Pontryagin Thom and Framed Cobordism — Examples

(A single framed point has degree $\pm 1$.) Let $(x,\phi )$ be a framed point and let $f=f_{(x,\phi )}$ be its Pontryagin-Thom map. By [F2], $f$ is smooth, the centre $y_0$ is a regular value and $f^{-1}(y_0)=\{x\}$; the differential of $f$ at $x$ is the framing composed with the orientation-preserving identification of ${\mathbb{R}}^n$ with $T_{y_0}{S}^n$, so its sign is $+1$ when the basis $\phi$ is positive and $-1$ when it is negative. By the regular-value formula [F3], $\mathrm{deg}(f)=+1$ in the first case and $\mathrm{deg}(f)=-1$ in the second: a positively framed point realizes $+1$ and its orientation reversal $-1$.

(Finite unions and the signed count.) For a finite framed $0$-manifold $P=\{(x_1,{\phi }_1),\dots ,(x_m,{\phi }_m)\}$ choose pairwise disjoint tubes around the points and use the normalized smooth single-point collapses of [F2] on them, sending their complement to the basepoint. Each collapse is constant near its tube boundary, so the resulting map $f_P:S^n\to S^n$ is smooth everywhere. Its centre preimage is $P$, and at $x_i$ its differential induces ${\phi }_i$, with local sign $\mathrm{sgn}(x_i)$ by step 1.1. Thus $y_0$ is regular, including when $P$ is empty. Since $n\ge 1$, both spheres are nonempty connected closed oriented manifolds, and $f_P$ is proper by compactness. The regular-value formula [F3] therefore gives $\mathrm{deg}{f}_P={\sum }_i\mathrm{sgn}(x_i)$, including the empty sum zero.

(Classification.) By the correspondence of [F4] two closed framed $0$-manifolds of $S^n$ are framed cobordant if and only if their Pontryagin-Thom maps are homotopic, and by [F4] homotopy classes of based self-maps of $S^n$ are classified by degree. By step 2.1 the degree of the Pontryagin-Thom map of $P$ is exactly the signed count, so the framed cobordism class of $P$ is determined by ${\sum }_x\mathrm{sgn}(x)$ and every integer occurs: for $m\in \mathbb{Z}$ take $\mathrm{\mid }m\mathrm{\mid }$ distinct points framed positively if $m>0$ and negatively if $m<0$, and the empty set for $m=0$. Hence framed cobordism classes of framed $0$-manifolds are classified by the signed count.

Using [A1] for the Pontryagin–Thom and regular-value degree suppliers, for a finite framed set in $S^n$, the centre of the collapse target has precisely that set as its regular preimage, with differential signs equal to its framing signs. The regular-value degree formula therefore gives degree equal to the signed count. Degree classifies based self-maps of $S^n$, and the fixed-codimension Pontryagin--Thom bijection transfers this classification to framed cobordism. A single positive point has degree $+1$ and its reversal degree $-1$.

(The equator and its framing.) The equator is a closed embedded $(m-1)$-submanifold of $S^m$; in the height coordinate $s=x_{m+1}$ its normal bundle is the rank-one bundle spanned by ${\mathrm{\partial }}_s$, and the restriction of ${\nu }_{\mathrm{e}\mathrm{q}}$ is a nonvanishing section, hence a framing. For $m=1$ the equator is the two-point set $\{(\pm 1,0)\}$ and ${\nu }_{\mathrm{e}\mathrm{q}}=(0,-1)$ at both points; with respect to the standard orientation of $S^1$, whose positive tangent is $(0,1)$ at $(1,0)$ and $(0,-1)$ at $(-1,0)$, the first framing is negative and the second positive.

Let $\sigma$ be the smooth step function and put $h(t)=2t\sigma (8t-1)$. For $0\le t\le 1\mathrm{/}2$, take $W=\{(x,t)\in S^m\times I:x_{m+1}=h(t)\}$. Near $t=0$, $h=0$, so $W$ is the product of the equator with time. For $1\mathrm{/}4\le t\le 1\mathrm{/}2$, $h(t)=2t$, and at the cap $(x,t)=(e_{m+1},1\mathrm{/}2)$ the derivative $h^{\mathrm{\prime }}=2$ makes the defining function $x_{m+1}-h(t)$ a submersion. In local pole coordinates the surface is the smooth graph $t=x_{m+1}\mathrm{/}2$; it has no cap boundary. Elsewhere on $W$ the height gradient in $S^m$ is nonzero. Thus $W$ is a compact neat embedded $m$-manifold with only the equatorial boundary at time zero and a literal product collar. The field $V=(-{\mathrm{\nabla }}_{S^m}{x}_{m+1},h^{\mathrm{\prime }}(t))$ is a nonvanishing normal field: its two components cannot vanish together on $W$. Near time zero it is $(-e_{m+1},0)$, the outward normal of the northern hemisphere, with no time dependence. Trivialize the quotient normal by sending $[V]$ to $1$. This frames $W$ and extends the equator framing throughout its end collar. Hence $W$ is a framed null-cobordism.

(Nullhomotopy and the one-dimensional count.) By step 1.3 and [F3] the Pontryagin-Thom map of $(S^{m-1},{\nu }_{\mathrm{e}\mathrm{q}})$ is based homotopic to the Pontryagin-Thom map of the empty framed submanifold, which is the constant map by [F2]; hence the framed equator is framed null-cobordant and its Pontryagin-Thom map $S^m\to S^1$ is nullhomotopic. For $m=1$ this is also visible in the classification of [F4]: the two equator points carry signs $-1$ and $+1$ by step 1.2, so their signed count is $0$ and their framed class is the class of the empty $0$-manifold.

Stabilizing this equator preserves its dimension $m-1$ and changes its ambient sphere from $S^m$ to $S^{m+1}$, hence its codimension from one to two. By [F5], its collapse class suspends from the zero element of ${\pi }_m(S^1)$ to zero in ${\pi }_{m+1}(S^2)$, and remains zero under iteration. For $m=1$ both this equator and a single positively framed point are zero-dimensional; by [F4] their classes are $0$ and $+1$, respectively. For $m>1$ a single point belongs to a different dimension and is not a generator of the equator's stable stem.

In the affine chart $[1:w]$ the target coordinate is $w=z_2\mathrm{/}{z}_1$. At $(z_1,0)\in U$, its transverse differential is $\delta z_2\mapsto \delta z_2\mathrm{/}{z}_1$, an invertible complex map, hence of real rank two. Thus $y$ is regular, and its fibre is exactly $U=\{\mathrm{\mid }{z}_1\mathrm{\mid }=1,z_2=0\}$. This circle bounds the hemisphere disk $\{x_4=0,x_3\ge 0\}$ in $S^3$, so it is the standard unknot. The target identification with $S^2$ is smooth: in the $w$ chart it is $(2\mathrm{\Re }w,2\mathrm{\Im }w,1-\mathrm{\mid }w{\mathrm{\mid }}^2)\mathrm{/}(1+\mathrm{\mid }w{\mathrm{\mid }}^2)$, the inverse stereographic formula, with the analogous formula in the other chart. Both affine charts also give smooth bundle sections $(1,w)\mathrm{/}\sqrt{1+\mathrm{\mid }w{\mathrm{\mid }}^2}$ and $(v,1)\mathrm{/}\sqrt{1+\mathrm{\mid }v{\mathrm{\mid }}^2}$; multiplying by $S^1$ gives the bundle charts. To supply numerability, let $a=\mathrm{\mid }{z}_1{\mathrm{\mid }}^2$ on unit representatives, which is well defined on the base. Put $b_1=\sigma (4a-1)$, $b_2=\sigma (3-4a)$ and ${\lambda }_i=b_i\mathrm{/}(b_1+b_2)$. Their denominator is positive for $0\le a\le 1$, their sum is one, and the supports lie in $a\ge 1\mathrm{/}4\subset \{z_1\mathrm{\ne }0\}$ and $a\le 3\mathrm{/}4\subset \{z_2\mathrm{\ne }0\}$. This is a supplied finite support-subordinate partition of unity. The regular-preimage definition gives the framing $h_{\ast }b$.

(The Hopf map generates ${\pi }_3(S^2)$.) The Hopf map is a numerable circle bundle and hence a Hurewicz, in particular Serre, fibration by [F3]; its long exact sequence by [F4] contains ${\pi }_3(S^1)\to {\pi }_3(S^3)\stackrel{h_{\ast }}{\to }{\pi }_3(S^2)\to {\pi }_2(S^1)$. By [F6] the outer groups vanish ($k=3\ge 2$ and $k-1=2\ge 2$), so $h_{\ast }$ is an isomorphism; by [F5] ${\pi }_3(S^3)\cong \mathbb{Z}$. Hence ${\pi }_3(S^2)\cong \mathbb{Z}$, generated by the class of $h$.

(Its Pontryagin-Thom map is the Hopf map up to homotopy.) By [F2] the Pontryagin-Thom map $f_{(U,h_{\ast }b)}:S^3\to S^2$ is smoothly homotopic to $h$; in particular their classes in ${\pi }_3(S^2)$ agree.

(Conclusion.) The framed unknot's Pontryagin-Thom class is the class of $h$ by step 2.1, which generates ${\pi }_3(S^2)\cong \mathbb{Z}$ by step 1.2, and the correspondence of [F7] identifies framed cobordism classes of framed links in $S^3$ with ${\pi }_3(S^2)$; so $(U,h_{\ast }b)$ represents a generator. Full AC supplies both the hypothesis of the numerable-bundle lifting theorem [F3] and the countable choice inherited by the regular-preimage and collapse constructions [F1, F2] and the correspondence [F7]; no Hopf invariant theory is developed.

The Hopf map $h(z_1,z_2)=[z_1:z_2]$ has fibre $U=\{z_2=0\}$. The two affine charts give smooth local sections $(1,w)\mathrm{/}\sqrt{1+\mathrm{\mid }w{\mathrm{\mid }}^2}$ and $(v,1)\mathrm{/}\sqrt{1+\mathrm{\mid }v{\mathrm{\mid }}^2}$; circle multiplication supplies bundle charts. With $a=\mathrm{\mid }{z}_1{\mathrm{\mid }}^2$ on unit representatives, the functions $\sigma (4a-1)$ and $\sigma (3-4a)$, divided by their positive sum, give a finite support-subordinate partition on these charts. Thus under AC the supplied numerable circle bundle is a Hurewicz fibration. In the $w=z_2\mathrm{/}{z}_1$ chart the normal differential along $U$ is multiplication by $1\mathrm{/}{z}_1$, of real rank two, so $[1:0]$ is regular and a positive basis gives the Hopf framing. The regular-preimage collapse lemma makes its Pontryagin–Thom map homotopic to $h$.

In real coordinates, $U=\{x_3=x_4=0\}\subset S^3$ bounds $D=\{x_4=0,x_3\ge 0\}$. Let $h(t)=2t\sigma (8t-1)$ and put $W=\{(x,t):x\in S^3,x_4=0,x_3=h(t),0\le t\le 1\mathrm{/}2\}$. It has the literal collar $U\times [0,1\mathrm{/}8)$ and no end at time one. At the cap $x_3=1,t=1\mathrm{/}2$, $h^{\mathrm{\prime }}=2$, so it is the smooth graph $t=x_3\mathrm{/}2$ in the sphere $x_4=0$; elsewhere the height gradient there is nonzero. Thus it is a compact neat disk with boundary only $U\times \{0\}$. Its normal quotient is framed by the ordered classes of $e_4$ and $({\mathrm{\nabla }}_{S^2}{x}_3,-h^{\mathrm{\prime }})$, which are smooth, nonzero and independent everywhere. On the product collar this pair is $(e_4,e_3)$: the normal of $D$ and the inward normal of $U$ within $D$. Taking the dual trivialization defines precisely ${\phi }_D$, constant throughout that collar. Hence this framing is framed null-cobordant, and its collapse is nullhomotopic by [F3].

(The two framed submanifolds have the same underlying manifold.) The Hopf framing $(U,h_{\ast }b)$ and the bounding framing $(U,{\phi }_D)$ are framings of the normal bundle of the same standard unknot $U\subseteq S^3$: the underlying closed $1$-submanifold is $U$ in both cases.

The unit circle agrees with the quotient-circle model of the supplied universal cover by $[t]\mapsto (\cos 2\pi t,\sin 2\pi t)$ is a homeomorphism from $\mathbb{R}\mathrm{/}\mathbb{Z}$ to the unit circle. For $j\ge 2$, $S^j$ is path-connected, locally path-connected (small chart balls suffice), and simply connected. The subgroup criterion Lifting criterion for maps from path-connected locally path-connected spaces therefore lifts each based map $S^j\to S^1$ to $\mathbb{R}$. Linear contraction of the based lift gives ${\pi }_j(S^1)=0$. The long exact sequence of the Hopf bundle consequently makes $h_{\ast }:{\pi }_3(S^3)\to {\pi }_3(S^2)$ an isomorphism, since the adjacent groups ${\pi }_3(S^1)$ and ${\pi }_2(S^1)$ vanish. Degree identifies ${\pi }_3(S^3)$ with $\mathbb{Z}$ and its identity with $1$, so $[h]=h_{\ast }[\mathrm{i}\mathrm{d}]$ generates ${\pi }_3(S^2)$ and is nonzero.

(Their Pontryagin-Thom classes differ.) By [F1] the Pontryagin-Thom class of $(U,h_{\ast }b)$ is the generator of ${\pi }_3(S^2)\cong \mathbb{Z}$, computed in steps 1.1 and 2.1. By [F2] the Pontryagin-Thom map of $(U,{\phi }_D)$ is nullhomotopic, so by [F3] the framed submanifold $(U,{\phi }_D)$ is framed null-cobordant and its Pontryagin-Thom class is the zero element of ${\pi }_3(S^2)$. In the isomorphism ${\pi }_3(S^2)\cong \mathbb{Z}$ of [F1] the generator is not zero, so the two classes differ.

(Conclusion.) The two framings of the same underlying submanifold $U$ have different Pontryagin-Thom classes; equivalently, by [F4], the two framed submanifolds are not framed cobordant. Hence the Pontryagin-Thom class is not determined by the underlying submanifold alone, and the framing is load-bearing data; the statement refuted is false.

Using [A1] for the Pontryagin–Thom and regular-value degree suppliers, for a finite framed set in $S^n$, the centre of the collapse target has precisely that set as its regular preimage, with differential signs equal to its framing signs. The regular-value degree formula therefore gives degree equal to the signed count. Degree classifies based self-maps of $S^n$, and the fixed-codimension Pontryagin--Thom bijection transfers this classification to framed cobordism. A single positive point has degree $+1$ and its reversal degree $-1$.

(The framed point generates ${\pi }_1(S^1)$.) By [F1] the Pontryagin-Thom map of the positively framed point $(p,\phi )$ is a smooth self-map of $S^1$ whose differential at $p$ has sign $+1$, so its degree is $+1$; by [F2] degree is an isomorphism ${\pi }_1(S^1)\to \mathbb{Z}$, hence the class of the framed point is the generator. A single point realizes $+1$ and its orientation reversal $-1$ in the classification of [F1].

(Its stabilization is a positively framed point.) Under equatorial stabilization the point becomes $i(p)\in S^2$ with the framing ${\nu }_{\mathrm{e}\mathrm{q}}\oplus \phi$, where ${\nu }_{\mathrm{e}\mathrm{q}}$ is the equatorial normal, by [F3]; the normal bundle of the point $i(p)$ in $S^2$ is all of $T_{i(p)}{S}^2$, so this framing is a basis of $T_{i(p)}{S}^2$. By [F4] the pair $({\nu }_{\mathrm{e}\mathrm{q}},\phi )$ is positive for the standard orientation of $S^2$ because $\phi$ is a positive basis of the equator, so the stabilization is again a single positively framed point. By [F1] its Pontryagin-Thom map has degree $+1$ and, by [F2], it generates ${\pi }_2(S^2)\cong \mathbb{Z}$.

(The suspension identity on the generator.) By [F3] the Pontryagin-Thom class of the stabilization is $E$ of the class of the framed point, which is the generator of ${\pi }_1(S^1)$ by step 1.2. By [F4] the suspension sends the class of the identity of $S^1$ to the class of the identity of $S^2$, and the class of the identity is the generator of ${\pi }_1(S^1)$ by [F2]; hence $E(\mathrm{P}\mathrm{T}(p,\phi ))$ is the identity class of ${\pi }_2(S^2)$, i.e. the degree-$+1$ self-map of $S^2$. This agrees with step 1.3, where the Pontryagin-Thom class of the stabilization was computed directly as the generator of ${\pi }_2(S^2)$: the identity $\mathrm{P}\mathrm{T}(\sigma (p,\phi ))=E(\mathrm{P}\mathrm{T}(p,\phi ))$ holds on this nonzero class.

(Iteration and conclusion.) Repeating the two computations one dimension higher: the stabilization of the positively framed point of $S^k$ is the positively framed point of $S^{k+1}$ (the same boundary-orientation computation as step 1.3, where the equatorial normal is prepended to a positive basis), and its Pontryagin-Thom class is the generator of ${\pi }_{k+1}(S^{k+1})\cong \mathbb{Z}$; by [F3] and [F4] these classes are related by the suspension isomorphisms $E:{\pi }_k(S^k)\to {\pi }_{k+1}(S^{k+1})$, which send generator to generator. The stabilized class is therefore a nonzero element of the zeroth stable stem ${\pi }_0^s$ (Stable stems of the sphere), and the example verifies the stabilization-to-suspension compatibility on a nonzero class.