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\newcommand{\id}{\mathop{\rm id}\nolimits} \)
PONTRYAGIN CONSTRUCTION
ISHAN LEVY
1. Introduction
The Pontryagin construction is a way of relating framed submanifolds to homotopy classes of maps
to a sphere. Here a framed submanifold is a submanifold with a trivialization of the normal
frame bundle. Now we don’t want to consider all framed submanifolds, but rather mod out by an
equivalence relation called cobordism. We say that \(N,N' \subset M\) are cobordant manifolds if \(N\times [0,\ee ] \cup N' \times [1-\ee ,1]\) can be extended
in the interval \(M\times [\ee ,1-\ee ]\) to a submanifold of \(M\times [0,1]\) with boundary \(N \cup N'\). In particular we would like to
consider framed submanifolds up to framed cobordism, where we require the extension
to be framed. I will use \(\simeq \) to denote homotopic maps and \(\sim \) to denote framed cobordant
submanifolds.
Throughout, we will assume that \(M\) is compact, \(f: M \to S^p\) (\(S^p\) oriented) a smooth map, \(y\) a regular value, we
naturally get a framed submanifold by looking at \(N_f=f^{-1}(y)\), and \(f_{|N_f}\) induces a bundle map on \(N_f\)’s normal bundle
and \(T_y\), trivializing it using a positively oriented basis of \(T_yS^p\).
We would like to prove:
Theorem 1.1. \(N_f\) is well defined up to framed cobordism class, and only depends on the
homotopy class of \(f\). Moreover, \(f \mapsto N_f\) gives a bijection between framed compact cobordism classes
of codimension \(p\) and \([M,S^p]\).
2. Well defined
We will begin by showing the first statement. First note that the cobordism class doesn’t depend
on the basis we chose for \(f\), only the orientation, since \(\GL _n(\RR )^+\) is connected (This can be proven by using
row/column operations carefully or using Graham-Schmidt to reduce to showing \(\SOr (n)\) is
connected, which is done by using induction and the fibration \(\SOr (n-1)\hookrightarrow \SOr (n) \to S^{n-1}\)). Then given two choices of
frames on \(N_f\), they are pullbacks of two different elements of the tangent frame bundle of \(y\),
so by choosing a smooth path on \(\GL _n(\RR )^+\) that is constant on \([0,\ee ]\cup [1-\ee ,1]\), we have framed \(y \times [0,1]\) in \(S^p \times [0,1]\), and by
considering the natural map induced by \(f\) from \(M\times [0,1] \to S^p \times [0,1]\), this framing of \(y \times [0,1]\) induces a cobordism
between the two frames of \(N_f\). Thus we will ignore the particular frame chosen at \(y\) from now
on.
We would like to show the cobordism class is well defined up to homotopy. Given a
homotopy, we would like to take the preimage of \(y\) on the homotopy to get a cobordism.
Unfortunately \(y\) is not necessarily a regular value of the homotopy. To fix this, the following
lemma:
Lemma 2.1. The cobordism class of \(f^{-1}(z)\) is constant for \(z\) in a neighborhood of \(y\).
Proof. The set of critical points is compact as \(M\) is, hence there is a convex neighborhood of \(y\)
consisting of regular values. Now choosing a family \(r_t\) of smooth rotations of the sphere that
takes \(y\) to \(z\), and is constant on \([0,\ee ]\cup [1-\ee ,1]\). Then consider the map \(r \circ f:M \times [0,1]\to S^p\times [0,1] \to S^p\). \(y\) is regular for \(r \circ f\), so we get a cobordism
between \(f^{-1}(y)\) and \(f^{-1}(z)\). □
Theorem 2.2. The cobordism class is well defined, and is only dependant on homotopy
class.
Proof. First note that if \(f \simeq g\), then we can assume the homotopy is constant on \([0,\ee ]\cup [1-\ee ,1]\), and choose \(z\) a
regular value of the homotopy satisfying the conditions of the previous lemma for \(f\) and \(g\) so
that \(f^{-1}(y) \sim f^{-1}(z) \sim g^{-1}(z)\sim g^{-1}(y)\). Now if \(z\) is another regular value, and \(r\) a rotation sending \(z\) to \(y\), \(r\circ f \simeq f\) so \(f^{-1}(y)\sim (r\circ f)^{-1}(y) = f^{-1}(z)\). □
3. Surjectivity
We would now like to show that for any framed submanifold \(N\), we can produce a map \(f\) with
\(N_f \sim N\).
Lemma 3.1 (Tubular Neighborhood Theorem). Let \(P\subset M\) be submanifold of codimension \(p\), with \(P\)
compact. Then there is a neighborhood of \(P\) diffeomorphic to the normal bundle of \(P\), with \(P\) as
the \(0\)-section.
Proof. By exponentiating the normal bundle, we get a local diffeomorphism \(P\times B_\ee \to M\), and since \(B_\ee \) is
diffeomorphic to \(\RR ^p\), it suffices to show that for small \(\ee \), this is injective. However, if \((p_i,x_i)\), \((q_i,y_i)\) are a
sequence of points for which it is not injective with the magnitude of the \(x_i,y_i\) going to \(0\), by
compactness of \(P\times \overline B_{\frac{\ee }{2}}\), we can extract a convergent subsequence, which contradicts local injectivity.
□
This Lemma holds for non-compact submanifolds but the proof is a bit more annoying.
Theorem 3.2. The map \(f \to N_f\) is surjective.
Proof. We consider a tubular neighborhood of a framed submanifold \(N\), giving a map \(f:\RR ^p\times N \to \RR ^p\). Now
consider \(S^p = y_0 \cup \RR ^p\), and smoothly extend \(f\) to \(M\) by setting all other values to \(y_0\). Then \(f^{-1}(0) = N\). □
4. Injectivity
We would now like to show that if we have a cobordism \(f^{-1}\sim g^{-1}\) via some framed submanifold \(P \subset M\times [0,1]\), \(f \simeq g\). To do
this, given the cobordism, we would like to use the proof of surjectivity on the cobordism to yield a
homotopy. However, this still leaves us to prove:
Lemma 4.1. If \(f^{-1}(y)=g^{-1}(y)=N\), \(f \simeq g\).
Proof. If \(f,g\) agree on a neighborhood of \(N\), then removing the neighborhood, we get a map to
\(\RR ^p\) instead of \(S^p\), which we can linearly homotopy without spoiling the overall smoothness. So
it suffices to deform \(f\) to agree with \(g\) in a neighborhood of \(N\). To do this, choose a tubular
neighborhood \(N \times \RR ^p\) that misses the antipode \(y_0\) of \(y\). Then we have maps \(F,G:N\times \RR ^p \to \RR ^p\) with \(DF_{|N\times 0} = DG_{|N\times 0}\), and we can assume
that \(DF_{|N\times 0}\) is the identity on each \(n\times \RR ^p\). We would like to linearly deform \(f\) to match \(g\), but we would like
to avoid adding new zeroes. To do this, note by compactness of \(N\), there is an \(\delta \) ball around \(0\)
such that when \(F,G\) are restricted to it, \(\Vert DF-I\Vert ,\Vert DG-I \Vert <\ee \). Then \(\Vert F(n,x)-x\Vert \leq \Vert c x^2\Vert \) for small \(\Vert x\Vert \) by Taylor’s theorem, so by multiplying
by \(\Vert x\Vert \) on either side and using Cauchy Schwarz, we get \(|F(n,x)\cdot x|\geq \Vert x\Vert ^2-c\Vert x\Vert ^3\) which is positive when \(\Vert x\Vert \) and \(c\) are small.
Then doing the same with \(G\), we find that \(F\) and \(G\) lie in the same half plane for small \(\Vert x\Vert \), so that
we can linearly deform \(F\) to match \(G\) locally without adding new \(0\)s. □
Theorem 4.2. If \(N_f \sim N_g\), \(f \simeq g\).
Proof. As in the proof of surjectivity, choose a tubular neighborhood of a cobordism and
construct a homotopy \(H\) such that \(H^{-1}(y)\) is the cobordism. Now by the previous lemma, \(f\simeq H_0 \simeq H_1\simeq g\). □
5. Applications
The Pontryagin construction can be viewed as a generalization of degree theory, and we can see
that the most trivial case of it does coincide with degree theory.
Theorem 5.1 (Theorem of Hopf). If \(M^n\) is compact, orientable, and connected, then \([M^n,S^n] \cong \ZZ \), where the
isomorphism is given by degree. If \(M^n\) is non-orientable, then \([M^n,S^n] \cong \ZZ /2\ZZ \) with the isomorphism given by degree
mod \(2\). In particular, \(\pi _n(S^n) \cong \ZZ \).
Proof. The codimension \(0\) compact framed submanifolds are finite collections of points with a
\(\pm 1\) orientation. Now it is clear that if \(M\) is orientable, then the cobordism class is only dependant
on degree, ie. the sum of these orientations. If \(M\) is not orientable, then points with positive or
negative orientation are the same up to cobordism, so degree mod \(2\) determines the cobordism
class. □