Collar gluing and seam smoothing give transitivity

(The glued space and the bi-collar.) Let $X:=W_1\bigsqcup W_2$ and let $\sim$ be the equivalence relation on $X$ generated by ${\theta }_1(0,x)\sim {\theta }_0^{\mathrm{\prime }}(0,x)$ for $x\in M_1$; put $W:=X\mathrm{/}\sim$ with the quotient topology. Define $\mathrm{\Theta }:M_1\times (-1,1)\to W,\mathrm{\Theta }(x,s)=\{\begin{array}{ll}[{\theta }_1(s,x)],& s\le 0,\\ [{\theta }_0^{\mathrm{\prime }}(s,x)],& s>0.\end{array}$ The two formulas agree at $s=0$ and each is continuous on its closed half, so $\mathrm{\Theta }$ is continuous; it is injective because ${\theta }_1,{\theta }_0^{\mathrm{\prime }}$ are injective and their images meet exactly in the identified seam; and it is open onto its image: for a relatively open subset of $M_1\times (-1,1)$ the two half-images are open in the collar neighbourhoods of $W_1$ and $W_2$, whose union is saturated. Hence $O:=\mathrm{\Theta }(M_1\times (-1,1))$ is open in $W$ and $\mathrm{\Theta }$ is a homeomorphism onto $O$. The images $A,B$ of $W_1\setminus (\mathrm{\partial }{W}_1{)}_1$ and $W_2\setminus (\mathrm{\partial }{W}_2{)}_0$ are open in $W$ and homeomorphic to those open submanifolds, and $W=A\cup B\cup O$.

(The smooth atlas.) For a chart $(V,\psi )$ of $M_1$ define the seam chart $\mathrm{\Psi }:\mathrm{\Theta }(V\times (-1,1))\to \psi (V)\times (-1,1),\mathrm{\Psi }(\mathrm{\Theta }(x,s))=(\psi (x),s)\mathrm{.}$ Take the atlas consisting of the charts of $W_1$ with domain disjoint from $(\mathrm{\partial }{W}_1{)}_1$, the charts of $W_2$ with domain disjoint from $(\mathrm{\partial }{W}_2{)}_0$, and all seam charts $\mathrm{\Psi }$. Its domains cover $W$ by step 1.1 and are open in $W$. Transitions between two charts from the same half are smooth by the given structures; a chart of the first half and one of the second have disjoint domains; two seam charts have transition $({\psi }_2\circ {\psi }_1^{-1},\mathrm{id})$, smooth in $M_1$ coordinates; and a seam chart with a chart of $W_1$ (or $W_2$) has overlap in $s<0$ (or $s>0$), where it equals the smooth expression $\alpha \circ {\theta }_1$ (or $\alpha \circ {\theta }_0^{\mathrm{\prime }}$) in the coordinates $(\psi (x),s)$. Thus the generated maximal atlas is a smooth structure of dimension $n+1$ with boundary; each point of the seam lies in the seam chart and has a Euclidean neighbourhood.

(Boundary, outer collars and bordism structure.) The seam is $O_0:=\mathrm{\Theta }(M_1\times \{0\})$, which every seam chart exhibits as an interior hypersurface $\{s=0\}$; hence the boundary of $W$ is the image of $(\mathrm{\partial }{W}_1{)}_0\bigsqcup (\mathrm{\partial }{W}_2{)}_1$. Put $(\mathrm{\partial }W{)}_0:=\mathrm{im}(\mathrm{\partial }{W}_1{)}_0$ and $(\mathrm{\partial }W{)}_1:=\mathrm{im}(\mathrm{\partial }{W}_2{)}_1$; these are open and closed in $\mathrm{\partial }W$ and cover it. The outer collars ${\theta }_0:[0,1)\times M_0\to W_1\subset W$ and ${\theta }_1^{\mathrm{\prime }}:(-1,0]\times M_2\to W_2\subset W$ are unchanged smooth embeddings onto open collar neighbourhoods of these parts with the required normalisation, so $(W,{\theta }_0,{\theta }_1^{\mathrm{\prime }})$ is a bordism from $M_0$ to $M_2$.

(Canonicity and conditional collar comparison.) The atlas of step 2.1 is determined by the supplied collars: changing a chart of $M_1$ changes a seam chart by $({\psi }_2\circ {\psi }_1^{-1},\mathrm{id})$, so the generated maximal atlas is unchanged. For the additional comparison assume ${\mathrm{A}\mathrm{C}}_{\omega }$. Normalize outgoing collars by $c(p,t)={\theta }_1(-t,p)$ and incoming collars by $c(p,t)={\theta }_0^{\mathrm{\prime }}(t,p)$. Apply the boundary-fixing comparison construction [F5] separately in $W_1$ and $W_2$, supported near their seam parts, to obtain $H_1,H_2$ intertwining the old and new collars near the seam. They descend to a bijection of the glued spaces because both fix the seam points; in the old source and new target seam coordinates the descended map is $(p,s)\mapsto (p,s)$ near $s=0$. Away from the seam it and its inverse are the smooth maps $H_i,H_i^{-1}$. Thus it is a diffeomorphism fixed off a neighbourhood of the seam. Only this comparison uses [F5]; construction and transitivity use the given collars directly.

(The orientations glue.) Assume both bordisms oriented. By [F2], ${\theta }_1$ is orientation-preserving for the product orientation $ds\wedge o_1$ of $(-1,0]\times M_1$, and ${\theta }_0^{\mathrm{\prime }}$ is orientation-preserving for the product orientation $ds\wedge o_1$ of $[0,1)\times M_1$: both use the same orientation $o_1$ of the seam and the interval coordinate first. In the bi-collar coordinates $(x,s)$ of step 1.1 this says that a basis $({\mathrm{\partial }}_s,{\mathrm{\partial }}_x)$ is positive in the orientation of $W_1$ for $s<0$ and in the orientation of $W_2$ for $s>0$. Therefore the chart system that declares all charts of the first half positive for the orientation of $W_1$ and all charts of the second half positive for the orientation of $W_2$ is consistent on overlaps: the transition across the seam is the identity in the coordinates $(x,s)$ with Jacobian determinant $+1$. By step 2.1 this defines one orientation of $W$. Equivalently, the induced orientations of the common boundary component are $+M_1$ as the outgoing face of $W_1$ and $-M_1$ as the incoming face of $W_2$, so they are opposite and glue. On the outer faces the induced orientations are $-M_0$ and $M_2$, unchanged from the two bordisms, and the outer collars remain orientation-preserving; hence $W$ with this orientation is an oriented bordism from $M_0$ to $M_2$.

(Compactness, Hausdorffness and second countability.) The finite disjoint union $X=W_1\bigsqcup W_2$ is compact Hausdorff: finite subcovers of its two summands combine, and separation is checked within a summand or by the disjoint summands. Let $q:X\to W$ be the quotient. It is closed: the saturation of a closed $C\subseteq X$ adds only the images of its intersections with the two closed seam parts under the seam-identifying homeomorphism, hence is closed. For distinct $u,v\in W$, the fibres $q^{-1}(u),q^{-1}(v)$ are finite and disjoint. Hausdorffness of $X$ gives disjoint open sets $U,V$ containing these two fibres (intersect and unite finitely many separating neighbourhoods). The open sets $W\setminus q(X\setminus U)$ and $W\setminus q(X\setminus V)$ contain $u,v$ and are disjoint, proving Hausdorffness. Compactness follows from [F6]. Choose countable bases of $W_1,W_2,M_1$; their restrictions give bases of $A,B$, and the products of the basis of $M_1$ with rational intervals give a countable basis of $O$. Their union is a countable basis of $W$ by [F6]. Together with steps 2.1, 3.1 and 4.1 this makes $W$ the required compact smooth bordism, oriented when the supplied bordisms are. No choice axiom is needed for transitivity.