Collapse of a framed neat cobordism in X times I

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.