The Pontryagin-Thom map of the standard framed equator

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.