Framed cobordism is an equivalence relation

(Reflexivity.) Let $(N,\phi )$ be a closed framed codimension-$k$ submanifold of $X$. Put $W:=N\times I\subseteq X\times I$ and $\epsilon :=\frac{1}{4}$. By [F2] and [F4], $W$ is a compact embedded submanifold; its boundary in $X\times I$ is $N\times \{0\}\bigsqcup N\times \{1\}=\mathrm{\partial }W$, and $W$ is transverse to $\mathrm{\partial }(X\times I)=X\times \{0,1\}$ because $TW$ contains the $I$-direction, so $W$ is neat. Its ends are the literal products $N\times [0,\frac{1}{4})$ and $N\times (\frac{3}{4},1]$. By [F2] the normal bundle $\nu (W\subseteq X\times I)$ is canonically ${\mathrm{pr}}_N^{\ast }\nu (N\subseteq X)$; let $\mathrm{\Psi }$ be the pullback of $\phi$ along ${\mathrm{pr}}_N$ under this identification. Then $\mathrm{\Psi }$ is a smooth bundle isomorphism and over each end collar it is the pullback of $\phi$, so $(W,\frac{1}{4},\mathrm{\Psi })$ is a framed cobordism from $(N,\phi )$ to $(N,\phi )$.

(Symmetry.) Let $(W,\epsilon ,\mathrm{\Psi })$ be a framed cobordism from $(N_0,{\phi }_0)$ to $(N_1,{\phi }_1)$. Put $\rho (x,t):=(x,1-t)$ and $W^{\mathrm{\prime }}:=\rho (W)$. By [F3] $W^{\mathrm{\prime }}$ is a compact neat embedded submanifold of $X\times I$ with $\mathrm{\partial }{W}^{\mathrm{\prime }}=N_1\times \{0\}\bigsqcup N_0\times \{1\}$, and its ends are the literal products $\rho (N_1\times (1-\epsilon ,1])=N_1\times [0,\epsilon )$ and $\rho (N_0\times [0,\epsilon ))=N_0\times (1-\epsilon ,1]$. The differential $d\rho$ is the identity on the $TX$ factor and multiplication by $-1$ on the $TI$ factor; along an end collar the $I$-direction is tangent to $W$ and to $W^{\mathrm{\prime }}$, so $d\rho$ induces, over $\rho$, the identity map $\nu (N_i\subseteq X)\to \nu (N_i\subseteq X)$ of the canonical product identifications. Define ${\mathrm{\Psi }}^{\mathrm{\prime }}$ over $W^{\mathrm{\prime }}$ by ${\mathrm{\Psi }}_{\rho (p)}^{\mathrm{\prime }}:={\mathrm{\Psi }}_p\circ (\overline{d{\rho }_p}{)}^{-1}$, where $\overline{d{\rho }_p}:\nu (W{)}_p\to \nu (W^{\mathrm{\prime }}{)}_{\rho (p)}$ is the quotient isomorphism induced by $d{\rho }_p$. Then ${\mathrm{\Psi }}^{\mathrm{\prime }}$ is a smooth bundle isomorphism $\nu (W^{\mathrm{\prime }}\subseteq X\times I)\to W^{\mathrm{\prime }}\times {\mathbb{R}}^k$ which on the collar $N_1\times [0,\epsilon )$ is the pullback of ${\phi }_1$ and on $N_0\times (1-\epsilon ,1]$ the pullback of ${\phi }_0$. Hence $(W^{\mathrm{\prime }},\epsilon ,{\mathrm{\Psi }}^{\mathrm{\prime }})$ is a framed cobordism from $(N_1,{\phi }_1)$ to $(N_0,{\phi }_0)$.

(Transitivity, construction.) Let $(W_1,{\epsilon }_1,{\mathrm{\Psi }}_1)$ be a framed cobordism from $(N_0,{\phi }_0)$ to $(N_1,{\phi }_1)$ and $(W_2,{\epsilon }_2,{\mathrm{\Psi }}_2)$ one from $(N_1,{\phi }_1)$ to $(N_2,{\phi }_2)$. Put $\epsilon :=\min ({\epsilon }_1,{\epsilon }_2)$, define ${\sigma }_1(x,t):=(x,t\mathrm{/}2)$ and ${\sigma }_2(x,t):=(x,(t+1)\mathrm{/}2)$, and set $W_1^{\ast }:={\sigma }_1(W_1)\subseteq X\times [0,\frac{1}{2}]$, $W_2^{\ast }:={\sigma }_2(W_2)\subseteq X\times [\frac{1}{2},1]$ and $W:=W_1^{\ast }\cup W_2^{\ast }$. By [F3] each $W_i^{\ast }$ is a compact neat embedded submanifold, with $\mathrm{\partial }{W}_1^{\ast }=N_0\times \{0\}\bigsqcup N_1\times \{\frac{1}{2}\}$ and $\mathrm{\partial }{W}_2^{\ast }=N_1\times \{\frac{1}{2}\}\bigsqcup N_2\times \{1\}$.

(Transitivity: $W$ is a neat submanifold carrying a framing.) Near $t=\frac{1}{2}$ one has $W_1^{\ast }\cap (X\times (\frac{1}{2}-\frac{\epsilon }{2},\frac{1}{2}])=N_1\times (\frac{1}{2}-\frac{\epsilon }{2},\frac{1}{2}]$ and $W_2^{\ast }\cap (X\times [\frac{1}{2},\frac{1}{2}+\frac{\epsilon }{2}))=N_1\times [\frac{1}{2},\frac{1}{2}+\frac{\epsilon }{2})$, so $W\cap (X\times (\frac{1}{2}-\frac{\epsilon }{2},\frac{1}{2}+\frac{\epsilon }{2}))=N_1\times (\frac{1}{2}-\frac{\epsilon }{2},\frac{1}{2}+\frac{\epsilon }{2})$ is a product piece. Every point of $W$ with $t\mathrm{\ne }\frac{1}{2}$ lies in $W_1^{\ast }\cap (X\times [0,\frac{1}{2}))$ or in $W_2^{\ast }\cap (X\times (\frac{1}{2},1])$, which are open pieces of the smooth submanifolds $W_i^{\ast }$; those two pieces and the product piece cover $W$, and on their overlaps (subsets of the product piece) the three descriptions agree. Hence $W$ is a compact neat embedded submanifold of $X\times I$ with $\mathrm{\partial }W=N_0\times \{0\}\bigsqcup N_2\times \{1\}$ and with the literal product ends $N_0\times [0,\frac{\epsilon }{2})$ and $N_2\times (1-\frac{\epsilon }{2},1]$. [F1, F3, step 1.3] Transport ${\mathrm{\Psi }}_1$ across ${\sigma }_1$ and ${\mathrm{\Psi }}_2$ across ${\sigma }_2$ as in step 1.2. Since ${\sigma }_i$ preserves the splitting $T(X\times I)=TX\oplus TI$ up to a positive scale in $TI$, the transported framings are smooth bundle isomorphisms $\nu (W_i^{\ast }\subseteq X\times I)\to W_i^{\ast }\times {\mathbb{R}}^k$, the first equal to the pullback of ${\phi }_1$ on the collar $N_1\times (\frac{1}{2}-\frac{\epsilon }{2},\frac{1}{2}]$ and the second equal to the pullback of ${\phi }_1$ on $N_1\times [\frac{1}{2},\frac{1}{2}+\frac{\epsilon }{2})$. These two collars cover the overlap region just described, where both transported framings are the pullback of ${\phi }_1$ under the canonical identification $\nu (W){\mathrm{\mid }}_{N_1\times \mathrm{\Theta }}\cong {\mathrm{pr}}^{\ast }\nu (N_1\subseteq X)$; away from the overlap each is smooth. Hence they define one smooth bundle isomorphism $\mathrm{\Psi }:\nu (W\subseteq X\times I)\to W\times {\mathbb{R}}^k$, which on the outer ends is the pullback of ${\phi }_0$ and of ${\phi }_2$. Therefore $(W,\frac{\epsilon }{2},\mathrm{\Psi })$ is a framed cobordism from $(N_0,{\phi }_0)$ to $(N_2,{\phi }_2)$.

(Conclusion.) Steps 1.1, 1.2 and 2.1 exhibit reflexivity, symmetry and transitivity of framed cobordism for arbitrary closed $X$, $k\ge 0$ and framed submanifolds, including the empty manifold and the rank-zero case $k=0$, where all framings are unique. By [F5] framed cobordism is an equivalence relation. No orientation, no metric and no choice beyond the inherited ${\mathrm{A}\mathrm{C}}_{\omega }$ is used: all constructions are explicit, and the only choice-dependent input is the smooth normal-bundle structure of [F1].