Product boundary formula for oriented manifolds

(Case $\mathrm{\partial }V=\mathrm{\varnothing }$.) If $\mathrm{\partial }V=\mathrm{\varnothing }$, the first clause of [F1] with $(M,N)=(W,V)$ states that $\mathrm{\partial }(W\times V)$ equals $\mathrm{\partial }W\times V$ and carries the product boundary orientation: the induced boundary orientation on $\mathrm{\partial }W$ first, then the orientation of $V$. In this case $W\times \mathrm{\partial }V=\mathrm{\varnothing }$, so the displayed formula has a single summand, with sign $+1$ as the first summand, and the identification is the canonical projection diffeomorphism.

(Case $\mathrm{\partial }W=\mathrm{\varnothing }$.) If $\mathrm{\partial }W=\mathrm{\varnothing }$, the second clause of [F1] with $(M,N)=(W,V)$ states that $\mathrm{\partial }(W\times V)$ equals $W\times \mathrm{\partial }V$ and carries $(-1{)}^m$ times the product orientation, namely $(-1{)}^m$ times (orientation of $W$)$\otimes$(induced boundary orientation of $\mathrm{\partial }V$). In this case $\mathrm{\partial }W\times V=\mathrm{\varnothing }$, so this is the second summand of the display.

(Both factors closed.) If $\mathrm{\partial }W=\mathrm{\varnothing }$ and $\mathrm{\partial }V=\mathrm{\varnothing }$, then $W\times V$ is a product of compact spaces, hence compact by [F3], and its boundary is empty; a compact smooth manifold with empty boundary is closed, so $W\times V$ is closed and both summands of the display are empty.

(Unoriented theory.) The identifications of steps 1.1 and 1.2 are diffeomorphisms of the underlying smooth manifolds and do not depend on the orientations: the projection $\mathrm{\partial }(W\times V)\to \mathrm{\partial }W\times V$ (when $\mathrm{\partial }V=\mathrm{\varnothing }$) and $\mathrm{\partial }(W\times V)\to W\times \mathrm{\partial }V$ (when $\mathrm{\partial }W=\mathrm{\varnothing }$) are canonical diffeomorphisms, and reading them in the unoriented theory gives the same disjoint-union identity without the sign $(-1{)}^m$.

(Assembly and the excluded corner case.) In the two corner-free cases steps 1.1 and 1.2 identify the boundary with the two summands of the display with the stated orientations, and step 1.3 covers the closed case; step 1.4 covers the unoriented theory. If both boundaries are nonempty, then near a point of $\mathrm{\partial }W\times \mathrm{\partial }V$ the space $W\times V$ is locally a product of two half-spaces, a quadrant with a corner, and is not a smooth manifold with boundary in the sense fixed on this page; the formula is therefore asserted only in the corner-free case, exactly as stated.