Freed's [notes](https://people.math.harvard.edu/~dafr/bordism.pdf) give the following definition of oriented bordism.

> **Definition**. Let $\Sigma_0$ and $\Sigma_1$ be the two *oriented* closed manifolds. An $ n $-dimensional *bordism* from $\Sigma_0$ to $\Sigma_1$ is a triple $(M, i_0, i_1)$ where
> 1. $M$ is an $ n $-dimensional *oriented* compact manifold with boundary, whose boundary $ \partial M $ inherits the boundary orientation and where we defined a partition of the connected components of $ \partial M $ into *in-boundaries* and *out-boundaries*;
> 2. $ i_0 $ and $ i_1 $ are smooth *orientation preserving* embeddings
$$
i_0 \colon \Sigma_0 \times [0,\varepsilon[ \to M \qquad i_1 \colon \Sigma_1 \times ]1-\varepsilon, 1] \to M  
$$
that define (orientation preserving) diffeomorphisms $ \Sigma_0 \cong i_0(\Sigma_0,0)\cong \partial_0M $ and $ \Sigma_1\cong i_1(\Sigma_1,1)\cong \partial_1M $ between the "abstract" in- and out-boundaries $ \Sigma_0 $ and $ \Sigma_1 $, and the (disjoint union of) the chosen in- and out-boundaries $ \partial_0M $ and $ \partial_1M $.

Now, take an *oriented* $ 2 $-dimensional closed manifold $ \Sigma_0 $ and let's try to define a bordism from $ \Sigma_0 \sqcup \overline{\Sigma}_0 $ to the null manifold $ \emptyset^2 $, where $ \bar\Sigma_0 $ is just $ \Sigma_0 $ with its orientation reversed.

Let $ M $ be the oriented manifold $ M = \Sigma_0 \times [0,1] $, where $ [0,1] $ carries its canonical "from left to right" orientation and where the boundaries of $ M $ acquire the induced "Stokes" orientation [1].


To define an oriented bordism the two maps
$$
i_0\colon \left(\Sigma_0\sqcup \bar\Sigma_0\right)\times [0,\epsilon[\to M\qquad i_1\colon \emptyset^2\times [1 - \epsilon, 1[\to M
$$
must be defined. Here $ (\Sigma_0 \sqcup \bar{\Sigma}_0) \times [0,\varepsilon[ $ is nothing but the disjoint union of two (times an unspecified number of) cylinders. One of them, $ \Sigma_0 \times [0,\varepsilon[ $, has the same orientation as $ M $, hence I can embed it into $ M $ in an orientation preserving way. On the other hand, the remaining one $ \bar\Sigma_0 \times [0,\varepsilon[ $ has the opposite orientation, and of course I cannot.

Is there something I'm missing? I see basically no other way of defining a bordism from $ \Sigma_0\sqcup \bar \Sigma_0 $ to the null manifold (and it's quite clear that one must exist).

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[1] We can depict the *manifold* $ M $ as a cylinder with two inward-pointing arrows stemming from its bases.