Every bordism class is realized by an embedded collapse

Representatives and the collapse element. A class in ${\mathrm{\Omega }}_n^O$ or ${\mathrm{\Omega }}_n^{SO}$ is represented by a closed smooth $n$-manifold $M$ with an orientation in the oriented case [F1]. By [F2], for every $r\ge n+1$ there is an embedding $M\hookrightarrow {\mathbb{R}}^{n+r}\subset S^{n+r}$, the latter its one-point compactification with closed tubular neighbourhood and normal bundle $\nu$. The collapse lemma [F3] produces the stable class $\alpha (M)\in {\pi }_n(M\mathrm{O})$ (or in ${\pi }_n(M\mathrm{S}\mathrm{O})$ with the induced orientation on $\nu$), independent of the embedding, the tube, the classifying map and the collapse data, and additive under disjoint union.

Represent a stable class by $g:S^{n+r}\to T_r$ with $r\ge \max (n+1,2)$. By [F5] its compact image lies in a finite Grassmannian Thom space. Write $E$ for the smooth nonbasepoint stratum, identified radially with the bundle total space. The closed zero preimage $K$ is compact. If it is empty, radial expansion already contracts $g$ to the basepoint. Otherwise choose compact neighbourhoods $K\subset \mathrm{int}C\subset C\subset V\subset \bar{V}\subset g^{-1}(E)$ and a bump $\lambda =1$ near $C$, supported in $V$ [F6]. Properly embed $E$ into Euclidean space and take a smooth neighbourhood retraction $R_E$. Approximate $jg$ there by a smooth $\mathrm{\Psi }$ with sufficiently small pointwise error [F6]. On the compact buffer $\bar{V}\setminus \mathrm{int}C$, the image stays away from the zero section; choose the error also small enough that all segments $jg+t\lambda (\mathrm{\Psi }-jg)$ remain inside the retraction domain and their retractions stay zero-free on that buffer. Retract these segments and paste with $g$ off $\mathrm{supp}\lambda$. The map is now smooth near all its zeros and fixes every original basepoint value. On a smaller compact neighbourhood of those zeros use the submersive perturbation family of [F6] with a bump parameter equal to $1$ near the zeros and supported in the smooth region. On the compact support outside that neighbourhood a sufficiently small parameter introduces no zero. Parametric transversality and density of good parameters [F6] give a parameter within this small ball; multiplying it by a scalar from $0$ to $1$ gives a based homotopy. We have thus replaced $g$ by a map smooth and transverse near its compact zero preimage $P$, without assuming smooth endpoints in the homotopy-perturbation supplier. By [F5], $P$ is a closed $n$-manifold with its specified pulled-back normal bundle.

Let $W$ be a compact bordism. Glue its two copies using the supplied collars and their signed normal coordinate to give its double a boundaryless smooth atlas; let $k:DW\hookrightarrow {\mathbb{R}}^D$ be an embedding [F2, F4]. In disjoint collars write $w=(x,t)$, $0\le t<3\delta$, and choose a smooth nondecreasing $\chi$ that is zero for $t\le \delta$ and equals $t$ for $t\ge 2\delta$, strictly increasing between the flat part and $2\delta$. Define $R(w)=(x,\chi (t))$ on collars and $R(w)=w$ elsewhere. Choose a smooth height $h:W\to I$ equal to $t$ on the incoming collar, $1-t$ on the outgoing collar, and strictly between $0,1$ off the boundary. Such a height is obtained by a partition of unity and these collar functions [F4]. Then $j=(kR,h):W\to {\mathbb{R}}^D\times I$ is injective: if $j(w_1)=j(w_2)$, equality of the first coordinate forces $R(w_1)=R(w_2)$, and where $R$ can identify distinct points the height separates their collar parameters. Its derivative is injective too: off the flat collar $dR$ is injective, and on any collar direction lost by $dR$ the derivative of $h$ is nonzero. Compactness makes $j$ an embedding, neat and product-shaped near its boundary. Extend the product collars slightly beyond $I$. Local normal addition has invertible derivative at the zero section; a finite cover and compactness give a uniform injective normal tube about $j(W)$, product-shaped at the two ends. The normal Gauss map classifies it in a finite Grassmannian. Collapsing this tube in the spatial one-point compactification gives a based homotopy $S^D\times I\to T_{D-n}$ whose endpoints are the two normally classified collapses. In the oriented case give ${\mathbb{R}}^D\times I$ the orientation $(-1{)}^n$ times spatial-then-time. Moving the time vector across the $n$ endpoint tangent vectors contributes $(-1{)}^n$, so the normal-first orientation of ${\nu }_W$ restricts to exactly the endpoint normal orientations. The outward-normal-first convention then gives the required incoming and outgoing bordism signs. By [F3] the endpoint stable classes are $\alpha (M_0),\alpha (M_1)$, independent of their embeddings. Hence the assignment descends to bordism classes and its pinch additivity makes it a homomorphism.

Compare this map with its normally classified collapse. On a small normal tube write $g=(b(x,v),a(x,v))$, with $a(x,0)=0$. Its normal differential $A_x$ identifies ${\nu }_P$ with $b(x,0{)}^{\ast }E$. The homotopy $b(x,tv)$ contracts the base coordinate to $b(x,0)$; write $Q_t$ for the smooth orthogonal projection onto the universal fibre over $b(x,tv)$ in its finite ambient Euclidean space. Solve ${\dot{U}}_t=[{\dot{Q}}_t,Q_t]U_t$, $U_0=1$. This is smooth orthogonal transport: differentiating $U_t^{-1}{Q}_t{U}_t$ gives zero, since $Q_t^2=Q_t$ implies $Q_t{\dot{Q}}_t{Q}_t=0$. The compact parameter set gives existence throughout $I$, and at $v=0$ the transport is the identity. Use $U_t$ to identify the fibres; this proves the needed smooth interval transport directly. All resulting fibre coordinates have the same invertible first derivative $A_x$ and uniform error $o(\mathrm{\parallel }v\mathrm{\parallel })$ on compact $P$. Thus on a sufficiently small tube their straight-line homotopy to $A_{x}v$ is zero-free for $v\mathrm{\ne }0$: its norm is bounded below by $\frac{1}{2}{\min }_x{\sigma }_{\min }(A_x)\mathrm{\parallel }v\mathrm{\parallel }$. Apply the same construction with a tube cutoff; outside the smaller tube the map and all changes stay away from the zero section by compactness. A radial Thom self-map $v\mapsto \min (\mathrm{\parallel }v\mathrm{\parallel }\mathrm{/}ϵ,1)v\mathrm{/}\mathrm{\parallel }v\mathrm{\parallel }$, continuously homotopic to the identity and fixing the basepoint, then collapses that exterior, for sufficiently small $ϵ>0$. On the linear tube the result is the collapse classified by $A_x$, with normal metric $\mathrm{\parallel }{A}_{x}v\mathrm{\parallel }$ and radius $ϵ$. Auxiliary tube and metric independence [F3] identify it with $\alpha (P)$. Therefore every stable class is in the image.

Oriented case and conclusion. The construction of steps 1.1, 2.1, 1.2 and 2.2 is carried out in $M\mathrm{S}\mathrm{O}$ whenever the manifold carries an orientation: the preimage $P$ of step 1.2 inherits the tangent orientation determined by the pulled-back normal orientation and the standard sphere orientation, and the classifying map lands in $B\mathrm{S}\mathrm{O}(r)$, so the same arguments identify the class with $\alpha (P)\in {\pi }_n(M\mathrm{S}\mathrm{O})$. Pinch additivity and the fixed-coordinate structure maps pass both constructions to the stable colimit [F3, F5, step 2.1, step 1.2]. Therefore the maps ${\mathrm{\Omega }}_n^O\to {\pi }_n(M\mathrm{O})$ and ${\mathrm{\Omega }}_n^{SO}\to {\pi }_n(M\mathrm{S}\mathrm{O})$ are well-defined surjective homomorphisms in every degree; for $n=0$ finite sets of points (with supplied signs in the oriented case), embedded in $S^r$, realize the degree-zero classes and the empty manifold the zero class by the based quotient convention of [F3].