Framed cobordism of framed submanifolds

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.