Pontryagin Thom and Framed Cobordism

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Let $S\subseteq X$ be a closed embedded smooth submanifold of a smooth manifold $X$ without boundary, with normal bundle $\nu (S)=TX{\mathrm{\mid }}_S\mathrm{/}TS$ of rank $k$, a smooth vector bundle over $S$ by Assuming countable choice, normal and conormal bundles are smooth vector bundles (Normal and conormal bundles of an embedded submanifold, Smooth embeddings).

A framing of $S$ in $X$ is a smooth bundle isomorphism $\phi :\nu (S)⟶S\times {\mathbb{R}}^k$ over ${\mathrm{i}\mathrm{d}}_S$ (Smooth vector bundles, rank, fibres, and trivial bundles); equivalently a trivialization of $\nu (S)$, equivalently a global frame of its sections, and in Milnor's metric language the same thing as a framing of the orthogonal complement $T{S}^{\perp }$. A framed submanifold is a pair $(S,\phi )$. When a Riemannian metric on $X$ is supplied, the orthogonal-complement model of the normal bundle is canonically identified with $\nu (S)$ by Assuming countable choice, an ambient metric identifies the two normal bundles, which is how Milnor's framings are read in the metric-free quotient convention used here; a change of ambient metric changes that comparison but not the framing data itself.

Rank $k=0$ and $S=\mathrm{\varnothing }$ are included (Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right): a framing of a rank-zero bundle is the unique bundle isomorphism onto $S\times {\mathbb{R}}^0=S$, and the empty framing is unique. A framing is an actual trivialization of $\nu (S)$, never merely a stable isomorphism of it: adding trivial summands to a normal bundle is a different construction, and the stable and unstable notions are kept apart throughout (Smooth vector bundles, rank, fibres, and trivial bundles).

The countable-choice hypothesis is inherited solely from the smooth normal-bundle structure of Assuming countable choice, normal and conormal bundles are smooth vector bundles; this definition itself selects nothing and proves nothing. The symbol $\nu (S)$ always denotes the quotient normal bundle $TX{\mathrm{\mid }}_S\mathrm{/}TS$, and the rank-$k$ trivialization that fixes the identification of the normal fibres with ${\mathbb{R}}^k$ is part of the data, not a choice made afterwards.

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Let $X$ be a closed smooth manifold and $k\ge 0$ (Smooth manifolds and their smooth charts, Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right). A framed cobordism from a closed framed codimension-$k$ submanifold $(N_0,{\phi }_0)$ of $X$ to another $(N_1,{\phi }_1)$ is data $(W,\epsilon ,\mathrm{\Psi })$ consisting of

• a compact neat embedded submanifold $W\subseteq X\times I$ of dimension $\dim X-k+1$ with $\mathrm{\partial }W=N_0\times \{0\}\bigsqcup N_1\times \{1\}$, the $N_i$ read in the slice $X\times \{i\}$ (Neat submanifolds of a manifold with boundary, Smooth embeddings; here $I=[0,1]$ with its product smooth structure and $X$ has been identified with $X\times \{i\}$);
• a number $\epsilon \in (0,\frac{1}{2})$ such that the ends of $W$ are exactly the products $W\cap (X\times [0,\epsilon ))=N_0\times [0,\epsilon ),W\cap (X\times (1-\epsilon ,1])=N_1\times (1-\epsilon ,1];$ equivalently, the product collar embeddings ${\theta }_0:N_0\times [0,\epsilon )\to W$, ${\theta }_0(x,s)=(x,s)$ and ${\theta }_1:N_1\times (1-\epsilon ,1]\to W$, ${\theta }_1(x,s)=(x,s)$ are part of the data, with ${\theta }_i(x,i)=(x,i)$ and images exactly the two ends;
• a framing $\mathrm{\Psi }:\nu (W\subseteq X\times I)\to W\times {\mathbb{R}}^k$ (Framings of a normal bundle) which, over each end collar $N_i\times {\mathrm{\Theta }}_i$ (where ${\mathrm{\Theta }}_0=[0,\epsilon )$ and ${\mathrm{\Theta }}_1=(1-\epsilon ,1]$), is the pullback of ${\phi }_i$ along the product projection, under the canonical identification $\nu (W\subseteq X\times I){\mid }_{N_i\times {\mathrm{\Theta }}_i}\cong {\mathrm{pr}}_{N_i}^{\ast }\nu (N_i\subseteq X)$ induced by the product structure: along the whole collar the $I$-direction is tangent to $W$, so the quotient normal of $W$ in $X\times I$ restricts there to the quotient normal of $N_i$ in $X$. In particular, at the end slice $t=i$ the restriction of $\mathrm{\Psi }$ corresponds to ${\phi }_i$; requiring the constancy over the whole collar is Milnor's normalisation $u^i(x,t)=(v^i(x),0)$.

Two closed framed codimension-$k$ submanifolds of $X$ are framed cobordant when such data exist. The relation is introduced here only as a relation; that it is reflexive, symmetric and transitive is proved in Framed cobordism is an equivalence relation.

No orientation of $X$ or of the $N_i$ is used, and no direction of the normal bundle is singled out: all signs are carried by the actual framings ${\phi }_0,{\phi }_1,\mathrm{\Psi }$. The product ends and the constant framings on them are data, not choices made afterwards, so the restriction of $\mathrm{\Psi }$ to each end is a literal equality with ${\phi }_i$, with no implicit inward-normal sign and no implicit straightening of a general collar; this is Milnor's definition of cobordism within $M$, in which the subset $N_0\times [0,\epsilon )\cup N_1\times (1-\epsilon ,1]$ extends to $W$. The empty manifold is allowed as $N_0$, as $N_1$ and as $W$, and $k=0$ is allowed; for $k=0$ the normal bundles are rank zero, the framings are unique, and the condition on $\mathrm{\Psi }$ is vacuous. A framed cobordism $(W,\epsilon ,\mathrm{\Psi })$ also yields an (unoriented) bordism $(W,{\theta }_0,{\theta }_1)$ in the sense of Unoriented smooth cobordism of closed manifolds after rescaling ${\theta }_i$ to the standard widths, since $W$ is a compact smooth manifold with boundary and the ${\theta }_i$ are collars onto the two boundary parts. The countable-choice hypothesis is inherited from the smooth normal-bundle structure through Framings of a normal bundle; the definition itself selects nothing.

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). For every closed smooth manifold $X$ and every $k\ge 0$, framed cobordism of closed framed codimension-$k$ submanifolds of $X$ (Framed cobordism of framed submanifolds) is an equivalence relation.

Reflexivity is realised by the cylinder $N\times I\subseteq X\times I$ together with the pullback of $\phi$ along ${\mathrm{pr}}_N$; symmetry is realised by the reflection $\rho (x,t)=(x,1-t)$ of $X\times I$ with the framing transported across $d\rho$; transitivity is realised by gluing two framed cobordisms along their common product end, the framings agreeing on the whole overlap collar.

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$) is inherited through the normal-bundle structure). Let $X$ be a closed smooth manifold and let $(N,\phi )$ be a closed framed codimension-$k$ submanifold of $X$, $k\ge 0$, with normal bundle $\nu =\nu (N\subseteq X)$ (Framings of a normal bundle).

Then $\phi$ induces a based homeomorphism ${\mathrm{\Phi }}_{\phi }:\mathrm{Th}(\nu )⟶N_{+}\wedge S^k,$ natural in the framed data, and the composite of any based map $c:X_{+}\to \mathrm{Th}(\nu )$ with the based projection $N_{+}\wedge S^k\to S^0\wedge S^k=S^k$ is a based map $X_{+}\to S^k$. The homeomorphism is independent of the metric used to form $\mathrm{Th}(\nu )$ up to the canonical radial homeomorphisms of Metric independence of the Thom space. For $k=0$, $\mathrm{Th}(\nu )=N_{+}\cong N_{+}\wedge S^0$ and the sphere-valued target is $S^0$. For $N=\mathrm{\varnothing }$, the Thom space and $N_{+}\wedge S^k$ are one-point spaces, and the composite $X_{+}\to S^k$ is constant at the basepoint.

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Let $X$ be closed and smooth and let $(N,\phi )$ be a closed framed codimension-$k$ submanifold, $k\ge 0$ (Framings of a normal bundle). A compatible chart $\mathrm{\Phi }$ for $(\nu (N),\mathrm{i}\mathrm{d})$, a supplied smooth metric and a sufficiently small radius give a collapse $c:X_{+}\to \mathrm{Th}(\nu (N))$ as in Pontryagin–Thom collapse with specified normal data. Compose it with the framing homeomorphism and projection of A framing identifies the Thom target with a sphere smash product: $X_{+}\stackrel{c}{\to }\mathrm{Th}(\nu (N))\stackrel{{\mathrm{\Phi }}_{\phi }}{\to }{N}_{+}\wedge S^k\stackrel{p}{\to }{S}^k\mathrm{.}$ Its based homotopy class is the Pontryagin–Thom class. The map is based at the disjoint point of $X_{+}$; its restriction to $X$ need not preserve any preselected point of $X$.

For $k\ge 1$ we use the following normalized smooth representative $f_{(N,\phi )}$. Fix an orientation-preserving stereographic coordinate $z:S^k\setminus \{\mathrm{\infty }\}\to {\mathbb{R}}^k$ with centre $y_0=z^{-1}(0)$ and positive basis $b_0=(d{z}_{y_0}{)}^{-1}(e_1,\dots ,e_k)$. Write $u={\phi }_s(v)$ in a compatible tube $\mathrm{\Phi }(s,v)$, and choose $r>0$ so that the chart is defined on $\{\mathrm{\mid }u\mathrm{\mid }\le 2r\}$; compactness of $N$ gives such a radius. Using The standard smooth step function, put $a(q)=1-\sigma \text{\hspace{0.17em}⁣}\left(\frac{q-r^2\mathrm{/}4}{3r^2\mathrm{/}4}\right)\mathrm{.}$ Thus $a=1$ for $q\le r^2\mathrm{/}4$, $a>0$ for $q<r^2$ and $a=0$ for $q\ge r^2$. Define $f_{(N,\phi )}(\mathrm{\Phi }(s,v))=z^{-1}\text{\hspace{0.17em}⁣}\left(\frac{u}{a(\mathrm{\mid }u{\mathrm{\mid }}^2)}\right)(\mathrm{\mid }u\mathrm{\mid }<r),$ and send all other points to $\mathrm{\infty }$. Near $N$ the target coordinate is exactly $u$. Near $\mathrm{\mid }u\mathrm{\mid }=r$, the coordinate at infinity obtained by inversion is $a(\mathrm{\mid }u{\mathrm{\mid }}^2)u\mathrm{/}\mathrm{\mid }u{\mathrm{\mid }}^2$, which is smooth and extends by zero because $a$ is smooth and vanishes identically for $\mathrm{\mid }u\mathrm{\mid }\ge r$. Hence the map is smooth everywhere and constant near the boundary of the larger tube. Its centre preimage is exactly $N$.

This representative has the preceding collapse class. Use the framing metric $\mathrm{\mid }v{\mathrm{\mid }}_{\phi }=\mathrm{\mid }{\phi }_s(v)\mathrm{\mid }$. In the disk model of Disk bundle, sphere bundle, and Thom space: the differential topology interface, the smooth profile above has normalized radius $b(t)=\frac{t}{\sqrt{a(t^2{)}^2+t^2}}(0\le t\le r),$ with $b(0)=0$, $b(r)=1$ and $b(t)>0$ for $t>0$. Interpolating $b(t)$ with $t\mathrm{/}r$ gives continuous disk-valued radial maps which agree at the boundary and never acquire an extra centre preimage. Closed pasting gives a based homotopy to the ordinary collapse, as in Continuity and smooth local representatives of collapse. Different metrics are compared by the radial Thom homeomorphisms; the next lemma proves tube independence.

For $k=0$, $N$ is clopen in $X$, and $f_{(N,\phi )}:X\to S^0$ sends $N$ to the nonbasepoint and $X\setminus N$ to the basepoint. For $N=\mathrm{\varnothing }$ it is constant at the basepoint. The framing is fixed data; a reflected framing changes the map by the corresponding sphere reflection. Countable choice is inherited only from the compatible-chart and normal-bundle machinery.

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$. Let $X$ be a closed smooth manifold and let $(N,\phi )$ be a closed framed codimension-$k$ submanifold of $X$, $k\ge 0$. Pontryagin-Thom maps of $(N,\phi )$ built from any two compatible tubular charts, any two supplied smooth metrics on $\nu (N\subseteq X)$, and any two sufficiently small positive radii are based homotopic as maps $X_{+}\to S^k$ (The Pontryagin-Thom map of a framed submanifold).

The framing is fixed data throughout: the auxiliary choices removed here are the chart, the metric and the radius, and the framing-induced homeomorphism ${\mathrm{\Phi }}_{\phi }$ is the same structure transported along the metric comparison. A bundle automorphism of the normal datum other than the identity changes ${\mathrm{\Phi }}_{\phi }$ and is a change of framed submanifold, not a change of tube data; the construction makes no claim of independence under such an automorphism.

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Let $(W,\epsilon ,\mathrm{\Psi })$ be a framed cobordism in $X\times I$, with the literal product ends and constant collar framings of Framed cobordism of framed submanifolds. Extend $W$ past both ends by the cylinders $N_0\times (-1,0]$ and $N_1\times [1,2)$ to a closed boundaryless embedded submanifold $\tilde{W}$ of $M=X\times (-1,2)$. The product collars make these extensions smooth; extend $\mathrm{\Psi }$ constantly on them. The boundary normal quotients are those of Neat submanifolds have boundary-adapted slice charts, or directly the quotients of these extended product tangent bundles.

A compatible tube $A$ of $\tilde{W}$ with datum $(\tilde{W}\times {\mathbb{R}}^k,{\mathrm{\Psi }}^{-1})$ exists by Compatible tubular charts realize a prescribed normal identification, now applied in the boundaryless manifold $M$. We can make this tube a product on smaller end collars as follows. Choose compatible tubes $B_i$ of $N_i$ in $X$ with datum $(N_i\times {\mathbb{R}}^k,{\phi }_i^{-1})$ and let $B$ be their products with time. Embed $M$ properly in Euclidean space by Every smooth manifold embeds in some finite-dimensional Euclidean space; The Euclidean tubular neighbourhood theorem supplies a smooth retraction $R$ of a Euclidean neighbourhood onto that image (normal addition followed by projection). Let $\chi$ be a smooth function of the base time, equal to one near the ends and supported within the original product collars, constructed using The standard smooth step function. On a small tube define $C(w,v)=R((1-\chi (w))A(w,v)+\chi (w)B(w,v)),$ reading $A,B,C$ in this Euclidean embedding and using $A$ where $\chi =0$. Both maps fix $w$ and induce ${\mathrm{\Psi }}_w^{-1}$ on the normal quotient; derivatives of $\chi$ multiply $B(w,0)-A(w,0)=0$. Thus $dC$ is invertible along zero: it is the identity on $T\tilde{W}$ and an isomorphism on the normal quotient. By The smooth inverse function theorem on manifolds, $C$ is locally a diffeomorphism there. It is injective on a sufficiently small uniform tube over compact $W$: otherwise distinct pairs with fibre coordinates tending to zero and equal images have convergent base subsequences; their limiting base points coincide because $C(w,0)=w$, contradicting local injectivity near that zero vector. Shrinking once more keeps the middle part away from $\mathrm{\partial }(X\times I)$; on the collars $C=B$ preserves time exactly. Consequently its restriction $\mathrm{\Phi }$ over $W$ is a neat compatible tube, product on smaller end collars. This is a local proof of the product-tube assertion used in Freed's Theorem 3.7 and the proof of Theorem 3.9, printed pp.26–27; boundaryless tube existence alone would not supply it.

In these framed coordinates choose $r>0$ with $\mathrm{\Phi }$ defined on $W\times \{\mathrm{\mid }v\mathrm{\mid }\le 2r\}$. For $k\ge 1$, let $z$ and $a$ be the stereographic coordinate and smooth radius profile of The Pontryagin-Thom map of a framed submanifold, and define the collapse of the framed cobordism by $c_W(\mathrm{\Phi }(w,v))=z^{-1}\text{\hspace{0.17em}⁣}\left(\frac{v}{a(\mathrm{\mid }v{\mathrm{\mid }}^2)}\right)(\mathrm{\mid }v\mathrm{\mid }<r),c_W=\mathrm{\infty }\text{ elsewhere}\mathrm{.}$ The same inversion-coordinate calculation proves smoothness, including across the cutoff sphere. On the product collars it is the product of the normalized collapse of $(N_i,{\phi }_i)$ with time; thus its endpoint restrictions are Pontryagin–Thom maps with compatible induced tube data. Extend by the constant basepoint on $\{\ast \}\times I$ to obtain a based homotopy $X_{+}\times I\to S^k$. For $k=0$, $W$ is clopen in $X\times I$; take its characteristic map to $S^0$, whose endpoint restrictions are the characteristic maps of $N_i$. For empty $W$ the map is constant. All formulas use supplied data and only the inherited countable-choice hypothesis.

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$. Let $X$ be a closed smooth manifold and let $(N_0,{\phi }_0)$, $(N_1,{\phi }_1)$ be closed framed codimension-$k$ submanifolds of $X$, $k\ge 0$. If they are framed cobordant (Framed cobordism of framed submanifolds), then their Pontryagin-Thom maps $X\to S^k$ (The Pontryagin-Thom map of a framed submanifold) are homotopic, indeed based homotopic as maps $X_{+}\to S^k$; a framed cobordism supplies an explicit homotopy $X\times I\to S^k$ whose restrictions at the two ends are the two Pontryagin-Thom maps up to based homotopy.

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Fix the standard orientation of $S^k$ and identify $S^k=D^k\mathrm{/}{S}^{k-1}$ with the one-point compactification ${\mathbb{R}}^k\cup \{\mathrm{\infty }\}$ of the zero section of the trivial rank-$k$ bundle over a point, so a chosen centre $y_0$ corresponds to $0$ (Trivial Thom spaces as suspension smash products, Disk bundle, sphere bundle, and Thom space: the differential topology interface). Let $X$ be a closed smooth manifold, let $f:X\to S^k$ be smooth and let $y\in S^k$ be a regular value, with $b=(b_1,\dots ,b_k)$ a positively oriented basis of $T_y{S}^k$ (Smooth manifolds and their smooth charts, Transversality to a point is the regular-value condition); read $b$ as the trivialization $\beta :T_y{S}^k\to {\mathbb{R}}^k$ sending $b_i$ to the $i$-th standard basis vector (Orientation of a finite-dimensional real vector space).

Then $N=f^{-1}(y)$ is a closed embedded submanifold of $X$ of codimension $k$, and the differential of $f$ factors through the normal quotient to a smooth bundle isomorphism $\nu (N\subseteq X)⟶(f{\mathrm{\mid }}_N{)}^{\ast }T{S}^k;$ composing it with $\beta$ gives a framing $f_{\ast }b:\nu (N\subseteq X)⟶N\times {\mathbb{R}}^k\mathrm{.}$ The pair $(N,f_{\ast }b)$ is the framed regular preimage of $(f,y,b)$.

Both assertions are the case $E={\mathbb{R}}^k$ of Transverse preimages carry the pulled-back normal structure, applied locally in a target chart centred at $y$, whose differential at $y$ is $\beta$, to the trivial rank-$k$ bundle over the one-point base: its Thom space is $S^k={\mathbb{R}}^k\cup \{\mathrm{\infty }\}$, its zero section is $\{0\}$, transversality of $f$ to $\{0\}$ is exactly regularity of $y$ (Transverse smooth maps, Transversality to a point is the regular-value condition), and clauses (i)-(ii) of that proposition give the closed embedded submanifold and the specified isomorphism of the normal quotient with the pulled-back target fibre; inserting the positive basis $\beta$ turns the target factor into ${\mathbb{R}}^k$ and is precisely the framing. The construction is independent of any auxiliary choice: only the derivative of $f$ along $N$, the value $y$ and the basis $b$ enter, so no chart of $S^k$ at $y$ is chosen and the framing is the composite displayed above. Regular values are not assumed to exist a priori; they are dense by Regular values form a dense $G_{\delta }$ set, and for $k\ge 1$ the framed cobordism class is shown to be independent of the regular value and positive basis in the two following lemmas.

Rank $k=0$ and $N=\mathrm{\varnothing }$ are included: for $k=0$ the sphere is $S^0$, the basis is empty, $\beta$ is the unique map $T_y{S}^0\to {\mathbb{R}}^0$, and $N=f^{-1}(y)$ is a clopen submanifold of $X$ of dimension $\dim X$, carrying the unique rank-zero framing; for $N=\mathrm{\varnothing }$ the normal bundle and the framing are empty. Regular-value independence does not extend to $k=0$: a constant map from a point to $S^0$ has the point and the empty set as its two regular fibres. They are not framed cobordant within the point, since a compact neat codimension-zero submanifold of $I$ is clopen, and one containing $0$ must also contain $1$. The countable-choice hypothesis is inherited exactly from Transverse preimages carry the pulled-back normal structure; the definition itself selects nothing.

Let $V$ be a finite-dimensional real vector space with an orientation (Orientation of a finite-dimensional real vector space). The set of positively oriented bases of $V$, with the topology it inherits from the linear isomorphisms $V\to {\mathbb{R}}^{\dim V}$, is path-connected; indeed any two positively oriented bases are joined by a smooth path of positively oriented bases, equivalently ${\mathrm{G}\mathrm{L}}^{+}(k,\mathbb{R})=\{A\in \mathrm{G}\mathrm{L}(k,\mathbb{R}):\det A>0\}$ is smoothly path-connected for every $k\ge 0$ (Invertible matrices and the general linear group ${\mathrm{GL}}_n(F)$, Paths, path-connected spaces and path components). In particular, for the oriented vector space $T_y{S}^k$ with its standard orientation, any two positive bases at $y$ can be joined by a continuous path of positive bases.