discussed in

\begin{itemize}%
\item [[John Thorpe]], \emph{Parallelizablility and flat manifolds}, 1965 ([[ThorpeParallelizable.pdf:file]])

\end{itemize}
\hypertarget{for_2manifolds}{}\subsubsection*{{For 2-manifolds}}\label{for_2manifolds}

The [[moduli space of framed surfaces]] is discussed in

\begin{itemize}%
\item [[Oscar Randal-Williams]], \emph{Homology of the moduli spaces and mapping class groups of framed, r-Spin and Pin surfaces} (\href{http://arxiv.org/abs/1001.5366}{arXiv:1001.5366})

\end{itemize}
\hypertarget{for_3manifolds}{}\subsubsection*{{For 3-manifolds}}\label{for_3manifolds}

\begin{itemize}%
\item [[Rob Kirby]], [[Paul Melvin]], \emph{Canonical framings for 3-manifolds}, Turkish Journal of Mathematics, volume 23, number 1,1999 ([[KirbyMelvon3Framings.pdf:file]])

\end{itemize}
The relation to ``[[2-framings]]'' is duscussed in

\begin{itemize}%
\item [[Michael Atiyah]], \emph{On framings of 3-manifolds} (\href{http://www.maths.ed.ac.uk/~aar/papers/atiyahfr.pdf}{pdf})

\end{itemize}
[[!redirects framed manifolds]]

[[!redirects framed cobordism]] [[!redirects framed cobordisms]]

[[!redirects parallelizable manifold]] [[!redirects parallelized manifold]] [[!redirects parallelizable manifolds]] [[!redirects parallelized manifolds]]

[[!redirects framing]] [[!redirects framings]]

[[!redirects parallelizable sphere]] [[!redirects parallelizable spheres]]

[[!redirects parallelizable]]

[[!redirects n-framing]] [[!redirects n-framings]] [[!redirects 1-framing]] [[!redirects 1-framings]] [[!redirects 3-framing]] [[!redirects 3-framings]] [[!redirects 4-framing]] [[!redirects 4-framings]] [[!redirects 5-framing]] [[!redirects 5-framings]]



\end{document}